1. Introduction

Resonator based on nano-scale electromechanical system (so called NMR), which is endowed with large quality factor, high frequency and small size, has recently received extensive attention. It not only serves as an ultra-sensitive tool of quantum measuring, bimolecular detection and semiconductor technology [1], but also offers the growing potential applications and fundamental research range from quantum state engineering [2], weak force detection [3] and nano-scale manipulation [4]. For the sake of pursuing new properties, the NMR is coupled to some other quantum elements such as optical cavities [5], quantum dots (QDs) [6, 7] and cold ions [8], which can provide a good platform for studying various interesting phenomena of quantum effects in a macroscopic system [9, 10]. For example, when a NMR is coupled to a qubit via the Jaynes-Cummings interaction in a hybrid optomechanical system, there exists two-color electromagnetically induced transparency [11], which can be applied to the generation of nonclassical states of the NMR or photon blockade. Recent theoretical [6, 12] and experimental [2, 13] researches have found that the NMR can be successfully cooled to near its quantum ground state. And in the prospective researches it may be used for quantum information processing [10], as quantum devices between hybird quantum systems [14,15], and for investigating the limits of quantum mechanics with macroscopic objects. Among these applications, it is intrinsic to prepare the system which contains the NMR into the nonclassical states and investigate the quantum nature of the produced states.

Entanglement is one of the most fascinating nonclassical properties of multi-particle systems. Theoretical and experimental investigations of quantum entanglement in the NMRs, or the NMR-cavity system have achieved great progress. In experiments, manipulating the quantum state of a NMR has been realized by embedding it in a superconducting qubit or an inductor-capacitor NMR [2], or an electromechanical circuit [16], which provides a great help to generate an entangled state between a NMR and another degree of freedom. Shortly thereafter, T. A. Palomaki et al. [17] have successfully achieved the entanglement between a NMR and a microwave cavity in an electromechanical circuit. Meanwhile, many proposals have been proposed which involve the entangled NMRs mediated by a cavity field, based on measurement [18] or a deterministic method of producing the arbitrary phonon states [19]. The entangled NMRs can also be prepared by using nonlinear dissipation process [20, 21] or optimized two-tone (or four-tone) driving of a cavity only with one driven auxiliary mode as the engineered reservoir [22], or enhancing intrinsic mechanical nonlinearities [23, 24]. However, entanglement decay dynamics between the NMRs can be affected by the squeezed vacuum reservoir in the nonlinear quantum scissor system [25]. Moveover, the nonclassical states of the NMR such as Bell state [26] and Fock state [27] can be prepared via intrinsic nonlinearities, and multiphoton blockade can also be realized by using nonlinear scissors [28, 29]. The entangled NMRs can be treated as quantum memory elements when they are embedded in a superconducting quantum device because the entangled-state can be read and written [30].

Another important aspect of nonclassical states is Einstein-Podolsky-Rosen (EPR) steering. It is closely related to entanglement and refers to the ability of one system to steer the states of another system through local measurements. The two systems are strongly entangled in which they are not just correlated, but correlated in a specific direction. Therefore, we may boldly say that EPR steering occurs in the asymmetrical system. But how to tell EPR steering? In 1989, Reid presented a testable criterion of EPR steering in the ream of continuous-variable systems based on the Heisenberg uncertainty relation [31]. Later, the first experimental evidence of this effect between two spatially separated and correlated light modes was confirmed by Ou et. al. [32]. Since Reid’s criterion is based on a position-momentum uncertainty relation involving product of variances of noncommuting observes, it can only detect correlations that appears up to second-order in the tested observes. The Reid’s criterion fails to test EPR steering in the non-Gaussian states. For arbitrary bipartite Gaussian states of continuous variable systems, a computable measure of EPR steering is introduced [33]. Several experiments have been investigated that EPR steering is a stronger form of correlations compared to entanglement and weaker to nonlocality [34] between only two parties [35] or multiparty [36, 37]. The steerability of a two-qubit Bell local state has been experimentally demonstrated by using polarization-entangled photons [34]. While, the multiparty EPR steering by utilizing optical networks and efficient detection has been observed in experiment [36].

