1. Introduction
The superposition principle represents one of the most fundamental differences between quantum mechanics and classical physics. It allows a quantum system to be in a superposition of two distinct states, which may exhibit features the two superimposed components do not possess, as a result of quantum interference between these components. Among various kinds of superposition states, Schrödinger cat states are of particular interest [1]. In quantum optics, the cat state is usually defined as a superposition of two coherent states with different phases or amplitudes of a harmonic oscillator. Although formed by states with classical analogs, cat states can show nontrivial quantum behaviors, e.g. negative quasiprobability distribution in certain regimes of phase space [2]. Due to these features, cat states are considered as an ideal candidate for exploration of the quantum-to-classical transition, arising from the coupling between the quantum system and the environment, which leads to the loss of the quantum coherence between the superposed components-decoherence [3]. The decoherence process of cat states has been investigated both theoretically and experimentally in cavity QED [2, 4] and ion traps [5].
The superposition of two pairs of quasiclassical states for two oscillators, referred to as the two-mode cat state [6] or entangled coherent state [7, 8], shows more striking nonclassical effects. The quantum interference between these superimposed two-mode components leads to oscillations of the joint quasiprobability distribution in four-dimensional phase space [9–11]. More importantly, due to the entanglement between the quasiclassical states, the features of these oscillators exhibiting correlations stronger than any classical model allows. Consequently, a combination of these correlations may violate Bell inequalities that are obeyed by local hidden variable theories [12–14], as has been shown both theoretically [9, 15, 16] and experimentally [10]. Because of the coupling with the environment, these correlations are lost quickly; the larger the system size, the higher the decaying rate. Investigation of the decoherence-induced evolution of the Bell signal for these entangled states is important for understanding the transition from quantum nonlocality to classical locality, and has relevance to quantum communication [17–20] and quantum computation [21–23] protocols with such states as quantum nonlocality is responsible for the advantage of quantum information processors over their classical counterparts [24]. The decoherence process for a photonic cat state delocalized in two cavities was studied in [9], but where the Bell signal, based on the displaced parities correlations−values of the joint Wigner function at different points in phase space [14], does not well manifest the quantum nonlocality; it drops below the classical upper bound 2 [25] before the disappearance of the entanglement. This is due to the fact that each of the displaced parities correlations for the classical mixture of two pairs of quasiorthogonal coherent states is no more than 0.5 so that the maximized Bell signal is 1. Violation of the Bell inequality requires considerable amount of entanglement between these coherent states to be remained.
In this paper we study the decoherence-induced nonlocality evolution for two-mode cat states with a new formalism based on the combination of the quantum-number measurement, cat-state-encoded qubit rotation, and phase-space displacement. With our formalism, the Bell inequality is violated as long as there remains entanglement between the two oscillators and the distance between the coherence is sufficiently large. Therefore, the Bell signal evolution well characterizes the decoherence process of the two-mode nonlocal correlations.
In section 2 , we construct a new kind of operators combined by a cat-encoded qubit rotation, phase-space displacement, and the quantum on–off measurement operator. The Bell signal, based on the correlations associated with such operators, exceeds the classical bound when there remains quantum coherence between the two pairs of quasiorthogonal coherent states of two oscillators. In section 3 , we study the decoherence-induced decaying process of the optimized Bell signal for a two-mode cat state by numerical simulation, showing that the extent of the Bell inequality violation well characterizes the amount of entanglement. We also compare our results with those based on previous formalisms.
