Detection of genuine tripartite entanglement and steering in hybrid optomechanics

Y. Xiang; F. X. Sun; M. Wang; Q. H. Gong; Q. Y. He

doi:10.1364/OE.23.030104

1. Introduction

The multipartite entangled state has attracted great interest, mainly due to its fundamental significance for understanding the nature of transition from quantum to classical regime, and potential application in certain quantum information tasks involving more than two parties [1–4]. Van Loock and Furusawa first developed the concept of fully N–partite inseparable state for continuous variable systems [5], which is demonstrated if the system cannot be described by any one of the biseparable forms. Another important type of multipartite quantum correlations is known as genuine multipartite entanglement [6–12]. By definition, a total number of N systems are genuinely N–partite entangled if and only if the density operator of the N–partite system cannot be represented in any possible mixtures of the bipartition. Recently, Shalm et al. pointed out that the full N–partite inseparability does not always imply genuine N–partite entanglement for general mixed states [13]. They derived criteria for detecting genuine tripartite entanglement, and experimentally generated genuine three-photon entangled state using the nonlinear process of cascaded spontaneous parametric downconversion. In addition, Armstrong et al. reported the realization of genuine tripartite entanglement of three intense light beams with respect to the quadrature phase amplitudes of distinct fields, setting a milestone for testing mesoscopic quantum mechanics [14].

However, it has been realized that entanglement in terms of inseparability does not demonstrate either Bell nonlocality [15] or “steering” [16, 17]. The concept of steering was first introduced by Schrödinger [18] to describe the “spooky action-at-a-distance” nonlocality raised in the Einstein–Podolsky–Rosen (EPR) paradox [19]. In 2007, Wiseman, Jones and Doherty formalized the concept of steering, and pointed that it is fundamentally different from both entanglement and Bell’s nonlocality in terms of violations of local hidden state (LHS) models [16, 17]. In fact, steering gives a way to quantify how measurements by Alice on her local particle A can collapse the wave packet of Bob’s particle B. Note the inherent asymmetry of steering that a system can be steerable in one way but not the other [20–22]. The realization of steering in experiments reveals an EPR paradox [23, 24]. Besides, steering is also potentially less susceptible to noise and decoherence than the strongest form of nonlocality, Bell’s nonlocality, in which all local hidden variable (LHV) models are falsified [25]. Therefore, the generation and observation of steering is more accessible to experiment [20, 26–33]. What’s more, steering is closely related to certain quantum information task such as one-sided device-independent quantum key distribution [34].

Recently, the concept of genuine N–partite EPR steering has been developed by He and Reid [35], where a steering nonlocality is necessarily shared among all N parties. To demonstrate it, one must eliminate the steering created by mixing states with two-party steering across different bipartitions. In 2014, Teh and Reid derived various criteria that can be used to demonstrate genuine tripartite entanglement and steering [36]. We notice that a latest experimental demonstration of genuine multipartite EPR steering that was defined in a different way by Cavalcanti and his co-workers [37]. In their scenario, one can certify different kinds of multi-partite entanglement of the ordinal state via the implementation of quantum state tomography by trusted parties. In this work, we will follow the definition given in Ref. [14, 35, 36]. It has been shown that the special multipartite entangled and steerable states can be created in the multimode optical system [14, 38] and multimode pulsed cavity optomechanical system [39]. We then ask can we generate robust genuine multipartite entanglement and steering in other promising systems, e.g. in a hybrid massive system.

Thanks to the rapid progresses of experimental techniques, optomechanical systems cooled near their ground states [40–42] provide a new physical platform to study quantum effects in mesoscopic massive systems. Particularly, the hybrid systems combining nano-mechanical oscillator and atomic ensemble or Bose-Einstein condensate and propagating fields via radiation pressure offer new possibilities of local information processing and network construction [43–47]. Recently, the generation of bipartite or tripartite quantum entangled states in optomechanical systems has been proposed [48–57].

In this paper, we propose a scheme to generate the genuine tripartite entanglement and genuine tripartite steering in the context of pulsed cavity optomechanics, and test them by a unified signature in terms of an EPR-type variance which manifests as reduced noise variance. We consider an ensemble of N identical two-level atoms located inside an optical cavity with a movable mirror. The cavity mode is driven by a short laser pulse, which couples with the mirror by a nonlinear parametric type interaction and with the atomic ensemble via a linear beamsplitter-type interaction. Using a set of experimentally realistic parameters, we can easily distinguish various types of quantum correlations among light-mirror-atom modes in this system, including the tripartite inseparability, genuine tripartite entanglement, the steering of the mirror by the collaboration of cavity field and atomic mode, and genuine tripartite steering. We further study the effects of mechanical thermal noise and damping, the relaxation time of the atomic mode, and the gains included in the criteria on the observation of those tripartite quantum correlations.

2. Dynamics of the system

The schematic diagram of the three-mode atomic optomechanical system is illustrated in Fig. 1, which can be described by the following Hamiltonian [49, 52, 53]

\begin{array}{l} H & = & ℏ Δ_{c} a_{c}^{†} a_{c} + ℏ ω_{m} b_{m}^{†} b_{m} + ℏ Δ_{a} c_{a}^{†} c_{a} + ℏ g_{0} a_{c}^{†} a_{c} (b_{m}^{†} + b_{m}) + ℏ g_{a} (c_{a}^{†} a_{c} + a_{c}^{†} c_{a}) \\ + i ℏ [E (t) a_{c}^{†} - E^{*} (t) a_{c}] . \end{array}

The first three terms give the energies of cavity field mode a_c, mirror mode b_m, and atomic mode c_a, where Δ_c = ω_c − ω_L, Δ_a = ω_a − ω_L are the detunings of the driving laser with respect to the cavity resonance and the atomic transition frequency, respectively. The fourth term denotes the optomechanical interaction, where g₀ is the single-photon coupling strength. This interaction between the cavity and mechanical modes involves the nonlinear optomechanical coupling, which can create quantum entanglement. The fifth term indicates a beamsplitter-type interaction between the cavity mode and atomic mode with coupling constant g_a. The last term represents the driving energy, where we assume the cavity is driven by a rectangular pulse with constant amplitude of E(t) over the short time interval τ and zero outside this interval. Note that there is no direct interaction between the mechanical mode and atomic mode.

