One-way Einstein-Podolsky-Rosen steering via atomic coherence

Wenxue Zhong; Guangling Cheng; Xiangming Hu

doi:10.1364/OE.25.011584

1. Introduction

Einstein-Podolsky-Rosen (EPR) steering, firstly introduced by Schrödinger [1], is the entanglement-based quantum effect that embodies the “spooky action at a distance” scrutinized by Einstein, Podolsky, and Rosen [2]. It describes the ability of one observer to nonlocally “steer” the other observer’s state through local measurements and represents a fundamental form of nonlocality in quantum theory, intermediate between entanglement [2] and Bell non-locality [3]. In the view of quantum information task, EPR steering could be regarded as the distribution of entanglement from an untrusted party essentially, while entangled states need that both parties trust each other and Bell nonlocality is presented on the premise that they distrust each other [4, 5]. EPR steering provides a novel insight on quantum nonlocality and exhibits an inherent asymmetric feature, which is at the heart of the so-called EPR paradox [6] and plays a key role in the one-sided device-independent quantum key distribution protocols [7], quantum teleportation [8], randomness generation [9], and subchannel discrimination [10]. In addition, the quantum information processes based on continuous variable (CV) can be unconditionally implemented in the crucial steps such as preparing, unitarily manipulating and measuring of quantum states via using continuous quadrature amplitudes of the quantized electromagnetic field [11]. For these reasons, especially, CV EPR steering has recently attracted significant interest both in theory and experiment [12–27]. The asymmetric Gaussian steering and non-Gaussian EPR steering could be produced via using the various optical nonlinear processes [12–15]. The CV steering effects in the optomechanical system have been investigated via combining the various hybrid system [16–18]. Moreover, the multipartite EPR steering has been extensively researched. Cavalcanti et al. [19] presented a unified criteria for multipartite quantum nonlocality for EPR steering, He et al. [20] have developed the concept of genuine N-partite EPR steering and derived the inequalities to demonstrate multipartite EPR steering for Gaussian CV states in loophole-free scenarios. And recently they [21] show the two-way steering as the resources for secure teleportation of coherent states beyond the no-cloning threshold. Kogias et al. [22] proposed an intuitive and computable quantification of EPR steering for the Gaussian states. Experimentally, the creation and characterization of CV EPR correlation have been demonstrated between optical beams in a time-gated fashion [23]. An experimental realization of two entangled Gaussian modes of light has been presented that in fact shows the steering effect in one direction but not in the other [24]. Eberle et al. [25] showed the low-resource generation of bipartite EPR steering by mixing a squeezed mode with a vacuum mode at a balanced beam splitter. Steinlechner et al. [26] have demonstrated unconditional EPR steering using continuous measurements of position- and momentum-like variables. Janousek et al. [27] have presented experimental observations of multiparty EPR steering for the development of multiparty quantum communication protocols with asymmetric observers. It is noted that most of the above schemes are focused on the other hybrid system than the atom-cavity system. Actually, the atoms have long lived coherence between their ground states, due to which the absorption is cancelled and the dispersion is remarkably enhanced under the condition driving [28,29].

On the other hand, atomic coherence has attracted much attention in quantum optics and laser physics in past decades. It can lead to various physical effects [28–32], such as laser without inversion [28], electromagnetically induced transparency [28,29], coherent population trapping [28,30], enhanced nonlinear processes [31], modification of spectra [32], and so on. Recently, great interest has been paid to the coherent manipulation of quantum correlations of the photons in near-resonantly driven systems [33–44]. In this case, the atoms are excited by the driving fields and then they emit new photons of different frequencies into the cavity modes. As a result, the fluctuations of photons from stimulated emission are highly correlated. For instance, electromagnetically induced transparency has been shown to generate the entangled source of light [33–35]. The schemes for entanglement preparation based on the correlated spontaneous emission laser have been proposed by many researchers [36–39]. The four-wave mixing process in an ensemble of atoms is found to be an efficient way for obtaining the quantum correlation, where the narrow-band entangled beams can be generated with the potential applications in long-distance communications [40–44]. In particular, based on the four-wave mixing process in two-level atomic system, Pielawa et al. presented to implement the two-step engineering to obtain the EPR entangled states via the dissipation of the atomic reservoir [40]. In this scheme, the dressed-state representation and the Bogoliubov modes are employed, where one of the transformed modes is coupled with the Rabi sidebands and the other is decoupled from the dressed atoms. As a consequence, the one-channel dissipation process establishes for the Bogoliubov modes and it is necessary to utilize two successive beams of atoms as a reservoir to achieve the EPR entanglement. However, the atomic coherent effects are currently limited to play a crucial role in entanglement generation. In addition, the non-resonant interactions are always applied in the above four-wave mixing processes [40–44]. For the above two-level scheme [40], the entanglement is obtained when the detunings exist, not only for driving-bare atom detuning, but also for the cavity mode-atom detunings. Obviously, the third-order nonlinear susceptibility of two-level four-wave mixing is relevant to the dressed-state population difference, which strongly depends on the detuning between the driving field and the atoms. When the driving field interacts resonantly with the two-level atomic ensemble, the populations of two dressed states are equal and no correlated photons are obtained.

Here we analyze the influence of atomic coherence on the steady-state one-way EPR steering in the four-level double-cascade atomic system. Our scheme is based on the full-resonant interactions in which all the fields are resonant with the different atomic transitions and the generated light exhibits the steerability property due to coherence-induced mixing process. In terms of the method of the dressed-state representation and the Bogoliubov modes, the asymmetric couplings occur between the transformed modes and the dressed atoms, in which only one of the Bogoliubov modes is involved into the interaction with the dressed atoms and the other is excluded to the atomic reservoir. It is such an asymmetric interaction induces the asymmetric dissipation channel, throught which one transformed mode undergoes annihilation due to absorption of the dressed atoms and the other still is not affected by the atoms. The results show that the one-way steering occurs from the Bogoliubov mode with more photons to the transformed mode with few photons, which is easily achieved via changing the intensity of an external field. There are some striking features in the present scheme. First, the one-way steering is established under the condition of full resonance, where the resonant interactions appear not only between the electromagnetical fields and the bare atoms, but also between the Bogoliubov modes and dressed atoms, which is in stark contrast to the two-level scheme [40]. Furthermore, it is easy to achieve the asymmetric steering only via modulating the intensity of one of driving fields. Second, the asymmetric correlation occurs the field modes with large frequency difference. The cavity modes are generated from the different transitions and so they strongly depend on the atomic transition frequencies. With a proper choice of the coupled atoms, the steering could occur between the fields between the optical and microwave fields. Last but not the least, the present scheme is based on the dissipation of the atomic reservoir, which is robust against the stochastic fluctuations and is independent of the initial states for the field modes and atoms.