2. Bell inequality based on rotated and displaced on–off correlations
The nonlocal state of two mesoscopic fields under consideration can be expressed as
$\begin{eqnarray}\left|{\psi }_{\pm }\right\rangle ={ \mathcal N }\left({\left|{\alpha }_{1}\right\rangle }_{1}{\left|{\alpha }_{2}\right\rangle }_{2}+{\left|-{\alpha }_{1}\right\rangle }_{1}{\left|-{\alpha }_{2}\right\rangle }_{2}\right),\end{eqnarray}$
where ${\left|\pm {\alpha }_{1}\right\rangle }_{1}{\left|\pm {\alpha }_{2}\right\rangle }_{2}$ denote the products of coherent states of the two fields stored in different cavities, whose amplitudes are ±α1 and ±α2, respectively, and ${ \mathcal N }={\left[2+2{{\rm{e}}}^{-2\left({\left|{\alpha }_{1}\right|}^{2}+{\left|{\alpha }_{2}\right|}^{2}\right)}\right]}^{-1/2}$ is the normalization factor, which approximates $1/\sqrt{2}$ when ${\left|{\alpha }_{1}\right|}^{2}+{\left|{\alpha }_{2}\right|}^{2}\gg 1$. The environmentally-induced decoherence gradually deteriorates the quantum coherence of this entangled coherent state and the nonlocal correlation between the two mesoscopic fields. After a time T, the field density operator is given by [6] $\begin{eqnarray}\begin{array}{c}{\rho }_{c}={{ \mathcal N }}^{2}[| {\alpha }_{1}^{{\prime} }{\rangle }_{1}\langle {\alpha }_{1}^{{\prime} }| \otimes | {\alpha }_{2}^{{\prime} }{\rangle }_{2}\langle {\alpha }_{2}^{{\prime} }| +| -{\alpha }_{1}^{{\prime} }{\rangle }_{1}\\ \quad \langle -{\alpha }_{1}^{{\prime} }| \otimes | -{\alpha }_{2}^{{\prime} }{\rangle }_{2}\langle -{\alpha }_{2}^{{\prime} }| \\ \quad +\,R(| {\alpha }_{1}^{{\prime} }{\rangle }_{1}\langle -{\alpha }_{1}^{{\prime} }| \otimes | {\alpha }_{2}^{{\prime} }{\rangle }_{2}\langle -{\alpha }_{2}^{{\prime} }| +| -{\alpha }_{1}^{{\prime} }{\rangle }_{1}\\ \quad \langle {\alpha }_{1}^{{\prime} }| \otimes | -{\alpha }_{2}^{{\prime} }{\rangle }_{2}\langle {\alpha }_{2}^{{\prime} }| )],\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{ccc}R & = & {{\rm{e}}}^{-2| {\alpha }_{1}{| }^{2}(1-{{\rm{e}}}^{-{\gamma }_{1}T})-2| {\alpha }_{2}{| }^{2}(1-{{\rm{e}}}^{-{\gamma }_{2}T})},\\ & & \pm {\alpha }_{j}^{{\prime} }=\pm {\alpha }_{j}{{\rm{e}}}^{-{\gamma }_{j}T/2},\end{array}\end{eqnarray}$
with γj being the decay rate of the jth cavity mode. To characterize the decoherence-induced nonlocality evolution, we introduce the following selective number phase gate $\begin{eqnarray}{O}_{j}({\phi }_{j})=\displaystyle \sum _{n\,=\,0}^{\infty }\left({\left|2n\right\rangle }_{j}\left\langle 2n\right|+{{\rm{e}}}^{{\rm{i}}{\phi }_{j}}{\left|2n+1\right\rangle }_{j}\left\langle 2n+1\right|\right),\end{eqnarray}$