The intrinsic asymmetry of the two entangled systems in certain direction will lead to the emergence of one-way EPR steering. In other words, they are steerable by one party but not the other. The one-way property of EPR steering provides a new understanding about the mysterious quantum physics. It has been discussed in several systems of continuous variables [37–40 ]. At the same time, Gaussian one-way EPR steering with two mode squeezed states has experimentally certified [41]. Furthermore, in a recent experimentally work [36], multipartite EPR steering with optical networks was observed, which can be extended to qubits, e.g., photonic, atomic, or otherwise.

In this paper, we consider that two spatially separated QDs are fixed on a NMR inside an optical cavity, where the QDs are driven by a strong laser field. We are interested in a regime where a photon and a phonon emission processes are accompanied with a laser photon absorption. If the QDs dynamics is faster than the NMR-cavity subsystem, we can trace over the QDs’ variables and obtain the dynamics of the cavity mode and the NMR mode. By tuning the frequency of the strong laser field, the differences of populations between the dressed states of the two QDs can be approximately obtained to unity, which is helpful in generation of the entanglement between the NMR mode and the cavity mode. The resonant interaction between the NMR-cavity subsystem and the dressed QDs can gain access to the optimal entanglement of the NMR-cavity subsystem. Meanwhile, the purity of the state of the NMR-cavity subsystem reaches a maximum and is near to 1. Through adjusting the parameters of the system, we can achieve the different interaction strengths of the cavity-dot coupling and the dot-NMR coupling. When we measure one of the two modes, the quantum vacuum fluctuation is brought in. In this case, one-way EPR steering behaviour exists. It can also be see the condition of observing the different types of EPR steering from the purity of the state of the cavity-NMR subsystem.

The organization of the paper is as follows. We begin in Sec. 2 with a description of the proposed schemes for the generation of entanglement and EPR steering between the cavity mode and the NMR mode, then derive the Markovian master equation in the interaction picture for the NMR-cavity subsystem. In Sec. 3, we study the generation and the properties of entanglement and purity of the considered system by using of Duan’s criterion [42]. Section 4 is devoted to a discussion of EPR steering and purity of the NMR-cavity subsystem with Reid’s criterion [31]. A summary is given in Sec. 5.

2. System and master equations

The QD-cavity system considered in this paper is schematically illustrated in Fig. 1(a). The two spatially separated QDs (1 and 2) are fixed on a NMR inside an optical cavity. Each of the QDs has an internal structure which can be described in terms of a two-level system, which consists of a ground level |g〉_1,2 and a excited state |e〉_1,2 with the separated energy ħω _1,2. The two QDs can be separated by 200 times their own size, with no direct tunnel and electrostatic couplings between them [43]. They can be located at the beam waist in the centre of the cavity and are driven by a strong laser with Rabi frequency Ω and wave vector $\overrightarrow{k}$. Both QDs are considered as the charge qubits and trapped on the surface of the NMR and they are coupled to the NMR through the electrostatic forces. The system may lose energy through cavity leakage with a photon decay rate of κ_a, through a heat bath of the NMR with a phonon decay rate of κ_b or through spontaneous emission of the QDs from the excited state into the ground state decay rate γ. For simplicity, we treat the driving external field classically and work in the interaction picture.

 

Fig. 1 (a) Schematic plot of two QDs that adsorb on the surface of the NMR inside a driven cavity. (b) Energy levels of the coupled QD-cavity system where the QD is red (the 1st QD) and blue (the 2nd QD) detuned from the laser.

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In a frame rotating at the driving laser frequency, under the dipole approximation and rotating wave approximation (RWA), the dynamics of the total system is governed by the Hamiltonian (we have taken ħ = 1 for conciseness of notation.)

Since the QDs are driven by a strong laser field, we can introduce dressed states which are eigenstates of the first term of the Hamiltonian H ₀ as described by Eq. (2), where eigenstates are [44]

We perform the unitary “dressing” transformation of the interaction Hamiltonian

We see that in the dressed-QDs picture, the NMR mode is tuned to the dressed-state transitions that occur at three characteristic frequencies, ω_m and ${\omega }_m\pm 2{\overline{\mathrm{\Omega }}}_j$. By matching the frequency of the NMR mode to one of the dressed states frequencies, we may manipulate the interaction between the driven system and the NMR mode.