This paper is organized as follows. In Sec. II, we describe the system and derive the master equation of the Bogoliubov modes. In Sec. III, we analyze the EPR steering criteria via using the transformed modes. And in Sec. IV we analyze the steering results for the same and different cavity decays. A summary is given in the last section.

2. Model and equation

We consider the resonant atom-cavity system, in which N_a driven cascade-type four-level atoms with the energy levels |l〉 (l = 1 – 4) are placed inside a two-mode cavity. As shown in Fig. 1(a), two strong coherent fields with frequencies ω_L₁ and ω_L₂ are applied to the transitions |1〉 – |2〉 and |2〉 – |4〉 of one atom, respectively. Two quantized fields with frequencies ω₁ and ω₂ are respectively generated from the atomic transitions |1〉 – |3〉 and |3〉 – |4〉. In the dipole approximation and the interaction picture, the master equation for the density operator of the atom-field system is written as [28]

\dot{ρ} = - \frac{i}{ℏ} [H_{0} + H_{I}, ρ] + ℒ ρ,

where

H_{0} = \frac{ℏ}{2} \sum_{μ = 1}^{N_{a}} [(Ω_{1} σ_{21}^{μ} + Ω_{1}^{*} σ_{12}^{μ}) + (Ω_{2} σ_{42}^{μ} + Ω_{2}^{*} σ_{24}^{μ})],

H_{I} = ℏ \sum_{μ = 1}^{N_{a}} [g_{1} (a_{1} σ_{31}^{μ} + a_{1}^{†} σ_{13}^{μ}) + g_{2} (a_{2} σ_{43}^{μ} + a_{2}^{†} σ_{34}^{μ})],

ℒ ρ = ℒ_{c} ρ + \sum_{j = 1, k = 2, 3} (ℒ_{j, k} ρ + ℒ_{k, j + 3} ρ) .

The term H₀ corresponds to the resonant interaction of the atoms with the driving fields and H_I represents the resonant coupling between the atoms and the quantized fields. Ω₁ and Ω₂ are complex Rabi frequencies of the driving fields and here they are taken to be real for simplificity. And g_l (l = 1, 2) are the coupling strengths of the atoms with the lth cavity field. ℒρ represents the relaxations of cavity modes and atoms, where

ℒ_{c} ρ = \sum_{l = 1, 2} \frac{κ_{l}}{2} (2 a_{l} ρ a_{l}^{†} - a_{l}^{†} a_{l} ρ - ρ a_{l}^{†} a_{l})

with κ_l (l = 1, 2) being the cavity loss rates, and

ℒ_{j, k} ρ = \sum_{μ = 1}^{N_{a}} \frac{γ_{j k}}{2} (2 σ_{j k}^{μ} ρ σ_{k j}^{μ} - σ_{k j}^{μ} σ_{j k}^{μ} ρ - ρ σ_{k j}^{μ} σ_{j k}^{μ})

with γ_jk being the atomic decay rates from state |k〉 to |j〉.

Fig. 1 (a) The resonantly-coupled four-level ladder atomic system. Two laser fields are resonantly applied to the atomic transitions |1〉 – |2〉 and |2〉 – |4〉, respectively, and two cavity modes are generated from the corresponding transitions |1〉 – |3〉 and |3〉 – |4〉. (b) The one-channel interaction between the dressed atoms and the Bogoliubov mode b₁. (c) The one-channel coupling of the dressed atoms with the Bogoliubov mode b₂. (d) The steady-state populations $ρ_{00}^{s s}$ and $ρ_{33}^{s s}$ for a single atom.

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Next we will utilize the present mixing system to establish the asymmetric quantum correlation between two field modes. To make the physical mechanism more clear, we employ the method of the dressed-state representation and the Bogoliubov mode. In order to do so, we will take the following steps. (i) The representation of coherence-induced dressed state. Assuming that the driving fields are much stronger than the cavity modes, Ω_l ≫ g 〈a_l〉, l = 1, 2, we diagonalize the Hamiltonian $H_{0}^{μ}$ for the μth atom and have the dressed states, i.e. eigenstates of $H_{0}^{μ}$ , with the form [45]

\begin{array}{l} | +^{μ} 〉 & = & \frac{1}{\sqrt{2}} (sin θ | 4^{μ} 〉 + | 2^{μ} 〉 + cos θ | 1^{μ} 〉), \\ | 0^{μ} 〉 & = - cos θ | 4^{μ} 〉 + sin θ | 1^{μ} 〉, \\ | -^{μ} 〉 & = & \frac{1}{\sqrt{2}} (sin θ | 4^{μ} 〉 - | 2^{μ} 〉 + cos θ | 1^{μ} 〉), \end{array}

where

θ = arctan \frac{Ω_{2}}{Ω_{1}}

. These dressed states |0^μ〉 and |+^μ〉, −^μ〉 have their eigenvalues

λ_{0, \pm} = 0, \pm \frac{ℏ d}{2}

with

d = \sqrt{Ω_{1}^{2} + Ω_{2}^{2}}

. In such a representation, the term H₀ can be viewed as the free Hamiltonian with

H_{0} = \frac{ℏ d}{2} \sum_{μ = 1}^{N_{a}} (σ_{+ +}^{μ} - σ_{- -}^{μ})

. (ii) Transformation of picture. We can implement picture theory and transform into the new interaction picture via using the unitary operator U = exp (−iH₀/t). Obviously, in the new picture the Hamiltonian H_I, with the terms such as exp(±idt/2) and exp(±idt), explicitly depends on the time. Here we can focus on the resonant case, where the cavity modes are resonant with the corresponding atomic transitions. Under the condition of g 〈a_l〉 ≪ Ω_l (l = 1, 2), we can make the secular approximation and then obtain the interaction Hamiltonian between the dressed atoms and the cavity modes

H_{I} = ℏ \sum_{μ = 1}^{N_{a}} [(g_{1} sin θ a_{1} - g_{2} cos θ a_{2}^{†}) σ_{30}^{μ} + (g_{1} sin θ a_{1}^{†} - g_{2} cos θ a_{2}) σ_{03}^{μ}] .