which produces a phase φj conditional on the photon-number of jth field mode being odd. This gate can be realized by dispersively coupling the field mode with an ancilla qubit that is driven by two subsequent pulses involving multiple frequency components [26–29]. When ${\left|2{\alpha }_{j}^{^{\prime} }\right|}^{2}\gg 0$, the two coherent states ${\left|{\alpha }_{j}^{{\prime} }\right\rangle }_{j}$ and ${\left|-{\alpha }_{j}^{^{\prime} }\right\rangle }_{j}$ are quasiorthogonal and can be considered as the basis states ${\left|{0}_{L}\right\rangle }_{j}$ and ${\left|{1}_{L}\right\rangle }_{j}$ for a logic qubit. In the basis $\left\{{\left|{\alpha }_{j}^{^{\prime} }\right\rangle }_{j},{\left|-{\alpha }_{j}^{^{\prime} }\right\rangle }_{j}\right\}$, the operation Oj(φj) produces a rotation around x axis by an angle of φj on the Bloch sphere, i.e. $\begin{eqnarray}\begin{array}{ccc}{O}_{j}({\phi }_{j}){\left|{\alpha }_{j}^{{\prime} }\right\rangle }_{j} & = & \cos \frac{{\phi }_{j}}{2}{\left|{\alpha }_{j}^{{\prime} }\right\rangle }_{j}-\mathrm{isin}\frac{{\phi }_{j}}{2}{\left|-{\alpha }_{j}^{^{\prime} }\right\rangle }_{j},\\ {O}_{j}({\phi }_{j}){\left|-{\alpha }_{j}^{^{\prime} }\right\rangle }_{j} & = & \cos \frac{{\phi }_{j}}{2}{\left|-{\alpha }_{j}^{^{\prime} }\right\rangle }_{j}-\mathrm{isin}\frac{{\phi }_{j}}{2}{\left|{\alpha }_{j}^{{\prime} }\right\rangle }_{j}.\end{array}\end{eqnarray}$
We here have discarded the trivial global phase factor ${{\rm{e}}}^{{\rm{i}}{\phi }_{j}/2}$. This corresponds to $\begin{eqnarray}\begin{array}{ccc}{O}_{j}({\phi }_{j}){\left|{+}_{{\phi }_{j}}\right\rangle }_{j} & = & {\left|{\alpha }_{j}^{{\prime} }\right\rangle }_{j},\\ {O}_{j}({\phi }_{j}){\left|{-}_{{\phi }_{j}}\right\rangle }_{j} & = & {\left|-{\alpha }_{j}^{^{\prime} }\right\rangle }_{j},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{ccc}{\left|{+}_{{\phi }_{j}}\right\rangle }_{j} & = & \cos \frac{{\phi }_{j}}{2}{\left|{\alpha }_{j}^{{\prime} }\right\rangle }_{j}+\mathrm{isin}\frac{{\phi }_{j}}{2}{\left|-{\alpha }_{j}^{^{\prime} }\right\rangle }_{j},\\ {\left|{-}_{{\phi }_{j}}\right\rangle }_{j} & = & \cos \frac{{\phi }_{j}}{2}{\left|-{\alpha }_{j}^{^{\prime} }\right\rangle }_{j}+\mathrm{isin}\frac{{\phi }_{j}}{2}{\left|{\alpha }_{j}^{{\prime} }\right\rangle }_{j}.\end{array}\end{eqnarray}$
The combination of this rotation and the displacement ${D}_{j}({\alpha }_{j}^{{\prime} })={{\rm{e}}}^{{\alpha }_{j}^{^{\prime} }{a}_{j}^{\dagger }-{\alpha }_{j}^{{\prime} * }{a}_{j}}$, with ${a}_{j}^{\dagger }$ and aj being the quantum-number rising and lowering operators, leads to $\begin{eqnarray}\begin{array}{ccc}{D}_{j}({\alpha }_{j}^{{\prime} }){O}_{j}({\phi }_{j}){\left|{+}_{{\phi }_{j}}\right\rangle }_{j} & = & {\left|2{\alpha }_{j}^{^{\prime} }\right\rangle }_{j},\\ {D}_{j}({\alpha }_{j}^{{\prime} }){O}_{j}({\phi }_{j}){\left|{-}_{{\phi }_{j}}\right\rangle }_{j} & = & {\left|0\right\rangle }_{j}.\end{array}\end{eqnarray}$