In this coupled system, there are two independent mechanisms for energy dissipation: the cavity decay and the two QDs dipole decay. The noise processes resulting from interactions between the system and the environment are assumed Markovian so that the time evolution of the system can be described by a master equation

Our aim is to study the correlation between the cavity mode and the NMR mode. we assume that the dynamics of the cavity photon and NMR phonon subsystems are slower than the QD dynamics, that is, Ω_m ≫ γ ≫ κ_a,b and Ω_m ≫ g,λ. In this regime, the two QDs possess the large number of degrees of freedom and are regarded as a reservoir because the reservoir variable $R_m^j(m=\pm ,z)$ can only be weakly perturbed by the NMR-cavity subsystem. Following the standard procedures, i.e. writing down the Liouville-von Neumann equation for the cavity-NMR subsystem density operator in the interaction picture with respect to the dressed QDs, performing the Born-Markov approximation and RWA, and tracing out the freedom degrees of the two QDs from Eq. (9), we can obtain the following master equation governing the dynamics of the NMR-cavity subsystem as

After obtaining the steady state solutions of the above equations, we can discuss the entanglement and one-way steering of the two-mode subsystem.

3. Entanglement between the cavity mode and the NMR mode

Up to now, we have achieved the master equation which governs the dynamics of the cavity-NMR subsystem and the expectation values of the cavity operators and the NMR operators. In the following, we will focus on quantum correlations of the NMR-cavity subsystem. The state of the cavity field and the NMR governed by Eq. (11) in phase space should be a two-mode Gaussian state since the master equation (11) only contains the quadratic terms of the bosonic operators a(b) and a ^†(b ^†). Two spatially separated quantum modes a and b can be fully described by means of field quadratures [46]: the amplitude quadrature, $X_1=(a{e}^{-i{\theta }_1}+a^{†}{e}^{i{\theta }_1})/\sqrt{2}$, $X_2=(b{e}^{-i{\theta }_2}+b^{†}{e}^{i{\theta }_2})/\sqrt{2}$ and the phase quadrature $Y_1=i(a^{†}{e}^{i{\theta }_1}-a^{†}{e}^{-i{\theta }_1})/\sqrt{2}$, $Y_2=i(b^{†}{e}^{i{\theta }_2}-b^{†}{e}^{-i{\theta }_2})/\sqrt{2}$ with θ_j(j = 1,2) being the phase angles of the modes, in analogy to the position q_j and momentum p_j of the original EPR variables. A major role in the theoretical and experimental manipulation of Gaussian states is played by unitary operations which preserve the Gaussian character of the states on which they act. For a two-mode Gaussian state, the quantum statistics properties of the two-mode field are completely determined by the covariance matrix of its Wigner characteristic function which is defined as [46]

In order to determine the entanglement of the field modes, an entanglement criterion for the two-mode Gaussian state is needed. Here, the summation of the quantum fluctuations proposed by Duan et al. in [42] is chosen. We introduce two operators $\widehat{u}=\varsigma {\widehat{X}}_1-\frac{1}{\varsigma }{\widehat{X}}_2$, $\widehat{v}=\varsigma {\widehat{Y}}_1+\frac{1}{\varsigma }{\widehat{Y}}_2$, where $\varsigma =\sqrt{(2n_2-1)/(2n_1-1)}$ is a state-dependent real number. According to Duan’s criterion, a two-mode Gaussian state is entangled if the sum of the variance $\mathrm{\Sigma }=〈{(\mathrm{\Delta }\widehat{u})}^2〉+〈{(\mathrm{\Delta }\widehat{v})}^2〉$ satisfy the following inequality:

It is evident that the entanglement condition Eq. (18) reduces to the following inequality

The two-mode Gaussian state are entangled as long as the entanglement parameter is satisfied with Eq. (19). It should be noted that the two-mode field with ϒ = −2 corresponds to the original EPR entanglement.

In an effort to study the property of the entanglement, another quantity which is investigating is the purity of the state of the NMR-cavity subsystem. The purity can be chosen to measure the mixedness of a state ρ and it is expressed as [47]

In general, for m-dimensional systems P ranges from $\frac{1}{m}$ for completely mixed states to 1 for pure states. The closer to 1 the quantity P is, the purer the state of the system is.