The Hamiltonian (6) describes the dressed atomic transitions |0〉 ↔ |3〉 accompanied by emission and absorption of photons. A transition |3〉 → |0〉 is accompanied by emission (absorption) of photon with frequency ω₁ (ω₂) and the reversible process is the amplification (absorption) of modes a₂ (a₁) and transition from |0〉 to |3〉. Such a coupling of dressed atoms with the cavity field has been presented in two-level mixing process, by which the two-mode squeezed vacuum state can be obtained via atomic reservoir [40]. However, there are differences between the present model and two-level scheme. Here the four-level scheme is focused on the full-resonant case, while the two-level scheme is based on the Rabi-sideband interaction. On the other hand, for the present scheme, the emission and absorption of photons strongly depend on the Rabi-frequency ratio of pump fields. But for two-level case, they are relevant to the detuning and Rabi frequency of the driving field. And the atomic relaxation term in the new picture is given by

ℒ_{a} ρ = \sum_{μ = 1}^{N_{a}} [\sum_{j = +, -, 0} (ℒ_{j 3} ρ + ℒ_{3 j} ρ) + \sum_{j, k = +, -, 0; j \neq k} ℒ_{j k} ρ + ℒ_{ph}^{+ -} ρ]

, where

ℒ_{ph}^{+ -} ρ = \frac{γ_{ph}}{4} (2 σ_{p}^{μ} ρ σ_{p}^{μ} - σ_{p}^{μ} σ_{p}^{μ} ρ - ρ σ_{p}^{μ} σ_{p}^{μ})

with

σ_{p}^{μ} = σ_{+ +}^{μ} - σ_{- -}^{μ}

. The parameters in ℒ_aρ formula are expressed as by γ₊₋ = γ₋₊ = (γ₁₂ cos² θ + γ₂₄ sin² θ)/4, γ₀₊ = γ₀₋ = γ₁₂ sin² θ/2, γ₊₀ = γ₋₀ = γ₂₄ cos² θ/2, γ₊₃ = γ₋₃ = γ₁₃ cos² θ/2, γ₃₊ = γ₃₋ = γ₃₄ sin² θ/2, γ₀₃ = γ₁₃ sin² θ, γ₃₀ = γ₃₄ cos² θ, and γ_ph = 2γ₊₋. In addition, the steady-state dressed populations

ρ_{33}^{s s}

and

ρ_{00}^{s s}

could be calculated at the absence of cavity modes as

\begin{array}{l} ρ_{33}^{s s} & = \frac{3 {sin}^{2} θ {cos}^{2} θ}{2 + 2 {cos}^{2} θ}, \\ ρ_{00}^{s s} & = \frac{{sin}^{2} θ + {sin}^{4} θ}{2 + 2 {cos}^{2} θ}, \end{array}

which have been plotted in Fig. 1(d). It is obvious that

ρ_{33}^{s s}

and

ρ_{00}^{s s}

strongly depends on the Rabi frequency ratio Ω₂/Ω₁ and especially for Ω₂ = Ω₁, the equal populaitons

ρ_{00}^{s s} = ρ_{33}^{s s}

is obtainable. (iii) The Bogoliubov field modes. Introducing a pair of Bogoliubov field modes [46]

\begin{array}{l} b_{1} & = a_{1} cosh r - a_{2}^{†} sinh r, \\ b_{2} & = a_{2} cosh r - a_{1}^{†} sinh r, \end{array}

we can rewrite the Hamiltonian (6) as

{\tilde{H}}_{I} = ℏ G \sum_{μ = 1}^{N_{a}} (b_{1} σ_{30}^{μ} + b_{1}^{†} σ_{03}^{μ}), (\frac{Ω_{2}}{Ω_{1}} > \frac{g_{2}}{g_{1}})

{\tilde{H}}_{I} = - ℏ G \sum_{μ = 1}^{N_{a}} (b_{2} σ_{03}^{μ} + b_{2}^{†} σ_{30}^{μ}), (\frac{Ω_{2}}{Ω_{1}} < \frac{g_{2}}{g_{1}})

with

G = \sqrt{| {(g_{1} sin θ)}^{2} - {(g_{2} cos θ)}^{2} |}

and the squeezing parameter tanhr = g₂ cot θ/g₁ (tanhr = g₁ tan θ/g₂) for the case of

\frac{Ω_{2}}{Ω_{1}} > \frac{g_{2}}{g_{1}} (\frac{Ω_{2}}{Ω_{1}} < \frac{g_{2}}{g_{1}})

. Obviously, the Bogoliubov mode b₁ is involved in Eq. (9) and only the transformed mode b₂ is included in Eq. (10). As a result, the one-channel interaction occurs between the transformed modes and the dressed atoms. Furthermore, for each atom the dressed state |0〉 induced by the applied fields plays a key role in establishing the dissipation process. When the population

ρ_{00}^{s s}

on state |0〉 is dominant, i.e.,

ρ_{00}^{s s} > ρ_{33}^{s s} (Ω_{2} / Ω_{1} > 1)

, which can be easily manipulated via adjusting the intensities of the applied fields, the atoms transit from the dressed levels |0〉 to |3〉 and the mean excitations from mode b₁ are absorbed by the atoms, as shown in Eq. (9) and Fig. 1(b). And so the dissipation of the atomic reservoir is existent, which could drive the Bogoliubov mode b₁ to reduce to their vacuum states. Meanwhile, the b₂ is excluded and so it should be in the steady squeezed state. For thc case of

ρ_{00}^{s s} < ρ_{33}^{s s} (Ω_{2} / Ω_{1} < 1)

, however, the mode b₂ should be dissipated by the dressed atoms, which is found in Eq. (10) and Fig. 1(c). The atoms jumping from |3〉 to |0〉 will annihilate the mean photons from mode b₂, and the Bogoliubov mode b₂ could be evolved into the vacuum states. In short, the one-channel quantum dissipation process could be established in the present scheme, which is quite important to realize the one-way EPR steering.