Fig. 1 Schematic diagram of a hybrid atomic optomechanical system. The cavity mode has a frequency ω_c and a decay rate κ, driven by a laser pulse of frequency ω_L. The movable mirror oscillates with frequency ω_m and dissipation rate γ_m. Each of the N identical two-level atoms includes a ground state |g_j〉 and an excited state |e_j〉, with transition frequency ω_a and relaxation time γ_a. The collective dipole lowering, raising, and population difference operators can be defined as $S^{-} = \sum_{j} | g_{j} 〉〈 e_{j} |$ , $S^{+} = \sum_{j} | e_{j} 〉〈 g_{j} |$ , and $S_{z} = \sum_{j} (| e_{j} 〉〈 e_{j} | - | g_{j} 〉〈 g_{j} |)$ , respectively. We assume that all the atoms are initially prepared in their ground state, so that S_z ≃ 〈S_z〉 ≃ −N. At the same time, the initial assumption κ ≫ g_a assures that the photons inside the cavity may escape from the cavity quickly before making an effective Rabi cycle between the two levels. Thus, most of atoms stay in the ground state such that the condition of low atomic excitation limit still stands. In this case, we may represent the collective dipole lowering and raising operators in terms of annihilation and creation operators $c_{a} = S^{-} / \sqrt{| 〈 S_{z} 〉 |}$ and $c_{a}^{†} = S^{+} / \sqrt{| S_{z} |}$ , respectively.

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Following the standard linearization method [49,52,53], we find that under the rotating wave approximation and putting the blue detuned laser pulse to both cavity and atomic resonance, the slowly varying parts of the fluctuation operators satisfy the following motion equations

\begin{array}{l} δ {\dot{a}}_{c} & = & - κ δ a_{c} - i g_{a} δ c_{a} - i g (δ b_{m}^{†} + δ b_{m}) - \sqrt{2 κ} a_{in}, \\ δ {\dot{b}}_{m} & = & - γ_{m} δ b_{m} - i g (δ a_{c} + δ a_{c}^{†}) - \sqrt{2 γ_{m}} b_{in}, \\ δ {\dot{c}}_{a} & = & - γ_{a} δ c_{a} - i g_{a} δ a_{c} - \sqrt{2 γ_{a}} c_{in}, \end{array}

where g and g_a are the effective coupling strengths of the cavity mode to the mirror and to the atomic mode, respectively. The Langevin noise operators corresponding to the light decay rates κ (associated with the coupling at the input cavity mirror, and assume no further loss processes inside the cavity), the mechanical dispassion γ_m, and the relaxation of atoms γ_a, have nonzero δ correlated functions,

〈 a_{in} (t) a_{in}^{†} (t^{'}) 〉 = ({\bar{n}}_{a} + 1) δ (t - t^{'})

,

〈 b_{in} (t) b_{in}^{†} (t^{'}) 〉 = ({\bar{n}}_{m} + 1) δ (t - t^{'})

,

〈 c_{in} (t) c_{in}^{†} (t^{'}) 〉 = ({\bar{n}}_{c} + 1) δ (t - t^{'})

, i.e., the cavity field, the mirror, and the atomic mode are in a nonzero temperature T with mean number of the thermal photons or phonons n̄_a,m,c = [exp(h̄ω_c,m,a/k_BT) − 1]⁻¹. Hereafter, we will drop the label δ on the fluctuation operators to simplify the notation.

In the bad cavity limit κ ≫ g_a, g, the cavity mode can be adiabatically eliminated from the dynamics of the mirror and atomic modes to leave all modes depending only on the input quantities. The adiabatic solutions for the fluctuation operator of the cavity mode is

a_{c} (t) \approx - i \frac{g_{a}}{κ} c_{a} (t) - i \frac{g}{κ} b_{m}^{†} (t) - \sqrt{\frac{2}{κ}} a_{in} (t) .

By substituting this result into the equations for ḃ_m and ċ_a in Eq. (2), we get

\begin{array}{l} {\dot{c}}_{a} & = & - (G_{a} + γ_{a}) c_{a} - \sqrt{{GG}_{a}} b_{m}^{†} + i \sqrt{2 G_{a}} a_{in} - \sqrt{2 γ_{a}} c_{in}, \\ {\dot{b}}_{m}^{†} & = & (G - γ_{m}) b_{m}^{†} + \sqrt{{GG}_{a}} c_{a} - i \sqrt{2 G} a_{in} - \sqrt{2 γ_{m}} b_{in}^{†}, \end{array}

where G = g²/κ (

G_{a} = g_{a}^{2} / κ

) is the effective coupling strength between the cavity mode and the mechanical (atomic) mode.

We may solve that set of equations by matrix diagonalization which gives two orthogonal superposition modes. The procedure is as follows. Equation (4) can be written in a matrix form as

\dot{\vec{Y}} (t) = M \vec{Y} (t) + \vec{η} (t),

where

\vec{Y} (t) = {(c_{a} (t), b_{m}^{†}) (t))}^{T}

, the drift matrix M is given by

M = (\begin{matrix} - G_{a} - γ_{a} & - \sqrt{{GG}_{a}} \\ \sqrt{{GG}_{a}} & G - γ_{m} \end{matrix}),

and

\vec{η} (t) = {(\begin{array}{l} i \sqrt{2 G_{a}} a_{in} - \sqrt{2 γ_{a}} c_{in}, & - i \sqrt{2 G} a_{in} - \sqrt{2 γ_{m}} b_{in}^{†} \end{array})}^{T}

. A diagonalization of the matrix above results in two real eigenvalues

λ_{u} = \frac{1}{2} [G - G_{a} - γ_{a} - γ_{m} + ε]

and

λ_{v} = \frac{1}{2} [G - G_{a} - γ_{a} - γ_{m} - ε]

, where

ε = \sqrt{G^{2} + {(G_{a} + γ_{a} - γ_{m})}^{2} - 2 G (G_{a} - γ_{a} + γ_{m})}

. The coefficients of the corresponding eigenvectors are

{(u_{1}, u_{2})}^{T} = {(1, (- λ_{u} - G_{a} - γ_{a}) / \sqrt{{GG}_{a}})}^{T}

and

{(v_{1}, v_{2})}^{T} = {(- 1, (λ_{v} + G_{a} + γ_{a}) / \sqrt{{GG}_{a}})}^{T}

. We can then directly obtain the time-dependent solutions of the modes c_a(t) and

b_{m}^{†} (t)