In the following, we investigate the condition for the occurrence of the entanglement and study the properties of the entanglement in the different cases. There is no direct interaction between the cavity and the NMR in the system that we have considered. They establish the quantum correlation through the two QDs. Therefore we consider that the two QDs essentially act as a bath of the cavity and the NMR. So now comes the question, why do we select two QDs rather than one? In the following, we discuss two cases where there is one QD and two QDs inside the cavity.

3.1. One QD inside the cavity

We will start the discussions on the special situation where only one QD adsorbs on the surface of the NMR. In this case, we are setting g ₂, λ ₂, s ₂, c ₂, and ${\overline{\mathrm{\Omega }}}_2$ to be zero in the system. Now, there is still no direct interaction between the cavity and the NMR. We numerically show the entanglement quantity ϒ and the purity P versus the phonon frequency ω_m under the above conditions in Fig. 2, for the other parameters with γ as: g ₁ =4γ, λ ₁ = 2.5γ, Ω=50γ, Δ₁ =98γ, κ_a = 0.4γ, κ_b = 0.001γ, $\overline{n}=0.5$. When these parameters are assumed to be the above values, ${\overline{\mathrm{\Omega }}}_1=70\gamma$. In Fig. 2, we can find that the entanglement quantity ϒ satisfies the inequality −1 < ϒ < 0 in the vicinity of ${\omega }_m=2{\overline{\mathrm{\Omega }}}_1$ and its minimum value is obtained when ${\omega }_m=2{\overline{\mathrm{\Omega }}}_1$. At the same time, the purity P reaches the maximum when the frequency of the NMR achieves $2{\overline{\mathrm{\Omega }}}_1$ (see Fig. 2), i.e., the maximal purity of the state is obtained. But the maximal purity is far away from 1, the system is prepared in the mixed state. This illustrates that the cavity mode and the NMR mode can be entangled when only one QD serves as the reservoir.

 

Fig. 2 The steady entanglement ϒ and the purity P characterizing the entanglement as a function of ω_m/γ for g ₁ /γ =4, λ ₁ /γ =2.5, Ω/γ =50, Δ₁ /γ =98, κ_a/γ =0.4, κ_b/γ =0.001, $\overline{n}=0.5$ when only one QD is in the cavity.

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The reason for generation of the entanglement can also be understood as following. When the frequency of the NMR ω_m approximates to the double of the generalized Rabi frequency of the strong laser with the 1st QD ${\overline{\mathrm{\Omega }}}_1$, we can perform the RWA to drop the terms oscillating at rapid frequencies in Eq. (8), then the 1st dressed QD and the NMR-cavity subsystem only remain the resonant interaction between them, which is described as

We introduce a two mode squeezing operator S(r) = exp(r^∗ab − ra ^† b ^†) with the parameter $r=\arctan \mathrm{h}\frac{g_1{s}_1^2}{{\lambda }_1{s}_1{c}_1}$. When we choose the related parameters meet the condition g ₁ g ₂ ≈ λ ₁ λ ₂, we can obtain the new bosonic operators ã and $\tilde{b}$ which are connected to the perviously mentioned operators a and b by the two mode squeezing transformation $ã=S^{†}(r)aS(r)=A(g_1{s}_1^2{a}^{†}+{\lambda }_1{s}_1{c}_{1}b)$, $\tilde{b}=S^{†}(r)bS(r)=A(g_2{s}_2^{2}a+{\lambda }_2{s}_2{c}_2{b}^{†})$ with the normalized parameter $A=1/\sqrt{{\lambda }_1^2{s}_1^2{c}_1^2-g_1^2{s}_1^4}$. So we perform the two-mode squeezing transformation S(r) to the interaction Hamiltonian $V_{\mathrm{I}}^1$ and it is transformed into the well-known Jaynes-Cummings form as