In the combination of the dressed-atom representation with the Bogoliubov field modes, we obtain the master equation for the density operator of system

\frac{d}{d t} \tilde{ρ} = - \frac{i}{ℏ} [{\tilde{H}}_{I}, \tilde{ρ}] + ℒ_{a} \tilde{ρ} + ℒ_{c} \tilde{ρ},

where ℒ_aρ̃ denotes the relaxation of the dressed atoms and has been written below Eq. (6), and ℒ_cρ̃ represents the relaxation of the Bogoliubov modes and has the form

\begin{matrix} ℒ_{c} \tilde{ρ} & = \sum_{l \neq m}^{l, m = 1, 2} [\frac{κ_{m}}{2} (b_{l}^{†} \tilde{ρ} b_{l} - b_{l} b_{l}^{†} \tilde{ρ}) + \frac{κ_{l}}{2} (N + 1) (b_{l} \tilde{ρ} b_{l}^{†} - b_{l}^{†} b_{l} \tilde{ρ}) \\ + \frac{κ_{l}}{2} M (b_{l} \tilde{ρ} b_{m} + b_{m}^{†} \tilde{ρ} b_{l}^{†} - b_{l} b_{m} \tilde{ρ} - b_{m}^{†} b_{l}^{†} \tilde{ρ})] + H . c ., \end{matrix}

with N = sinh² r and M = sinh r cosh r. For the same loss rates κ₁ = κ₂ = κ, the common formula is given by

ℒ_{c} \tilde{ρ} = \frac{κ}{2} \sum_{l = 1, 2} (N (b_{l}^{†} \tilde{ρ} b_{l} - b_{l} b_{l}^{†} \tilde{ρ}) + (N + 1) (b_{l} \tilde{ρ} b_{l}^{†} - b_{l}^{†} b_{l} \tilde{ρ})) + κ M (b_{1} \tilde{ρ} b_{2} + b_{2} \tilde{ρ} b_{1} - b_{1} b_{2} \tilde{ρ} - \tilde{ρ} b_{1} b_{2}) + c . c .

. When the atoms decay much faster than the cavity modes, we can adiabatically eliminate the dressed atomic variables, in which the dressed atoms are in the steady-state. By tracing out the dressed atomic variables, ρ_c = Tr_atomρ, the master equation of density operator for two cavity modes is derived as

\frac{d}{d t} {\tilde{ρ}}_{c} = \sum_{l = 1, 2} [\frac{A_{l}}{2} (b_{l}^{†} {\tilde{ρ}}_{c} b_{l} - b_{l} b_{l}^{†} ρ_{c}) + \frac{B_{l}}{2} (b_{l} {\tilde{ρ}}_{c} b_{l}^{†} - b_{l}^{†} b_{l} {\tilde{ρ}}_{c})] + ℒ_{c} {\tilde{ρ}}_{c},

where

A_{1} = 2 G^{2} N_{a} ρ_{33}^{s s} / Γ, B_{1} = 2 G^{2} N_{a} ρ_{00}^{s s} / Γ; A_{2} = B_{2} = 0, (\frac{Ω_{2}}{Ω_{1}} > \frac{g_{2}}{g_{1}})

A_{1} = B_{1} = 0; A_{2} = 2 G^{2} N_{a} ρ_{00}^{s s} / Γ, B_{2} = 2 G^{2} N_{a} ρ_{33}^{s s} / Γ, (\frac{Ω_{2}}{Ω_{1}} < \frac{g_{2}}{g_{1}})

with Γ = (γ₁₃ + (γ₂₄ + γ₃₄) cos² θ) /2. The A_l terms describe the gains of the Bogoliubov modes and the B_l terms indicate the absorbs, and

ρ_{00}^{s s}

,

ρ_{33}^{s s}

are the steady-state dressed populations of each atom and they are given by Eq. (7). Clearly, when the dressed populations satisfy

ρ_{33}^{s s} < ρ_{00}^{s s}

, the gain is smaller than the absorption for Bogoliubov mode b₁ with A₁ < B₁, which means that the photons from field mode b₁ are absorbed by the dressed atoms with the atomic transition from |0〉 to |3〉. Obviously, the one-channel quantum dissipation process is established and it can lead to the annihilation of mode b₁. On the contrary, for the case of

ρ_{33}^{s s} > ρ_{00}^{s s}

, similar behavior occurs for Bogoliubov mode b₂ with A₂ < B₂, and then mode b₂ is driven into the state with little photons due to the dissipation of atomic reservoir. It is for the one-channel dissipation process that the steady-state asymmetric quantum correlations can be achieved. Furthermore, only when the dissipation process is dominant, the stability conditions are satisfied and the steady-state solutions are obtainable. In order to clearly observe the physical mechanism, we firstly concentrate on the case of the balanced cavity losses (κ₁ = κ₂ = κ). Alternatively, combining the master equation and the average value formula 〈O〉 = Tr (ρO), we obtain the following differential equations with respect to the Bogoliubov field modes

\begin{array}{l} \frac{d}{d t} 〈 b_{1}^{†} b_{1} 〉 & = - 2 μ_{1} 〈 b_{1}^{†} b_{1} 〉 + (A_{1} + κ N), \\ \frac{d}{d t} 〈 b_{2}^{†} b_{2} 〉 & = - 2 μ_{2} 〈 b_{2}^{†} b_{2} 〉 + (A_{2} + κ N), \\ \frac{d}{d t} 〈 b_{1} b_{2} 〉 & = - (μ_{1} + μ_{2}) 〈 b_{1} b_{2} 〉 - κ M, \end{array}

and

〈 b_{1}^{†} b_{2}^{†} 〉 = 〈 b_{1} b_{2} 〉

. The other terms such as

〈 b_{1}^{2} 〉

,

〈 b_{2}^{2} 〉

,

〈 b_{1}^{†} b_{2} 〉

and their conjugate terms are not shown here due to the zero steady-state values. The parameters in the above equations are μ₁ = (B₁ + κ − A₁) /2 and μ₂ = (B₂ + κ − A₂) /2, where A_l and B_l are shown in Eqs. (14,15). Obviously, the solutions for the above equations at the steady-state are given by

\begin{array}{l} 〈 b_{1}^{†} b_{1} 〉 & = (A_{1} + κ N) / 2 μ_{1}, \\ 〈 b_{2}^{†} b_{2} 〉 & = (A_{2} + κ N) / 2 μ_{2}, \\ 〈 b_{1} b_{2} 〉 & = - κ M / (μ_{1} + μ_{2}) . \end{array}

For the unbalanced loss rates (κ₁ ≠ κ₂), the motion equations for quantum correlation are complex and so they are shown in Appendix A.