based on the two eigenvalues and their corresponding eigenvectors, as given by

\begin{array}{l} c_{a} (t) & = & \frac{e^{λ_{u} t} v_{2} + e^{λ_{v} t} u_{2}}{v_{2} + u_{2}} c_{a} (0) + \frac{e^{λ_{u} t} - e^{λ_{v} t}}{v_{2} + u_{2}} b_{m}^{†} (0) \\ + i \frac{v_{2} \sqrt{2 G_{a}} - \sqrt{2 G}}{v_{2} + u_{2}} e^{λ_{u} t} \int_{0}^{t} d t^{'} a_{in} (t^{'}) e^{- λ_{u} t^{'}} + i \frac{u_{2} \sqrt{2 G_{a}} + \sqrt{2 G}}{v_{2} + u_{2}} e^{λ_{v} t} \int_{0}^{t} d t^{'} a_{in} (t^{'}) e^{- λ_{v} t^{'}} \\ - \frac{\sqrt{2 γ_{a}}}{v_{2} + u_{2}} [u_{2} e^{λ_{v} t} \int_{0}^{t} d t^{'} c_{in} (t^{'}) e^{- λ_{v} t^{'}} + v_{2} e^{λ_{u} t} \int_{0}^{t} d t^{'} c_{in} (t^{'}) e^{- λ_{u} t^{'}}] \\ + \frac{\sqrt{2 γ_{m}}}{v_{2} + u_{2}} [e^{λ_{v} t} \int_{0}^{t} d t^{'} b_{in}^{†} (t^{'}) e^{- λ_{v} t^{'}} - e^{λ_{u} t} \int_{0}^{t} d t^{'} b_{in}^{†} (t^{'}) e^{- λ_{u} t^{'}}], \\ b_{m}^{†} (t) & = & \frac{(e^{λ_{u} t} - e^{λ_{v} t}) u_{2} v_{2}}{v_{2} + u_{2}} c_{a} (0) + \frac{e^{λ_{v} t} v_{2} + e^{λ_{u} t} u_{2}}{v_{2} + u_{2}} b_{m}^{†} (0) \\ + i u_{2} \frac{v_{2} \sqrt{2 G_{a}} - \sqrt{2 G}}{v_{2} + u_{2}} e^{λ_{u} t} \int_{0}^{t} d t^{'} a_{in} (t^{'}) e^{- λ_{u} t^{'}} - i v_{2} \frac{u_{2} \sqrt{2 G_{a}} + \sqrt{2 G}}{v_{2} + u_{2}} e^{λ_{v} t} \int_{0}^{t} d t^{'} a_{in} (t^{'}) e^{- λ_{v} t^{'}} \\ + \frac{u_{2} v_{2} \sqrt{2 γ_{a}}}{v_{2} + u_{2}} [e^{λ_{v} t} \int_{0}^{t} d t^{'} c_{in} (t^{'}) e^{- λ_{v} t^{'}} - e^{λ_{u} t} \int_{0}^{t} d t^{'} c_{in} (t^{'}) e^{- λ_{u} t^{'}}] \\ - \frac{\sqrt{2 γ_{m}}}{v_{2} + u_{2}} [u_{2} e^{λ_{u} t} \int_{0}^{t} d t^{'} b_{in}^{†} (t^{'}) e^{- λ_{u} t^{'}} + v_{2} e^{λ_{v} t} \int_{0}^{t} d t^{'} b_{in}^{†} (t^{'}) e^{- λ_{v} t^{'}}] . \end{array}

Substituting Eq. (7) into Eq. (3), we can get the time-dependent solution of the cavity field a_c(t).

Equations (3) and (7) manifest that the motions of all the three modes get correlated to the input noises a_in, $b_{in}^{†}$ , c_in with exponentially shaped envelopes, indicating that it is convenient to introduce a set of normalized temporal pulse-shape amplitudes. We can introduce a set of normalized temporal modes and determine how each mode at the end of the pulse t = τ is related to the initial modes. All technical details are illustrated in the Appendix.

To explain how to define the temporal modes, we consider a simplified model where we neglect the existence of the mechanical and atomic decoherence effects, i.e., γ_a = γ_m = 0. Note that this approximation is valid when the total duration τ of protocol is much smaller than the effective mechanical and atomic decoherence time 1/γ_mn̄_m and 1/γ_an̄_c. By applying the standard cavity input-output relations, $a_{out} (t) = a_{in} (t) + \sqrt{2 κ} a_{c} (t)$ , we can define a set of normalized temporal light modes (assuming G > G_a) [49, 52, 53],

\begin{array}{l} A_{in} & = & \sqrt{\frac{2 (G - G_{a})}{1 - e^{- 2 (G - G_{a}) τ}}} \int_{0}^{τ} d t a_{in} (t) e^{- (G - G_{a}) t}, \\ A_{out} & = & \sqrt{\frac{2 (G - G_{a})}{e^{2 (G - G_{a}) τ} - 1}} \int_{0}^{τ} d t a_{out} (t) e^{(G - G_{a}) t}, \end{array}

which obey the canonical commutation relations

[A_{i}, A_{i}^{†}] = 1

with i = (in, out). To evaluate the input and output quadratures of the oscillator and atomic ensemble, we define B_in = b_m(0), B_out = b_m(τ), C_in = c_a(0), C_out = c_a(τ). Then, we can determine how the output field of each mode is related to the input fields after an interaction time τ:

\begin{array}{l} A_{out} & = & - e^{r} A_{in} - i \sqrt{e^{2 r} - 1} (α B_{in}^{†} + β C_{in}), \\ B_{out} & = & (α^{2} e^{r} - β^{2}) B_{in} + α β (e^{r} - 1) C_{in}^{†} + i α \sqrt{e^{2 r} - 1} A_{in}^{†}, \\ C_{out} & = & (α^{2} - β^{2} e^{r}) C_{in} - α β (e^{r} - 1) B_{in}^{†} + i β \sqrt{e^{2 r} - 1} A_{in}, \end{array}

where

α = \sqrt{G / (G - G_{a})}

,

β = \sqrt{G_{a} / (G - G_{a})}

are the parameters related to the coupling strength, and r = (G − G_a)τ is the normalized interaction time parameter as an effective squeezing parameter. The general expressions for A_out, B_out, and C_out including mechanical damping γ_m, thermal noise n₀, n̄_m, and relaxation time of atoms γ_a are given in the Appendix.