When we take the frequency of the NMR ω_m = 140γ, i.e., ${\omega }_m=2\overline{\mathrm{\Omega }}$, we can calculate the population from Eq. (35) in the appendix and obtain the values $P_1^{+}=0.03$ and $P_1^{-}=0.97$. We can say that $P_1^{-}\approx 1$ and regard the dressed state | − 〉₁ as a quantum pure state, which is helpful for the entanglement between the two modes a and b. Here, we only consider the decay of the 1st QD. In this case, the Hamiltonian ${\tilde{V}}_{\mathrm{I}}^1$ in Eq. (22) represents the interaction between the combined mode ã and the 1st QD which can be treated as a quantum reservoir. Since the combined mode $\tilde{b}$ is decoupled from another combined mode ã and the 1st QD, only the mode ã interacts with the 1st QD. With the help of the dissipation of the 1st QD, the combined mode ã will be in a vacuum state and the NMR-cavity subsystem will be in a mixed entangled state when the system reaches the steady state.

3.2. Two QDs inside the cavity

For the case of using one auxiliary QD, the entanglement degree of the NMR-cavity subsystem still needs to be significantly improved. Generating high-purity quantum states is an important step toward quantum information processing. When there are two QDs that adsorb on the surface of the NMR, we restrict ourselves to the case that both Δ₁ and Δ₂ with the definitions below Eq. (2) are satisfied with Δ₁ = −Δ₂ (see Fig. 1(b)), which indicates that the laser field is red-detuned to the 1st QD and blue-detuned to the 2nd QD, the Rabi frequencies of the detuned field are ${\overline{\mathrm{\Omega }}}_1={\overline{\mathrm{\Omega }}}_2=\overline{\mathrm{\Omega }}$. So we can achieve the approximate interaction Hamiltonian from Eq. (8) in the same way by using the RWA and performing the two-mode squeezing transformation

From the Eq. (23), we can see that the subsystem of the cavity and the NMR is in two-mode squeezing state after we take some approximations. And the NMR-cavity subsystem is very close to the original EPR entangled state, i.e., ϒ is very close to −2. There are two kinds of processes in this NMR-cavity subsystem due to the combined interactions of the QDs with the cavity and the NMR: (1) through the emission of a photon with the cavity mode or the absorption of a phonon with the NMR mode accompanied by the transition of the 1st QD from the dressed state |−〉₁ to the dressed state |+〉₁ as described by the term $\tilde{a}{R}_{+-}^1$ in Eq. (23); (2) through the absorption of a photon with the cavity mode or the emission of a phonon with the NMR mode accompanied by the transition of the 2nd QD from the dressed state |+〉₂ to the dressed state |−〉₂ as described by the term $\tilde{b}{R}_{-+}^2$ in Eq. (23). It is important to note that, due to the combined interactions of the two QDs with the NMR and the cavity, the photon and the phonon created and (or) annihilated in these processes mentioned above and their counterparts are correlated, which leads to the entanglement between the NMR mode and the cavity mode.

It is well known that the optical squeezing of atomic resonance fluorescence in a cavity can be achieved when the atom is in a pure quantum state [48, 49], which reminds us to pay close attention to the population of the two QDs. In our system, it could build a squeezed vacuum reservoir for the NMR-cavity subsystem due to the coupling between the NMR and the cavity and the two QDs as described in Eq. (23). The population on states |+〉₁ and |−〉₂ of the two QDs may preclude the preparation of a squeezing vacuum reservoir. When we adjust the detunings Δ_j(j = 1,2) between the transition frequency of the j-th QD and the frequency of the laser to satisfy with conditions Δ₁ ≈ 2Ω and Δ₂ ≈ −2Ω, we find that the populations of the two QDs $P_j^{\pm }$ on the dressed states |+〉_j and |−〉_j are $P_1^{-}\gg P_1^{+}$ and $P_2^{+}\gg P_2^{-}$, meaning that the two QDs are nearly trapped in the quantum pure state (the states |−〉₁ and |+〉₂).

We still only take into consideration the decay of the two QDs, which is the same as before. From Eq. (23), it is even easy to know that the two combined modes ã and $\tilde{b}$ will be in the two-mode vacuum sate. Going back to the original representation, the steady state is the two-mode squeezed vacuum state. In this case, entanglement between the cavity mode and the NMR mode reaches the maximum.

The above explanation about the reason for the generation and the properties of the entanglement do not take into account the dissipations of the NMR and the cavity. Next, we think about all important factors which may affect the entanglement, including the dissipations and the environment temperature. We scale the parameters with γ as: g ₂ =1.1γ, λ ₂ =1.6γ, Ω=50γ, Δ₂ =−98γ, κ_a =0.01γ, the other parameters are the same with that in Fig. 2.