3. EPR steering criteria for the Bogoliubov field modes

Now we pay attention to the EPR steering criteria for the Bogoliubov field modes. Defining the two quadratures of each cavity field as $X_{l} = a_{l} + a_{l}^{†}$ and $Y_{l} = - i (a_{l} - a_{l}^{†})$ , it is found that the Heisenberg uncertainty principle requires V(X_l) V(Y_l) ≥ 1, where the variances are defined such that V(A) = 〈A²〉 − 〈A〉², V(A, B) = 〈AB〉 − 〈A〉〈B〉. According to criterion in [6], the criteria of EPR steering for bipartite Gaussian states are expressed as

S_{12} = V_{inf} (X_{1}) V_{inf} (Y_{1}) < 1, (2 \Rightarrow 1)

or

S_{21} = V_{inf} (X_{2}) V_{inf} (Y_{2}) < 1, (1 \Rightarrow 2)

where the inferred variances V_inf (O₁₍₂₎) = V (O₁₍₂₎) − V² (O₁, O₂) /V(O₂₍₁₎) and the variables are with respect to the original modes a_l. We note that the above criteria are sufficient and necessary to detect steering for the Gaussian measurements.The first condition S₁₂ < 1 means the steerability from cavity mode 2 to mode 1, and the second S₂₁ < 1 indicates the steering from mode 1 to mode 2. Correspondingly, for the field modes in the Gaussian state, the simplified criteria are given by

{| 〈 a_{1} a_{2} 〉 |}^{2} > 〈 a_{1}^{†} a_{1} 〉 (〈 a_{2}^{†} a_{2} 〉 + 1 / 2) (2 \Rightarrow 1)

and

{| 〈 a_{1} a_{2} 〉 |}^{2} > 〈 a_{2}^{†} a_{2} 〉 (〈 a_{1}^{†} a_{1} 〉 + 1 / 2) (1 \Rightarrow 2)

, as shown in [18,19]. For the real parameters expressed by Eq. (16), it is true for

〈 a_{1} a_{2} 〉 = 〈 a_{1}^{†} a_{2}^{†} 〉

, i.e., 〈a₁a₂〉 is a real number.

Here we focus on the two-mode Gaussian state with respect to the Bogoliubov modes. It is noted that the bosonic operator (b₁, b₂) comes in pairs in the master equation (13). When the cavity field initially is in a vacuum state, it should be in a two-mode Gaussian state in the subsequent time. In terms of Eq. (8), we substituting the non-zero quantum correlations into the steering criteria and we obtain the steering criteria related to the Bogoliubov field modes b_l, which are derived as

{(〈 b_{1} b_{2} 〉 + M / 2)}^{2} > (〈 b_{1}^{†} b_{1} 〉 - N / 2) (〈 b_{2}^{†} b_{2} 〉 + 1 / 2 - N / 2), (2 \Rightarrow 1)

{(〈 b_{1} b_{2} 〉 + M / 2)}^{2} > (〈 b_{2}^{†} b_{2} 〉 - N / 2) (〈 b_{1}^{†} b_{1} 〉 + 1 / 2 - N / 2) . (1 \Rightarrow 2)

Here we have used the condition

〈 b_{1} b_{2} 〉 = 〈 b_{1}^{†} b_{2}^{†} 〉

. Defining the variables T₁₂ and T₂₁, we have the following inequalities

T_{12} = {(〈 b_{1} b_{2} 〉 + M / 2)}^{2} - (〈 b_{1}^{†} b_{1} 〉 - N / 2) (〈 b_{2}^{†} b_{2} 〉 - N / 2 + 1 / 2) > 0, (2 \Rightarrow 1)

T_{21} = {(〈 b_{1} b_{2} 〉 + M / 2)}^{2} - (〈 b_{2}^{†} b_{2} 〉 - N / 2) (〈 b_{1}^{†} b_{1} 〉 - N / 2 + 1 / 2) > 0, (1 \Rightarrow 2)

where the detailed calculations are shown in Appendix B. When one of the above inequalities holds the asymmetric steering occurs, which is called as one-way steering and plays a crucial role in the secure quantum network and quantum communication. According to the present measurement, for T₁₂ > 0, the steerability occurs from cavity mode 2 to mode 1, and for T₂₁ > 0, it is possible to realize the steering from mode 1 to mode 2. Especially, when no coupling occurs between the Bogoliubov field modes b_l and the dressed atoms, the modes b_l (l = 1, 2) are in a two-mode squeezed vacuum state, i.e., the original cavity modes a_l are in the vacuum state. It is found that

〈 b_{1}^{†} b_{1} 〉 = 〈 b_{2}^{†} b_{2} 〉 = N

, and

〈 b_{1} b_{2} 〉 = 〈 b_{1}^{†} b_{2}^{†} 〉 = - M

. Obviously, the above two inequalities are not reasonable and there is no steering based on the present measuring method. For the case of two Bogoliubov field modes being in the vacuum state (the original cavity modes being in the squeezed state), Inequalities (22) and (23) was established, which corresponds to two-way EPR steering appears. Significantly, the one-way steering via atomic coherence could be obtained in the present system via using the above criteria in Eqs. (22,23).

4. Steady-state one-way Steering for the field modes

4.1. EPR steering for the balanced cavity losses

Now we analyze the EPR steering features of two field modes for the case of κ₁ = κ₂ = κ. Without loss of generality, we scale the Rabi frequencies, and cavity loss rates in units of atomic decay rates γ_ij and meanwhile we take γ₁₂ = γ₁₃ = γ₂₄ = γ₃₄ = γ and define the parameter $C_{l} = g_{l}^{2} N_{a} (l = 1, 2)$ . In Fig. 2, we plot T₁₂₍₂₁₎ as a function of the ratio Ω₂/Ω₁ of Rabi frequencies for two pump fields under the condition of Ω₂/Ω₁ > g₂/g₁. The corresponding parameters are chosen as κ = 0.1γ, C₁ = 0.48γ², C₂ = 0.48γ² (a), C₂ = 0.72γ² (b), C₂ = 0.96γ² (c), C₂ = 1.2γ² (d). For the same coupling strength, the values of T₁₂ and T₂₁ are both less than zero and so no steering occurs for two field modes in terms of the present measurement. Fortunately, when the coupling strengths satisfy g₂ > g₁, it is possible to obtain one-way EPR steering at the steady state. As shown in Figs. 2(b)–2(d), the variable T₁₂ could be greater than zero in the regime Ω₂/Ω₁ > 2, but T₂₁ is always less than zero in the whole region. Furthermore, T₁₂ is dramatically increases and then gradually decreases with the increasing of Ω₂/Ω₁, and the maximal values $T_{12}^{max}$ is observed with $T_{12}^{max} = 0.0103$ (b), 0.2814 (c) and 1.0659 (d). This means that the steering from mode 2 to mode 1 is existent and the steering from mode 1 to mode 2 is absent. As a consequence, the steady-state one-way steering could be achievable. In addition, we also test the steering features via using the criteria in Eqs. (18,19). We find that the value of S₁₂ is always less than 1 for the case of T₁₂ > 0, and then S₁₂ > 1 when T₁₂ < 0. For the case of the maximal value of T₁₂, the variable S₁₂ is minimal. Thus the simplified criteria in Eqs. (22,23) can be used to judge the steering of the field modes in our scheme.