The simultaneous coupling of all three modes can result in a tripartite quantum correlation [52, 53]. We investigate the possibility to generate different types of quantum correlations (tripartite inseparability, genuine tripartite entanglement, tripartite steerability, and genuine tripartite steering) in this hybrid atomic optomechanics, and analyze the effects of imperfections on the quantum correlation. The type of quantum correlation can be determined by using the reduced noise variance of the quadrature of mirror mode based on the outcomes of measurements performed on the cavity field and atomic modes, compared with different thresholds for each type. By examining properties of the variances of suitable linear combinations of the quadrature operators for the three modes, from the solution for output field operators given by Eq. (9) and the general one with finite γ_m, γ_a and thermal environment noise n̄_m shown in Appendix Eq. (18), we find that the strongest correlations between the mirror mode and cavity field occur in the X − P combination of the corresponding quadratures, whereas the strongest correlations between the mirror and atomic modes occur in the X − X combination of the corresponding quadratures [52, 53]. With these combinations we can test an EPR-type variance parameter

E_{m | ac (g)} = Δ_{inf, ac}^{2} X_{m} + Δ_{inf, ac}^{2} P_{m} = Δ^{2} (X_{m} + g_{a} P_{a} + g_{c} X_{c}) + Δ^{2} (P_{m} + h_{a} X_{a} + h_{c} P_{c}),

which can be minimized by optimizing the gain factors g_a,c and h_a,c. Here, the quadrature operators of the output modes are defined by

X_{j} = (O_{out} + O_{out}^{†}) / \sqrt{2}

and

P_{j} = (O_{out} - O_{out}^{†}) / \sqrt{2 i}

, where j ∈ {a, m, c}, O ∈ {A, B, C}, with the Heisenberg uncertainty relation ΔX_jΔP_j ≥ 1/2. We can similarly define the quadratures for the input temporal modes, and express the quadrature components X_j and P_j of the output modes in terms of those associated with the input modes. We assume that the input modes are in thermal states characterized by the variances

Δ^{2} X_{a}^{in} = Δ^{2} P_{a}^{in} = n_{a_{0}} + 1 / 2

,

Δ^{2} X_{c}^{in} = Δ^{2} P_{c}^{in} = n_{c_{0}} + 1 / 2

, and

Δ^{2} X_{m}^{in} = Δ^{2} P_{m}^{in} = n_{m_{0}} + 1 / 2

, where the variance is defined as Δ²O = 〈O²〉 − 〈O〉². Here, n_a₀, n_c₀, and n_m₀ are the initial occupation numbers of thermal photons or phonons in the cavity field, atomic mode, and mirror, respectively.

3. Different types of tripartite entanglement

First, a tripartite system {m, n, l} is genuinely entangled iff the density operator of the system cannot be represented in the biseparable form $ρ = P_{1} \sum_{i} η_{i}^{(1)} ρ_{m n}^{i} ρ_{l}^{i} + P_{2} \sum_{j} η_{j}^{(2)} ρ_{n l}^{j} ρ_{m}^{j} + P_{3} \sum_{k} η_{k}^{(3)} ρ_{m l}^{k} ρ_{n}^{k}$ , where ∑_j P_j = 1 and $\sum_{i} η_{i}^{(j)} = 1$ . P_j = {P₁, P₂, P₃} are probabilities for the system to be in a state with given bipartition. The bipartition ρ_mnρ_l indicates the subsystems m and n may be entangled or not, but both of them are separated with subsystem l. To show the existence of genuine tripartite entanglement among light-mirror-atom (a − m − c) in the proposed system, we need verify the elimination of all the possible mixtures of biseparable model by violating the following inequality [14, 36]

E_{m | ac (g)} \geq min {1 + | g_{a} h_{a} + g_{c} h_{c} |, | g_{a} h_{a} | + | 1 + g_{c} h_{c} |, | g_{c} h_{c} | + | 1 + g_{a} h_{a} |} .

Second, to demonstrate the steering genuinely shared among three parties m, n, and l, we need to falsify the following description of the statistics based on the hybrid LHS model, where the averages are given as $〈 X_{m} X_{n} X_{l} 〉 = P_{1} \sum_{i} η_{i}^{(1)} 〈 X_{m} X_{n} 〉 {〈 X_{l} 〉}_{ρ} + P_{2} \sum_{j} η_{j}^{(2)} 〈 X_{n} X_{l} 〉 {〈 X_{m} 〉}_{ρ} + P_{3} \sum_{k} η_{k}^{(3)} 〈 X_{m} X_{l} 〉 {〈 X_{n} 〉}_{ρ}$ [36]. The subscript ρ denotes that the average statistics is consistent with that of a quantum density matrix, i.e., 〈X〉_ρ = Tr(ρX), and its uncertainty variances must satisfy the Heisenberg uncertainty relation. Whereas there are no such constraints for products of moments, written without the subscript. Thus, to demonstrate the genuine tripartite steering shared among light-mirror-atom, the following inequality should be violated to eliminate the steering created by mixing states with two-party steering across different bipartitions [14,35,36]

E_{m | ac (g)} \geq min {1, | g_{a} h_{a} |, | g_{c} h_{c} |} .

As the parameter E_m|ac(g) must be smaller than the minimum of three bounds given by the three possible biseparable forms to demonstrate the quantum states that the entanglement or steering genuinely shared among three parties, it is in general difficult to realize in practical experiments. We can, however, still have the other two types of tripartite quantum correlation: tripartite inseparability, which is certified by the violation of the quantum separable state model [5, 14]

E_{m | ac (g)} \geq 1 + | g_{a} h_{a} | + | g_{c} h_{c} |,

and tripartite steering of the mirror mode m by the cavity mode a and atomic mode c, revealed by the violation of LHS model [14, 36]

E_{m | ac (g)} \geq 1 .

If genuine tripartite steering can be observed, the state necessarily acquires the property of tripartite inseparability, genuine tripartite entanglement, and steering of one mode by the other two. Note that all criteria above are sufficient but not necessary to certify the corresponding types of tripartite quantum correlation [58]. The parameter E_m|ac(g) in each inequality can be minimized by different optimal gain factors g_a,c and h_a,c. In other words, one must set different gain factors varying with time in the experiments to achieve different types of tripartite quantum entanglement tested by the violation of inequalities (11–14).

To save the trouble for real-time adjustment of gain factors in practice, we can test the reduced variance parameter with fixed gain factors $g_{a} = g_{c} = 1 / \sqrt{2}$ and $h_{a} = - h_{c} = 1 / \sqrt{2}$ , i.e. change inequality (10) to

E_{m | ac} = Δ^{2} (X_{m} + \frac{P_{a} + X_{c}}{\sqrt{2}}) + Δ^{2} (P_{m} + \frac{X_{a} - P_{c}}{\sqrt{2}}) .