The dependences of the steady-state variance of the entanglement quantity ϒ and the purity P characterize the steady entanglement of the NMR-cavity subsystem on the frequency ω_m of the NMR is shown in Fig. 3. We choose the parameters in Fig. 3, then we find the populations of the two QDs $P_{\pm }^j$ on the dressed states |+〉_j and |−〉_j are the value $P_1^{+}=P_2^{-}=0.03$, $P_2^{+}=P_1^{-}=\mathrm{0.97.}$ That is, $P_1^{-}\gg P_1^{+}$ and $P_2^{+}\gg P_2^{-}$. It can be approximated considered that the two QDs are nearly trapped in the quantum pure states |−〉₁ (or |+〉₂), is good for obtaining the great entanglement. It is indicated Fig. 3 that the value of ϒ can be reduced to be smaller than 0 in the vicinity of ω_m=140γ and the smallest value is found when ω_m=140γ which is equivalent to the condition ${\omega }_m=2\overline{\mathrm{\Omega }}$. We are only interested in the vicinity of ${\omega }_m=2\overline{\mathrm{\Omega }}$. In this limit, the interaction Hamiltonian between the dressed QDs and the NMR-cavity subsystem in the Eq. (8) contains terms that are time independent and thus corresponding to the resonant interaction of the NMR-cavity subsystem with the dressed QDs. It also contains terms that have an explicit time dependence of the forms $\exp [i(2\overline{\mathrm{\Omega }}+{\omega }_m)t]$ and exp(iω_mt). These terms correspond to the nonresonant interactions of the NMR-cavity subsystem with the dressed QDs, which are fast oscillating terms. From Fig. 3, we can see that the optimal entanglement of the NMR-cavity subsystem is generated and its value reaches almost −1.5 when the condition ${\omega }_m=2\overline{\mathrm{\Omega }}$ is well satisfied. Compared with Fig. 2, there is a stronger entanglement between the cavity mode and the NMR mode in Fig. 3.

 

Fig. 3 The steady variance ϒ and the purity P characterizing the entanglement as a function of ω_m/γ for g ₂ /γ =1.1, λ ₂ /γ =1.6, Ω/γ =50, Δ₂ /γ =−98, κ_a/γ =0.01 when two QDs are in the cavity. The other parameters are the same with that in Fig. 2.

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Next, we observe the purity P and find that it can reach the maximum value when the subsystem obtains the optimal entanglement. In comparison with Fig. 2, the maximal purity in Fig. 3 (blue dashed line) is improved greatly and is very close to 1, i.e., the subsystem is very near to the pure state. The nearer to the pure state the subsystem approximates, the stronger the entanglement of the subsystem gets.

Another information indicated from Fig. 3 is that the influence of the environment temperature. We can see from Fig. 3 that the increase of the entanglement ϒ and the decrease of the purity P come with the increase of the mean phonon number $\overline{n}$ of the phonon environment. The effect of the temperature of the phonon reservoir is to damage the entanglement between the cavity mode and the NMR mode.

4. EPR steering

Another important quantum quantity that we are interested in is the EPR steering of the two modes. The criteria of EPR steering, which is equivalent to a definition by M. D. Reid [31], is based on Heisenberg Uncertainty Relations for conditional measurements of the amplitude and phase quadrature X_j and P_j of the two modes. A state is steerable from the mode b to a if the correlations lead to the condition

${\mathrm{\Delta }}_{\inf }^2{X}_1$ denotes the referred variance of the mode a’s measurements conditioned on the mode b’s results. The average errors of the inferences are given by [50]

In our model, ${〈X_1〉}_{\min }=〈a^{†}a〉+\frac{1}{2}$, ${〈X_2〉}_{\min }=〈b^{†}b〉+\frac{1}{2}$, $〈a^{†}b〉=〈a{b}^{†}〉=0$. Substituting the above formulae to Eq. (24), we can know that the EPR steering parameter V_{a / b} is satisfied [40]

Conversely, EPR steering from the mode a to b is certified if the EPR steering parameter V_b|a is satisfied