Fig. 2 The variable T₁₂₍₂₁₎ versus the ratio Ω₂/Ω₁ of Rabi frequencies for two pump fields for Ω₂/Ω₁ > g₂/g₁. The corresponding parameters are chosen as κ = 0.1γ, C₁ = 0.48γ², C₂ = 0.48γ² (a), C₂ = 0.72γ² (b), C₂ = 0.96γ² (c), C₂ = 1.2γ² (d).

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Now we discuss the results for the case of Ω₂/Ω₁ < g₂/g₁. Figure 3 shows the variables T₁₂₍₂₁₎ depending on the ratio Ω₂/Ω₁, where the parameters are given by κ = 0.1γ, C₂ = 18.75γ², C₁ = 48γ² (a) and C₁ = 75γ² (b). It is clearly seen from Fig. 3(a) that the value of T₂₁ is larger than zero about for 0.5 < Ω₂/Ω₁ < 0.625, but the value of T₁₂ is always less than 0 for the whole parameter region via using the criteria of Eqs. (22,23). As we increase the parameter C₁ the variance T₂₁ greater than zero is still achievable. In the present situation, it is feasible to steer mode 1 from mode 2 and it is impossible to steer mode 2 from mode 1, and thus the one-way 1 → 2 steering is obtainable at the steady-state. In addition, we also note from Fig. 3(a) that for Ω₂/Ω₁ < 0.5, the values of T₁₂ and T₂₁ are simultaneously less than zero, which also occur in Fig. 3(b). That means that there is no steering between the two cavity fields by the above measure used. It is found that the mean excitations from mode 1 is not more than that in mode 2 when the ratio Ω₂/Ω₁ is relatively small, as shown in Fig. 4(b), and the absorption effect of the engineered reservoir does not overcome the dissipation effect of the thermal reservoir. So the quantum correlation between the two fields is not strong enough to generate steering.

Fig. 3 The variables S₁₂₍₂₁₎ and T₁₂₍₂₁₎ with respect to Ω₂/Ω₁, where the parameters are given by κ = 0.1γ, C₂ = 18.75γ², C₁ = 48γ² (a) and C₁ = 75γ² (b) for the case of Ω₂/Ω₁ < g₂/g₁.

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Fig. 4 The average photon number difference $〈 b_{1}^{†} b_{1} 〉 - 〈 b_{2}^{†} b_{2} 〉$ versus Ω₂/Ω₁, where the parameters are κ = 0.1γ, C₁ = 0.48γ², C₂ = 0.96γ² (a), κ = 0.1γ, C₁ = 75γ², C₂ = 18.75γ² (b), C₁ = C₂ = 5γ², κ₁ = 0.05γ, κ₂ = 0.1γ (c), and C₁ = C₂ = 50γ², κ₁ = 0.08γ, κ₂ = 0.04γ (d). The inset maps in (a,b) are the number difference for the same coupling strengths and cavity losses with κ = 0.1γ, C₁ = C₂ = 0.48γ².

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Physically, the one-channel quantum dissipation process induced by the coherent driving plays an important role in realizing the EPR one-way steering. As a matter of fact, in the regime Ω₂/Ω₁ > g₂/g₁, the Bogoliubov mode b₁ is coupled to the dressed atoms and the mode b₂ does not participate in the interaction, as shown by Eq. (8). Obviously, mode b₁ undergoes the dissipation process, where the dressed atoms acting as the reservoir absorb in the average excitations from mode b₁, and this leads to $〈 b_{1}^{†} b_{1} 〉 = (A_{1} + κ N) / (B_{1} - A_{1} + κ)$ . At the moment the mode b₂ is in the squeezed state and the mean photon number is $〈 b_{2}^{†} b_{2} 〉 = N$ . From Fig. 4(a) we observe the result of $〈 b_{2}^{†} b_{2} 〉 > 〈 b_{1}^{†} b_{1} 〉$ for g₂ > g₁, which gives rise to the steering from mode 2 to mode 1. However, For the coupling strengths of g₂ = g₁, the average photon $〈 b_{1}^{†} b_{1} 〉$ is larger than that of mode b₂ for Ω₂/Ω₁ > 1 shown by the inset in Fig. 4(a), which are completely opposite to the situations for realizing one-way steering, and so it is impossible to obtain the steering. For clearness we present the above two cases in Tables 1. In addition, under the condition of Ω₂/Ω₁ < g₂/g₁, the effective dissipation process occurs in the b₂ basis, in which the atoms undergo the dynamics of Eq. (9) and annihilates the average excitations from mode b₂. Meanwhile, the Bogoliubov mode b₁ is decoupled to the dressed atoms and so it should be in the squeezed state. It is natural that the inequality $〈 b_{1}^{†} b_{1} 〉 > 〈 b_{2}^{†} b_{2} 〉$ holds at g₂ < g₁, as shown in Fig. 4(b), which leads to the steering from mode 1 to mode 2. But the average photon number in mode b₁ is less than that in mode b₂ for g₂ ≥ g₁, as shown by the inset in Fig. 4(b) and so no steering occurs for the present case. The various cases are shown in Table 2.

Table 1. Possibility and Impossibility of one-way EPR steering for Ω₂/Ω₁ > 1.

View Table | View all tables in this article

Table 2. Possibility and Impossibility of one-way EPR steering for Ω₂/Ω₁ < 1.