Then the three modes are verified to exhibit tripartite inseparability, genuine tripartite entanglement, tripartite steering of the mirror by the collaboration of the other two modes, and genuine tripartite quantum steering, if the EPR-type tripartite entanglement parameter given by Eq. (15) satisfy E_m|ac < 2, E_m|ac < 1, E_m|ac < 1, and E_m|ac < 0.5, respectively. As a result, one can simply test a single parameter to determine four types of tripartite entanglement of a given state, with different thresholds for each type. Notice that the linear combinations of the quadrature components involved in the variances satisfy the commutation relation

[X_{m} + (P_{a} + X_{c}) / \sqrt{2}, P_{m} + (X_{a} - P_{c}) / \sqrt{2}] = 0

, which indicates the variances of the position and momentum of the mirror mode can be inferred simultaneously with a 100% precision with E_m|ac = 0, and hence demonstrate a perfect EPR paradox for the macroscopic and massive oscillator.

4. The effects of damping and thermal noise

Useful quantum resource must be immune to the dissipation and thermal noise. In this section we will investigate how the tripartite entanglement measured by the parameter shown in Eq. (15) is affected by the initial mechanical thermal noise n_m₀, and the decoherence induced by the reservoirs with decay rates γ_a, γ_m with or without additional mechanical thermal noise n̄_m. We will also show that the influence of the optimal gain factors g_a,c, h_a,c within the tripartite entanglement parameter E_m|ac(g) given in Eq. (10). All those imperfections will decrease the created tripartite entanglement. However, we will see that within a proper parameter regime, all these detrimental effects can be significantly suppressed such that a maximal entanglement can be sustained.

4.1. Sensitivity to the initial thermal noise

We first illustrate the influence of the initial state preparation by considering the sensitivity to the initial thermal noise when the mirror and atomic modes are not coupled to reservoirs, i.e., γ_m = γ_a = 0. The tripartite entanglement parameter can be obtained as given by

\begin{array}{l} E_{m | ac} & = & {[γ \sqrt{e^{2 r} - 1} - \frac{e^{r}}{\sqrt{2}}]}^{2} (2 n_{a_{0}} + 1) + {[α γ (e^{r} - 1) - \frac{α}{\sqrt{2}} \sqrt{e^{2 r} - 1} + 1]}^{2} (2 n_{m_{0}} + 1) \\ + {[β γ (e^{r} - 1) - \frac{β}{\sqrt{2}} \sqrt{e^{2 r} - 1} + \frac{1}{\sqrt{2}}]}^{2} (2 n_{c_{0}} + 1), \end{array}

where

γ = α - β / \sqrt{2}

.

We find that the tripartite entanglement is less sensitive to the initial thermal noise in the steering modes n_a₀ and n_c₀, and the influence is mainly induced by the initial thermal noise n_m₀ in the steered mechanical mode [53]. Hence, we can focus on the reduced tripartite entanglement parameter $E_{m | ac} = 2 {(e^{r} - \sqrt{e^{2 r} - 1})}^{2} (n_{m_{0}} + 1)$ with n_a₀ = n_c₀ = 0. Figure 2 illustrates the behavior of the tripartite entanglement with respect to the relative strength of the coupling constants r = (G − G_a)t and the initial thermal phonon number n_m₀ for a fixed coupling strength $α = \sqrt{2}$ . We see that even when the initial thermal noise is large, the tripartite entanglement parameter can be made negligibly small by increasing the squeezing parameter r. Tripartite inseparability (E_m|ac < 2), genuine tripartite entanglement and the steering of mirror by the cavity field and atomic modes (E_m|ac < 1), genuine tripartite steering (E_m|ac < 0.5) can be observed when $r > ln \frac{2 + n_{m_{0}}}{2 \sqrt{1 + n_{m_{0}}}}$ , $r > ln \frac{3 + 2 n_{m_{0}}}{2 \sqrt{2 + 2 n_{m_{0}}}}$ , and $r > ln \frac{5 + 4 n_{m_{0}}}{4 \sqrt{1 + n_{m_{0}}}}$ , respectively. For n_m₀ ≫ 1, the thresholds of squeezing parameter are reduced as $r > ln \sqrt{n_{m_{0}} / 4}$ , $r < ln \sqrt{n_{m_{0}} / 2}$ , and $r > ln \sqrt{n_{m_{0}}}$ , respectively.

Fig. 2 The effect of initial mechanical thermal noise n_m₀ on the EPR-type tripartite entanglement parameter E_m|ac. Left: the 3D plot of E_m|ac given in Eq. (16) as function of n_m₀ and squeezing parameter r = (G − G_A)τ for fixed coupling strength $α = \sqrt{2}$ ; Right: the contour plot of E_m|ac. The value of E_m|ac smaller than the thresholds 2, 1, and 0.5, reveals the tripartite inseparability, genuine tripartite entanglement and the steering of the mirror by the cavity field and atomic modes, and genuine tripartite steering, respectively. Here, the cavity field and the atomic mode are assumed initially in a vacuum state n_a₀ = n_c₀ = 0.

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4.2. Sensitivity to the decoherence

We then add the relaxation of the atom γ_a and the mechanical damping γ_m in expressions of output modes (see Appendix for the expressions with relaxation times), which usually arise because the system is coupled to a bath, to examine the robustness of the tripartite entanglement against the mechanical and atomic decoherence.

Figure 3 shows that both γ_a and γ_m in general have a destructive effect on the tripartite entanglement even in the absence of thermal noise in the reservoir n̄_m,a,c = 0 (i.e. temperature T = 0). However, it is interesting to see that in the presence of decoherence, different types of tripartite entanglement are preserved at smaller squeezing parameter r rather than at larger r. This can be understood by noticing that a strong squeezing leads to a high sensitivity of the variances to the external noise. Even for very large decay rates γ_a and γ_m, one still finds the observation of genuine tripartite steering E_m|ac < 0.5 and genuine tripartite entanglement E_m|ac < 1 at a relatively low level of squeezing nearby r ∼ 0.5. This indicates that the observation of genuine tripartite entanglement and steering immune to the mechanical damping to the heat bath becomes possible for lower and more accessible values of squeezing parameter r.

Fig. 3 The effect of the relaxation of the atom γ_a when γ_m = 0 (a) and the mechanical damping γ_m when γ_a = 0 (b) on the EPR-type tripartite entanglement parameter E_m|ac. Here, we set the coupling strength $α = \sqrt{2}$ , the initial thermal noise n_{m₀,a₀,c₀} = 0 and the reservoir bath n̄_m,a,c = 0.

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A crucial requirement to quantum information processing is the preservation of multipartite entanglement within a noisy environment. Figure 4 shows the decoherence effect due to the decay of the mirror into a general thermal bath with nonzero excitation number n̄_m. As we expect, the detection of the tripartite entanglement is clearly more challenging. However, one can still finds that in the presence of the thermal noise or damping, the genuine tripartite steering is preserved at small r rather than at large r. This is also because the detection of entanglement is less sensitive to the decoherence for lower value of squeezing parameter.