When one tries to guess the outcome about the mode a based on the outcomes of measurements about the mode b, we find the collapse of the mode a occurs and there is a violation of the uncertainty principle on the mode a. Then, we say the mode a can steer the mode b, and in the meantime, if the mode b cannot steer the mode a, one-way steering occurs. That is, only the meet of only one of the inequalities (28) and (29) reveals occurrence of one-way EPR steering. EPR Steering can occur in strongly entangled system and implies a direction between the two parties involved, while entanglement without EPR steering generally has no direction. We go back and look at the criterion of quantum entanglement in Eq. (19), which can be transformed into the form

Compared with Eqs. (28)–(30), we can know that EPR steering is an asymmetric form and not all entangled states are steerable.

We would like to point out that there are no parameter values at which only one of the two EPR steerings could be below zero when the mean phonon number of the NMR’s mode is equal to the mean photon number of the cavity’s mode. We assign the same values to all parameters as Fig. 3, expect for g ₁ = 3.8γ, g ₂ = 1.2γ, λ ₂ = 1.8γ and κ_a = 0.1γ. In this case, the cavity-dot coupling strength is different from the coupling strength between the NMR and the two QDs. Consequently, there are different mean occupations of the cavity mode and the NMR mode as shown in Fig. 4(a). This suggests that a kind of one-way EPR steering behavior might exist in the system. Then, we will discuss the effect of EPR steering that provides the information as to how a given mode steers the other modes to be entangled.

 

Fig. 4 (a) The mean photon 〈a ^† a〉 of the cavity mode and the mean phonon 〈b ^† b〉 of the NMR mode as the functions of ω_m/γ. (b) The one-way EPR steering parameters V _a|b and V _b|a characterizing EPR steering as a function of ω_m/γ. The parameters are estimated follows: g ₁ /γ =3.8, g ₂ /γ =1.2, λ ₂ /γ =1.8, κ_a/γ =0.1. The other parameters are the same with that in Fig. 3.

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In Fig. 4(b) we plot the EPR steering parameters V _a|b and V _b|a versus ω_m. we can see the EPR steering parameters V _a|b and V _b|a are less than zero in the vicinity of ω_m = 140γ, i.e., ${\omega }_m=2\overline{\mathrm{\Omega }}$. Through detailed numerical calculation, the range is: 138.8γ < ω_m < 141.4γ. In this range, the two modes a and b can steer each other. When ω_m moves slightly farther from $2\overline{\mathrm{\Omega }}$, i.e., 137.6γ < ω_m < 138.8γ or 141.4γ < ω_m < 143γ, the EPR steering parameters V _a|b < 0 and V _b|a > 0, which means one-way EPR steering occurs between the two modes and the phonon mode b can steer the photon mode a, but the photon mode a cannot steer the phonon mode b. That is to say, the phonon mode b is more capable for EPR steering than the photon mode a. When ω_m moves far away from $2\overline{\mathrm{\Omega }}$, i.e., ω_m < 137.6γ or ω_m > 143γ, both V _a|b and V _b|a are greater than zero, which denotes that no-way EPR steering appears.

The reason for generation of one-way EPR steering can also be understood as following. For V _b|a or V _a|b, we can consider that the quantum vacuum fluctuation is brought in when we measure the NMR mode b or the cavity mode a, which leads to that EPR steering is strictly stronger than entanglement. And in the condition of the parameters in Fig. 4, we find that EPR steering parameters V _b|a and V _b|a are greater than zero when ω_m moves far away from $2\overline{\mathrm{\Omega }}$. In this case, the cavity mode a and the phonon mode b cannot steer each other, but the entanglement between the cavity mode a and the phonon mode b still exists (see the inset of Fig. 5(a)).

 

Fig. 5 (a) The functions $\frac{1}{P_a}+\frac{1}{P_b}$ and $\frac{1}{P}+1$ of three purities depend on ω_m/γ. The inset is quantity ϒ characterizing the entanglement versus ω_m/γ. (b)the global purity P and the marginal purities P_a, P_b as the functions of ω_m/γ. The parameters are the same with that in Fig. 4.