View Table | View all tables in this article

4.2. EPR steering for the unbalanced cavity losses

The EPR steering effect in the case κ₁ ≠ κ₂ is shown in Figs. 5–6. For the sake of simplicity, we assume the same strengths and atomic decay rates. We firstly focus on the case of κ₁ < κ₂ and plot the variables T₁₂₍₂₁₎ with respect to the ratio Ω₂/Ω₁ in Fig. 5, where the parameters are C₁ = C₂ = 5γ², κ₁ = 0.05γ, κ₂ = 0.1γ (a) and κ₂ = 0.15γ (b). It is found the value of T₂₁ is always larger than zero and yet the value of T₁₂ is always less than 0 for Ω₂/Ω₁ > 1. Therefore the one-way 1 → 2 steering is achievable for such a case. Under the condition of Ω₂/Ω₁ > 1, the Bogoliubov mode b₁ is involved into the interaction with the dressed atoms. But the steady-state average photon numbers satisfy $〈 b_{1}^{†} b_{1} 〉 > 〈 b_{2}^{†} b_{2} 〉$ , as shown in Fig. 4(c). This is because that the cross coupling occurs between two Bogoliubov modes b₁ and b₂ due to the asymmetric cavity loss rates, which transfer the photon numbers from modes b₂ to b₁. Similarly, the variables T₁₂₍₂₁₎ are plotted for κ₁ > κ₂ in Fig. 6, in which the parameters are chosen as C₁ = C₂ = 50γ², κ₂ = 0.04γ, κ₁ = 0.08γ (a) and κ₁ = 0.1γ (b). At present the variable T₁₂ is always greater than zero and T₂₁ is always smaller than zero in the case of Ω₂/Ω₁ < 0.84, where the steady condition is satisfied. This implies that there is the steerability from mode 2 to mode 1, which induced by the asymmetric photons $〈 b_{2}^{†} b_{2} 〉 > 〈 b_{1}^{†} b_{1} 〉$ shown in Fig. 4(d).

Fig. 5 The variables T₁₂₍₂₁₎ depending on Ω₂/Ω₁ for κ₁ < κ₂, where the parameters are C₁ = C₂ = 5γ², κ₁ = 0.05γ, κ₂ = 0.1γ (a) and κ₂ = 0.15γ (b).

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Fig. 6 The variables T₁₂₍₂₁₎ as a the function of Ω₂/Ω₁ for κ₁ > κ₂, in which the parameters are C₁ = C₂ = 50γ², κ₂ = 0.04γ, κ₁ = 0.08γ (a) and κ₁ = 0.1γ (b).

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Before conclusion, we can see the remarkable characters of our scheme, which is in contrast to the schemes in second-harmonic generation [14] and in optomechanical system [18]. First, here we use the resonantly-driven atomic system to prepare the steady-state asymmetric steering and the steerability could be easily controlled via adjusting the ratio of Rabi frequencies of external fields. As shown in [28,29], the atoms have long lived coherence between their level states under the condition driving, which could lead to the emitted photons with strong correlation. But the one-way steering in the optomechanical system could be obtainable only when the detunings exist [18], where the detunings are complicated. Second, for the present scheme the narrow-band asymmetric correlation is induced by the four-wave mixing process and it may be more useful for the long-distance quantum communication. While the correlated photons are broadband in the schemes in [14]. Third, the asymmetric correlation occurs the field modes with different frequencies. Here the frequencies of cavity modes strongly depend on the atomic transition frequencies. With a proper choice of the coupled atoms, the steering could occurs between the optical and optical fields, and optical and microwave fields. However, the frequency difference of harmonic photons is usually fixed in the scheme [14] and the correlation occurs between the optical and microwave fields in optomechanical system [19]. Last, combining the dressed-state representation and transformed Bogoliubov modes, the coherence-induced dissipation effect plays a crucial role in establishing the asymmetric correlation, which is robust against the stochastic fluctuations and is independent of the initial states for the field modes and atoms.

5. Conclusion

In conclusion, we have presented to use the quantum dissipation process induced by the coherent excitation to achieve the steady-state one-way EPR steering of two field modes. A resonantly-driven four-level atomic system in double-ladder configuration is under our consideration, in which the four-wave mixing process leads to emission into two cavity modes. Fortunately, the effective one-channel dissipation process in the Bogoliubov modes b_l could be established via combining the dressed-state representation and the Bogoliubov field modes, where the dressed atoms act as a spin reservoir and absorb in the average excitations from modes b₁ or b₂. It is such an asymmetric dissipation process that gives rise to the asymmetric quantum correlation. It is found that the one-way steering (from mode 1 to mode 2 or from mode 2 to mode 1) occurs under the full-resonant interaction, which is strongly dependent on the ratio Ω₂/Ω₁, the cavity losses, coupling strengths and atom number.

A. Appendix: Steady-state solution for the unbalanced cavity losses

In terms of the master equation and the average value formula 〈O〉 = Tr (ρO), the following differential equations for the case of κ₁ ≠ κ₂ are derived as

\frac{d}{d t} 〈 b_{1} 〉 = - μ_{1} 〈 b_{1} 〉 + η 〈 b_{2}^{†} 〉,

\frac{d}{d t} 〈 b_{2} 〉 = - μ_{2} 〈 b_{2} 〉 - η 〈 b_{1}^{†} 〉,

\frac{d}{d t} 〈 b_{1}^{†} b_{1} 〉 = - 2 μ_{1} 〈 b_{1}^{†} b_{1} 〉 + η (〈 b_{1} b_{2} 〉 + 〈 b_{1}^{†} b_{2}^{†} 〉) + (A_{1} + κ_{2} N),

\frac{d}{d t} 〈 b_{2}^{†} b_{2} 〉 = - 2 μ_{2} 〈 b_{2}^{†} b_{2} 〉 - η (〈 b_{1} b_{2} 〉 + 〈 b_{1}^{†} b_{2}^{†} 〉) + (A_{2} + κ_{1} N),

\frac{d}{d t} 〈 b_{1} b_{2} 〉 = - (μ_{1} + μ_{2}) 〈 b_{1} b_{2} 〉 - η (〈 b_{1}^{†} b_{1} 〉 - 〈 b_{2}^{†} b_{2} 〉) - M (κ_{1} + κ_{2}) / 2,

with μ₁ = (B₁ + κ₁ (N + 1) − A₁ − κ₂N) /2, μ₂ = (B₂ + κ₂ (N + 1) − A₂ − κ₁N) /2 and η = (κ₂ − κ₁) M/2. The steady-state solutions are derived as