Fig. 4 Same to Fig. 2 but the mirror in contact with a nonzero temperature T reservoir with mean number of the thermal phonons n̄_m = n_m₀ = n, showing the decoherence effects due to mechanical coupling to a heat bath. Here, we set $α = \sqrt{2}$ , γ_m = 6 × 10⁻⁵κ, and γ_a = 10⁻⁴κ. The cavity field and the atomic modes are in the ordinary zero temperature environment n̄_a,c = n_a₀,c₀ = 0.

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5. The effect of optimal gain factors included in the criteria

We see that the test using the tripartite entanglement parameter E_m|ac with fixed gains $h_{a} = g_{c} = - h_{c} = 1 / \sqrt{2}$ is very sensitive to the mechanical thermal noise and decoherence. Those fixed gains are not the optimal factors to indicate the existence of tripartite quantum correlations. The following results however indicate a more promising prediction of four types of tripartite entanglement if the criteria given in inequalities (11–14) involving an asymmetric choice of amplitude weightings are used.

The sensitivity of the tripartite entanglement to the decoherence decay rate γ_m, γ_a is greatly reduced, as shown in Fig. 5(a). With fixed gains, the reduced variance E_m|ac goes up at larger squeezing parameter r because of a large sensitivity to the decoherence for a strong squeezing, as shown by the blue dashed curve. Therefore, it may lead to missing the test of the existence of tripartite entanglement. However, E_m|ac(g) → 0 (which manifests a perfect EPR paradox for the mirror mode) can be observed over a wide range of large r with optimal gain factors g_a,c, h_a,c (shown in the inset of the figure). This makes the detection of tripartite entanglement becomes more feasible.

To confirm the presence of the four types tripartite entanglement, we need to compare the value of E_m|ac(g) (black solid) with four different thresholds for each type with respect to the gain factors, to get falsification of inequalities (11–14). Notice from Fig. 5(b) that with optimal choice of gains, we can always produce inseparable tripartite entangled state (evidenced by the black curve is always lower than the red solid curve), genuine tripartite entangled state (confirmed by the black curve is lower than the cyan solid curve), the tripartite steering of the mirror mode by the measurements on the cavity field and atomic modes (tested by the blue dotted bound) in the entire range of r > 0. The genuine tripartite steering (lower than the yellow dashed bound) can be also observed for a relatively relaxed range of r compared with that showing E_m|ac < 0.5 (blue dashed curve) in Fig. 5(a).

Fig. 5 (a) The effect of gains in inequalities (11–14) on the tripartite entanglement parameter E_m|ac(g) (black solid) comparing with the parameter E_m|ac shown in Fig. 3 when n = 0 (blue dashed). The inset shows the optimal gains that can minimize the reduced noise variance of the mirror E_m|ac(g), in terms of the measurements performed on the cavity field and atomic modes. (b) the tripartite entanglement parameter E_m|ac(g) (black solid) versus four different thresholds determined by the corresponding gains for each type of tripartite entanglement. Here, we set $α = \sqrt{2}$ , γ_m = 6 × 10⁻⁵κ, and γ_a = 10⁻⁴κ, n̄_a,m,c = n_{a₀,m₀,c₀} = 0.

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According to different experimental systems, we could choose different gain factors to achieve the best genuine tripartite entangled and steerable states. But considering the implementation of experiments, optimizing all gain factors varying with time brings significant complication. In the scheme discussed here, choosing a combination of $g_{a} = h_{a} = 1 / \sqrt{2}$ and $g_{c} = - h_{c} = 1 / \sqrt{2}$ , one can simply measure a single parameter E_m|ac to distinguish which kind of tripartite quantum correlations created in the experiment, with different bounds 2, 1, 0.5 for each type. In fact, the optimal gain factors to minimize E_m|ac(g) are close to $\pm 1 / \sqrt{2}$ , as shown in the inset of Fig. 5(a), so that setting $g_{a} = h_{a} = 1 / \sqrt{2}$ and $g_{c} = - h_{c} = 1 / \sqrt{2}$ is a good choice to get a smaller value of E_m|ac. We can see that from the blue curve in Fig. 5(a), for lower, but more accessible values of squeezing parameter r, one can observe all four types tripartite quantum entanglement and determine the type of tripartite entangled states by simply testing a unified parameter E_m|ac, with different thresholds for each type.

6. Conclusion

In summary, we demonstrate a scheme for the generation of tripartite entangled states showing tripartite inseparability, genuine tripartite entanglement, steering of the mirror mode by the cavity field and atomic modes, and genuine tripartite steering in a hybrid atomic optomechanical system. Using the linearization approach, we derive analytical expressions for the input-output relations between the quadratures of the three modes in the presence of mechanical damping, the relaxation of atoms, and thermal noises. We can determine the type of the tripartite entanglement in terms of reduced noise variance of the mirror mode by the measurements performed on the other two modes, with different thresholds for each type. This allow us to distinguish various types of tripartite quantum correlations of a given state in a way that is easily measured in experiment. We also analyze the robustness of tripartite entanglement against different kinds of imperfect factors. The realization of the proposed scheme may be helpful for the study of quantum interfaces and quantum memories [59].

Appendix: the solutions for the output modes with decoherence

We now introduce a set of normalized temporal atomic output and input modes

\begin{array}{l} A_{in}^{1} & = & Γ_{1} \int_{0}^{τ} d t a_{in} (t) e^{- λ_{u} t}, A_{in}^{2} = Γ_{2} \int_{0}^{τ} d t a_{in} (t) e^{- λ_{v} t}, A_{in}^{3} = Γ_{3} \int_{0}^{τ} d t a_{in} (t) e^{λ_{u} t}, \\ B_{m}^{1} & = & Γ_{1} \int_{0}^{τ} d t b_{in} (t) e^{- λ_{u} t}, B_{m}^{2} = Γ_{2} \int_{0}^{τ} d t b_{in} (t) e^{- λ_{v} t}, B_{m}^{3} = Γ_{3} \int_{0}^{τ} d t b_{in} (t) e^{λ_{u} t}, \\ C^{1} & = & Γ_{1} \int_{0}^{τ} d t c_{in} (t) e^{- λ_{u} t}, C^{2} = Γ_{2} \int_{0}^{τ} d t c_{in} (t) e^{- λ_{v} t}, C^{3} = Γ_{3} \int_{0}^{τ} d t c_{in} (t) e^{λ_{u} t}, \end{array}

where

Γ_{1} = \sqrt{2 λ_{u} / (1 - e^{- 2 λ_{u} τ})}

,

Γ_{2} = \sqrt{2 λ_{v} / (1 - e^{- 2 λ_{v} τ})}

,

Γ_{3} = \sqrt{2 λ_{u} / (e^{2 λ_{u} τ} - 1)}

.