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Since the purity can be used to estimate the property of entanglement, we can try to investigate the purity of the state of the NMR-cavity subsystem when discussing EPR steering. For the system, it is in a two-mode Gaussian state and 〈a ²〉 = 〈b ²〉 = 〈a ^† b〉 = 0. For the NMR-cavity subsystem the the NMR-cavity consisting of (a) and the NMR (b), the information can be contained in the reduced density operators ρ_a = Tr_b(ρ), ρ_b = Tr_a(ρ) through separately measuring the cavity or the NMR. Then, the marginal purities P_a (the cavity) and P_b (the NMR) can be calculated after tracing over its variables to get

From the entanglement criteria (30), we can easily know that the inequality between the global purity P and the two marginal purities P_a, P_b of the state of the NMR-cavity subsystem

In this section, we obtain n ₂ > n ₁ after we select on the parameters in Fig. 4. In this case, when one-way EPR steering happens, we can find another inequality about the global purity P according to the EPR steering criteria Eqs. (28) and (29)

Similarly, if two-way EPR steering exists in the system, the global purity satisfies the inequality

And when no-way steering exists in the system, the inequality is P < P_b. Fig. 5(b) plots that the global purity P (magenta dot line) and the two marginal purities P_a (black solid line), P_b (blue dot-dashed line) as the functions of the frequency of the NMR ω_m. When comparing Fig. 4(b) with Fig. 5(b), we find that there is two-way EPR steering in the subsystem when $P_a<P<\frac{1}{2(n_2-n_1)+1}$ within the scope of 138.8γ < ω_m < 141.4γ. And in the range of 137.6γ < ω_m < 138.8γ and 141.4γ < ω_m < 143γ, one-way EPR steering occurs, in the meantime, P_b < P < P_a. Moreover, when the global purity P is less than the two marginal purities P_a and P_b, two-way EPR steering takes place. In other words, the inequality of the three purities Eq. (33) is completely consistent with the EPR steering criterions Eqs. (28) and (29) in the NMR-cavity subsystem.

In the above discussion, we have considered the situation that a strong laser field drives two different QDs. If we use two different strong laser fields to drive two identical QDs, respectively, where the detunings of the transition frequency of the j-th QD with the corresponding strong driving field are also in opposite number, we can draw the same conclusion with the previous discussion. Moreover, with the advance of nanotechnology, the parameters considered here are experimentally achievable. For example, in the system of double QDs integrated with superconducting microwave cavities [51], the QD-cavity couplings are g/2π ∼ 20 − 100MHz, QD decay rate is γ/2π ≈ 70MHz, and the cavity dissipation rate can reach κ/2π ≈ 2MHz. The QD embedded in a NMR can provide the QD-NMR interaction strength up to λ/2π ~ 72MHz [52]. Finally, to certify the Gaussian EPR steering, the detection of intermode correlations is required, which is operable via transferring the state of mechanical motion into the microwave field and then followed by the two-quadrature heterodyne measurement, as the procedures demonstrated in the probe of entanglement between the mechanical motion and microwave field in [17]. Although the measurement apparatus does not possess perfect quantum efficiency and detection is affected by the mechanical thermal motion, it can be still practicable to characterize the inequality of EPR steering with mechanical motion initially cooled to ground state in our system.

5. Conclusion

In summary, we have studied the entanglement and EPR steering dynamics between the cavity mode and the NMR mode. The cavity mode and the NMR mode are entangled by interacting with double two-level QDs commonly driven by a strong laser field. When the QDs dynamics are faster than the NMR-cavity subsystem, one needs to adjust the laser frequency such that a laser photon absorption is accompanied by the generation of a phonon and an optical cavity photon and simultaneously the laser frequency is red-detuned to the 1st QD and blue-detuned to the 2nd QD. In this case, we can obtain the optimal entanglement between the cavity mode and the NMR mode and the maximal purity of the state of the NMR-cavity subsystem when the matching condition that the frequency of the NMR is resonant with the dressed QDs is satisfied. Furthermore, when the average occupations of the cavity mode and the NMR mode are different, the quantum vacuum fluctuation is brought in by the measurement of one of the two mode and gives rise to one-way EPR steering. In the meantime, the global purity and the two marginal purities can be used to distinguish the condition for no-way, one-way or two-way EPR steering.