〈 b_{1}^{†} b_{1} 〉 = \frac{1}{P} [μ_{2} (μ_{1} + μ_{2}) (A_{1} + κ_{2} N) - 2 μ_{2} η D + η^{2} (A_{1} + A_{2} + (κ_{1} + κ_{2}) N)],

〈 b_{2}^{†} b_{2} 〉 = \frac{1}{P} [μ_{1} (μ_{1} + μ_{2}) (A_{2} + κ_{1} N) - 2 μ_{1} η D + η^{2} (A_{1} + A_{2} + (κ_{1} + κ_{2}) N)],

〈 b_{1} b_{2} 〉 = \frac{1}{P} [- 2 D μ_{1} μ_{2} - η μ_{2} (A_{1} + κ_{2} N) + η μ_{1} (A_{2} + κ_{1} N)],

where D = M (κ₁ + κ₂) /2 and P = 2 (μ₁ + μ₂) (μ₁μ₂ + η²), and the other parameters are zero.

B. Appendix: Derivation of the steering criteria in Eqs. (22,23)

According to Eq. (8), we have obtained the following equalities for the initial modes $a_{1} = b_{1} cosh r + b_{2}^{†} sinh r$ , and $a_{2} = b_{2} cosh r + b_{1}^{†} sinh r$ . And we have

\begin{array}{r} 〈 a_{1} a_{2} 〉 = (2 N + 1) n_{12} + M (n_{1} + n_{2} + 1), \\ 〈 a_{1}^{†} a_{1} 〉 = N (n_{1} + n_{2} + 1) + 2 M n_{12} + n_{1}, \\ 〈 a_{2}^{†} a_{2} 〉 = N (n_{1} + n_{2} + 1) + 2 M n_{12} + n_{2}, \end{array}

with 〈b₁b₂〉 = n₁₂,

〈 b_{1}^{†} b_{1} 〉 = n_{1}

and

〈 b_{2}^{†} b_{2} 〉 = n_{2}

. As shown in Section III, for the Gaussian state with the real parameters, the simplified criteria are

{〈 a_{1} a_{2} 〉}^{2} > 〈 a_{1}^{†} a_{1} 〉 (〈 a_{2}^{†} a_{2} 〉 + 1 / 2) (2 \Rightarrow 1)

and

{〈 a_{1} a_{2} 〉}^{2} > 〈 a_{2}^{†} a_{2} 〉 (〈 a_{1}^{†} a_{1} 〉 + 1 / 2) (1 \Rightarrow 2)

. And so we substitute Eq. (32) into the simplified criteria and obtain the following formulas

\begin{array}{r} {(n_{12} + M / 2)}^{2} > (n_{1} - N / 2) ((n_{2} + 1 / 2) - N / 2), (2 \Rightarrow 1) \\ {(n_{12} + M / 2)}^{2} > (n_{2} - N / 2) ((n_{1} + 1 / 2) - N / 2) . (1 \Rightarrow 2) \end{array}

Funding

National Natural Science Foundation of China (NSFC) (Grants Nos. 11474118, 11565013, 61178021 and 11165008); Foundation of Young Scientist of Jiangxi Province, China (Grant No. 20142BCB23011); Scientific Research Foundation of Jiangxi Provincial Department of Education (Grant No. GJJ160511).

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	Populations	Coupling	Dissipation	Photons	2 → 1steering	1 → 2steering
g₂ > g₁	$ρ_{33}^{s s} < ρ_{00}^{s s}$	b ₁	Yes	$〈 b_{1}^{†} b_{1} 〉 < 〈 b_{2}^{†} b_{2} 〉$	Possible	No
g₂ ≤ g₁	$ρ_{33}^{s s} < ρ_{00}^{s s}$	b ₁	Yes	$〈 b_{1}^{†} b_{1} 〉 < 〈 b_{2}^{†} b_{2} 〉$	No	No

	Populations	Coupling	Dissipation	Photons	2 → 1steering	1 → 2steering
g₂ ≥ g₁	$ρ_{33}^{s s} > ρ_{00}^{s s}$	b ₂	Yes	$〈 b_{1}^{†} b_{1} 〉 < 〈 b_{2}^{†} b_{2} 〉$	No	No
g₂ < g₁	$ρ_{33}^{s s} > ρ_{00}^{s s}$	b ₂	Yes	$〈 b_{1}^{†} b_{1} 〉 < 〈 b_{2}^{†} b_{2} 〉$	No	Possible

	Populations	Coupling	Dissipation	Photons	2 → 1steering	1 → 2steering
g₂ > g₁	$ρ_{33}^{s s} < ρ_{00}^{s s}$	b ₁	Yes	$〈 b_{1}^{†} b_{1} 〉 < 〈 b_{2}^{†} b_{2} 〉$	Possible	No
g₂ ≤ g₁	$ρ_{33}^{s s} < ρ_{00}^{s s}$	b ₁	Yes	$〈 b_{1}^{†} b_{1} 〉 < 〈 b_{2}^{†} b_{2} 〉$	No	No

	Populations	Coupling	Dissipation	Photons	2 → 1steering	1 → 2steering
g₂ ≥ g₁	$ρ_{33}^{s s} > ρ_{00}^{s s}$	b ₂	Yes	$〈 b_{1}^{†} b_{1} 〉 < 〈 b_{2}^{†} b_{2} 〉$	No	No
g₂ < g₁	$ρ_{33}^{s s} > ρ_{00}^{s s}$	b ₂	Yes	$〈 b_{1}^{†} b_{1} 〉 < 〈 b_{2}^{†} b_{2} 〉$	No	Possible

One-way Einstein-Podolsky-Rosen steering via atomic coherence

Abstract

1. Introduction

2. Model and equation

3. EPR steering criteria for the Bogoliubov field modes

4. Steady-state one-way Steering for the field modes

4.1. EPR steering for the balanced cavity losses

4.2. EPR steering for the unbalanced cavity losses

5. Conclusion

A. Appendix: Steady-state solution for the unbalanced cavity losses

B. Appendix: Derivation of the steering criteria in Eqs. (22,23)

Funding

References and links

Cited By

Figures (6)

Tables (2)

Equations (33)

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