To evaluate the input and output quadratures of the oscillator and atomic ensemble, we define B_in = b_m(0), B_out = b_m(τ), C_in = c_a(0), C_out = c_a(τ). Then, we can determine how the output field of each mode is related to the input fields after an interaction time τ:

\begin{array}{l} C_{out} & = & \frac{e^{λ_{u} τ} v_{2} + e^{λ_{v} τ} u_{2}}{v_{2} + u_{2}} C_{in} + \frac{e^{λ_{u} τ} - e^{λ_{v} τ}}{v_{2} + u_{2}} B_{in}^{†} + i \frac{v_{2} \sqrt{2 G_{a}} - \sqrt{2 G}}{Γ_{3} (v_{2} + u_{2})} A_{in}^{1} \\ + i \frac{e^{λ_{v} τ} (u_{2} \sqrt{2 G_{a}} + \sqrt{2 G})}{Γ_{2} (v_{2} + u_{2})} A_{in}^{2} - \frac{1}{Γ_{3} (v_{2} + u_{2})} (v_{2} \sqrt{2 γ_{a}} C^{1} + \sqrt{2 γ_{m}} B_{m}^{1 †}) \\ + \frac{e^{λ_{v} τ}}{Γ_{2} (v_{2} + u_{2})} (\sqrt{2 γ_{n}} B_{m}^{2 †} - u_{2} \sqrt{2 γ_{a}} C^{2}), \\ B_{out} & = & \frac{(e^{λ_{u} τ} - e^{λ_{v} τ}) u_{2} v_{2}}{v_{2} + u_{2}} C_{in}^{†} + \frac{e^{λ_{v} τ} v_{2} + e^{λ_{v} τ} u_{2}}{v_{2} + u_{2}} B_{in} - i \frac{u_{2} v_{2} \sqrt{2 G_{a}} - u_{2} \sqrt{2 G}}{Γ_{3} (v_{2} + u_{2})} A_{in}^{1 †} \\ + i \frac{e^{λ_{v} τ} (v_{2} u_{2} \sqrt{2 G_{a}} + v_{2} \sqrt{2 G})}{Γ_{2} (v_{2} + u_{2})} A_{in}^{2 †} - \frac{u_{2}}{Γ_{3} (v_{2} + u_{2})} (v_{2} \sqrt{2 γ_{a}} C^{1 †} + \sqrt{2 γ_{m}} B_{m}^{1}) \\ + \frac{v_{2} e^{λ_{v} τ}}{Γ_{2} (v_{2} + u_{2})} (u_{2} \sqrt{2 γ_{a}} C^{2 †} - \sqrt{2 γ_{m}} B_{m}^{2}), \\ A_{out} & = & \frac{1}{u_{2} + v_{2}} (\frac{i \sqrt{2} f_{1} v_{2}}{Γ_{3}} + i \sqrt{2} f_{2} u_{2} Γ_{3} Γ_{5}) C_{in} + \frac{1}{u_{2} + v_{2}} (\frac{i \sqrt{2} f_{1}}{Γ_{3}} - i \sqrt{2} f_{2} Γ_{3} Γ_{5}) B_{in}^{†} \\ + \frac{e^{λ_{u} τ}}{λ_{u} (u_{2} + v_{2})} (h_{1} A_{in}^{1} - i f_{1} v_{2} \sqrt{γ_{a}} C^{1} - i f_{1} \sqrt{γ_{m}} B_{m}^{1 †}) \\ + \frac{1}{λ_{u} (u_{2} + v_{2})} (- h_{1} A_{in}^{3} + i f_{1} v_{2} \sqrt{γ_{a}} C^{3} + i f_{1} \sqrt{γ_{m}} B_{m}^{3 †}) \\ + \frac{2 e^{λ_{u} τ} Γ_{4}}{(λ_{u} + λ_{v}) (u_{2} + v_{2})} (- h_{2} A_{in}^{2} - i f_{2} u_{2} \sqrt{γ_{a}} C^{2} + i f_{2} \sqrt{γ_{m}} B_{m}^{2 †}) \\ + \frac{2}{(λ_{u} + λ_{v}) (u_{2} + v_{2})} (h_{2} A_{in}^{3} - i f_{2} u_{2} \sqrt{γ a} C^{3} - i f_{2} \sqrt{γ_{m}} B_{m}^{3 †}) - A_{in}^{3}, \end{array}

where

Γ_{4} = \sqrt{\frac{λ_{u} (e^{2 λ_{v} τ} - 1)}{λ_{v} (e^{2 λ_{u} τ} - 1)}}

,

Γ_{5} = \frac{e^{(λ_{u} + λ_{v}) τ} - 1}{λ_{u} + λ_{v}}

,

f_{1} = - \sqrt{G} u_{2} - \sqrt{G_{a}}

,

f_{2} = \sqrt{G} v_{2} - \sqrt{G_{a}}

,

h_{1} = G_{a} v_{2} - G u_{2} - \sqrt{{GG}_{a}} (1 - v_{2} u_{2})

,

h_{2} = - G_{a} u_{2} + G v_{2} - \sqrt{{GG}_{a}} (1 - v_{2} u_{2})

.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) under Grants No. 11274025 and No. 61475006. We thank fruitful discussions with Z. Ficek and M. D. Reid on the properties of multipartite steering. Q.Y.H. also thanks Prof. C. P. Sun at Beijing Computational Science Research Center for hospitality.

References and links

1. M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999). [CrossRef]

2. R. Cleve, D. Gottesman, and H.-K. Lo, “How to share a quantum secret,” Phys. Rev. Lett. 83, 648–651 (1999). [CrossRef]

3. P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: A quantum teleportation network,” Phys. Rev. Lett. 84, 3482–3485 (2000). [CrossRef] [PubMed]

4. H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continuous variables,” Nature 431, 430–433 (2004). [CrossRef] [PubMed]

5. P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003). [CrossRef]

6. R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, “Operational framework for nonlocality,” Phys. Rev. Lett. 109, 070401 (2012). [CrossRef] [PubMed]

7. J. D. Bancal, N. Gisin, Y. C. Liang, and S. Pironio, “Device-independent witnesses of genuine multipartite entanglement,” Phys. Rev. Lett. 106, 250404 (2011). [CrossRef] [PubMed]

8. S. M. Giampaolo and B. C. Hiesmayr, “Genuine multipartite entanglement in the XY model,” Phys. Rev. A 88, 052305 (2013). [CrossRef]

9. J. D. Bancal, J. Barrett, N. Gisin, and S. Pironio, “Definitions of multipartite nonlocality,” Phys. Rev. A 88, 014102 (2013). [CrossRef]

10. F. Levi and F. Mintert, “Hierarchies of multipartite entanglement,” Phys. Rev. Lett. 110, 150402 (2013). [CrossRef] [PubMed]

11. D. L. Deng, Z. S. Zhou, and J. L. Chen, “Svetlichny’s approach to detecting genuine multipartite entanglement in arbitrarily-high-dimensional systems by a Bell-type inequality,” Phys. Rev. A 80, 022109 (2009). [CrossRef]

12. M. Huber, H. Schimpf, A. Gabriel, C. Spengler, D. Brup, and B. C. Hiesmayr, “Experimentally implementable criteria revealing substructures of genuine multipartite entanglement,” Phys. Rev. A 83, 022328 (2011). [CrossRef]

13. L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2013). [CrossRef]

14. S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. - A. Bachor, M D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015). [CrossRef]

15. R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989). [CrossRef] [PubMed]

16. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007). [CrossRef] [PubMed]

17. S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007). [CrossRef]

18. E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935). [CrossRef]

19. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935). [CrossRef]

20. V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012). [CrossRef]

21. Q. Y. He, Q. H. Gong, and M. D. Reid, “Classifying directional Gaussian entanglement, Einstein-Podolsky-Rosen steering, and discord,” Phys. Rev. Lett. 114, 060402 (2015). [CrossRef] [PubMed]

22. I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015). [CrossRef] [PubMed]

23. M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989). [CrossRef] [PubMed]

24. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. A 80, 032112 (2009). [CrossRef]

25. J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964).

26. B. Hage, A. Samblowski, and R. Schnabel, “Towards Einstein-Podolsky-Rosen quantum channel multiplexing,” Phys. Rev. A 81, 062301 (2010). [CrossRef]

27. D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845–849 (2010). [CrossRef]

28. D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012). [CrossRef] [PubMed]

29. A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein-Podolsky-Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).

30. B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. M. Wiseman, R. Ursin, and A. Zeilinger, “Loophole-free Einstein–Podolsky–Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012). [CrossRef]

31. S. Steinlechner, J. Bauchrowitz, T. Eberle, and R. Schnabel, “Strong Einstein-Podolsky-Rosen steering with unconditional entangled states,” Phys. Rev. A 87, 022104 (2013). [CrossRef]

32. K. Sun, J. S. Xu, X. J. Ye, Y. C. Wu, J. L. Chen, C. F. Li, and G. C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014). [CrossRef] [PubMed]

33. C. M. Li, K. Chen, Y. N. Chen, Q. Zhang, Y. A. Chen, and J. W. Pan, “Genuine high-order Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 115, 010402 (2015). [CrossRef] [PubMed]

34. C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301(R) (2012). [CrossRef]

35. Q. Y. He and M. D. Reid, “Genuine multipartite Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 111, 250403 (2013). [CrossRef]

36. R. Y. Teh and M. D. Reid, “Criteria for genuine N-partite continuous-variable entanglement and Einstein-Podolsky-Rosen steering,” Phys. Rev. A 90, 062337 (2014). [CrossRef]

37. D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P. H. Souto Ribeiro, and S. P. Walborn, “Detection of entanglement in asymmetric quantum networks and multipartite quantum steering,” Nat. Commun. 6, 7941 (2015). [CrossRef] [PubMed]

38. M. Wang, Q. H. Gong, and Q.Y. He, “Collective multipartite Einstein-Podolsky-Rosen steering: more secure optical networks,” Opt. Lett. 39, 6703–6706 (2014). [CrossRef] [PubMed]

39. M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015). [CrossRef] [PubMed]

40. A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010). [CrossRef]

41. J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011). [CrossRef] [PubMed]

42. J. D. Teufel, T. Donner, D. L Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011). [CrossRef] [PubMed]

43. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008). [CrossRef] [PubMed]

44. T. P. Purdy, D. W. C. Brooks, T. Botter, N. Brahms, Z.-Y. Ma, and D. M. Stamper-Kurn, “Tunable cavity optomechanics with ultracold atoms,” Phys. Rev. Lett. 105, 133602 (2010). [CrossRef]

45. R. Kanamoto and P. Meystre, “Optomechanics of a quantum-degenerate Fermi gas,” Phys. Rev. Lett. 104, 063601 (2010). [CrossRef]

46. S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012). [CrossRef]

47. A. Carmele, B. Vogell, K. Stannigel, and P. Zoller, “Opto-nanomechanics strongly coupled to a Rydberg super-atom: coherent versus incoherent dynamics,” New J. Phys. 16, 063042 (2014). [CrossRef]

48. C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307(R) (2008). [CrossRef]

49. S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011). [CrossRef]

50. L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A 85, 022327 (2012). [CrossRef]

51. L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013). [CrossRef] [PubMed]

52. Q. Y. He and Z. Ficek, “Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system,” Phys. Rev. A 89, 022332 (2014). [CrossRef]

53. M. Wang, Q. H. Gong, Z. Ficek, and Q.Y. He, “Role of thermal noise in tripartite quantum steering,” Phys. Rev. A 90, 023801 (2014). [CrossRef]

54. Y. Yan, G. X. Li, and Q. L. Wu, “Entanglement and Einstein-Podolsky-Rosen steering between a nanomechanical resonator and a cavity coupled with two quantum dots,” Opt. Express 23, 21306–21322 (2015). [CrossRef] [PubMed]

55. Y. D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91, 013807 (2015). [CrossRef]

56. Q. Lin and B. He, “Optomechanical entanglement under pulse drive,” Opt. Express 23, 24497–24507 (2015). [CrossRef] [PubMed]

57. T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013). [CrossRef] [PubMed]

58. For two-mode Gaussian states, the condition E_A|B(g) < 1 is necessary and sufficient to confirm steering of A by B [16, 17, 24], and the minimum value of E_A|B(g) can be used to quantify the Gaussian EPR steering [22].

59. K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010). [CrossRef]

Detection of genuine tripartite entanglement and steering in hybrid optomechanics

Abstract

1. Introduction

2. Dynamics of the system

3. Different types of tripartite entanglement

4. The effects of damping and thermal noise

4.1. Sensitivity to the initial thermal noise

4.2. Sensitivity to the decoherence

5. The effect of optimal gain factors included in the criteria

6. Conclusion

Appendix: the solutions for the output modes with decoherence

Acknowledgments

References and links

Cited By

Figures (5)

Equations (18)

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