When ${\left|2{\alpha }_{j}^{^{\prime} }\right|}^{2}\gg 0$, ${\left|2{\alpha }_{j}^{{}^{{\prime} }}\right\rangle }_{j}$ and ${\left|0\right\rangle }_{j}$ are quasiorthogonal and thus can be distinguished by the quantum number on–off measurement, with the corresponding operator $\begin{eqnarray}{M}_{j}=\displaystyle \sum _{n\,=\,1}^{\infty }{\left|n\right\rangle }_{j}\left\langle n\right|-{\left|0\right\rangle }_{j}\left\langle 0\right|.\end{eqnarray}$
The outcomes of the measurement of Mj are 1 and −1 for the states ${\left|2{\alpha }_{j}^{{}^{{\prime} }}\right\rangle }_{j}$ and ${\left|0\right\rangle }_{j},$ respectively. This measurement can be realized by a π/2-rotation of the ancilla qubit conditional on the vacuum state ${\left|0\right\rangle }_{j}$ of the corresponding cavity [30, 31]. Therefore, after the rotation Oj(φj) and displacement ${D}_{j}({\alpha }_{j}^{{\prime} })$, the on–off measurement Mj is approximately equivalent to the measurement of the effective spin of the cat state qubit along the axis with an angle of φj to the z axis on the yz plane, i.e. $\begin{eqnarray}{\sigma }_{j,{yz}}({\phi }_{j})\simeq {\left|{+}_{{\phi }_{j}}\right\rangle }_{j}\left\langle {+}_{{\phi }_{j}}\right|-{\left|{-}_{{\phi }_{j}}\right\rangle }_{j}\left\langle {-}_{{\phi }_{j}}\right|,\end{eqnarray}$
where $\begin{eqnarray}{\sigma }_{j,{yz}}({\phi }_{j})={O}_{j}^{\dagger }({\phi }_{j}){D}_{j}^{\dagger }({\alpha }_{j}^{{\prime} }){M}_{j}{D}_{j}({\alpha }_{j}^{{\prime} }){O}_{j}({\phi }_{j}).\end{eqnarray}$
To investigate the evolution of the nonlocality, we use the correlation between ${\sigma }_{1,{\phi }_{1}}$ and ${\sigma }_{2,{\phi }_{2}}$, defined as2 ), this correlation is14 ) is due to quantum entanglement between the coherent states of the two bosonic modes. The Bell inequality, based on correlations defined in this way, is
$\begin{eqnarray}C({\phi }_{1},{\phi }_{2})=\left\langle {\sigma }_{1,{yz}}({\phi }_{1})\otimes {\sigma }_{2,{yz}}({\phi }_{2})\right\rangle .\end{eqnarray}$
For the state of equation ( $\begin{eqnarray}\begin{array}{c}C({\phi }_{1},{\phi }_{2})={{ \mathcal N }}^{2}\left\{\left(\cos {\phi }_{1}-2{{\rm{e}}}^{-4{\left|{\alpha }_{1}^{^{\prime} }\right|}^{2}}{\cos }^{2}\frac{{\phi }_{1}}{2}\right)\right.\\ \quad \times \,\left(\cos {\phi }_{2}-2{{\rm{e}}}^{-4{\left|{\alpha }_{2}^{^{\prime} }\right|}^{2}}{\cos }^{2}\frac{{\phi }_{2}}{2}\right)\\ \quad +\,\left(\cos {\phi }_{1}+2{{\rm{e}}}^{-4{\left|{\alpha }_{1}^{^{\prime} }\right|}^{2}}{\sin }^{2}\frac{{\phi }_{1}}{2}\right)\\ \quad \times \,\left(\cos {\phi }_{2}+2{{\rm{e}}}^{-4{\left|{\alpha }_{2}^{^{\prime} }\right|}^{2}}{\sin }^{2}\frac{{\phi }_{2}}{2}\right)\\ \quad +\,2R\left[{{\rm{e}}}^{-2\left({\left|{\alpha }_{1}^{^{\prime} }\right|}^{2}+{\left|{\alpha }_{2}^{^{\prime} }\right|}^{2}\right)}-\sin {\phi }_{1}\sin {\phi }_{2}\right.\\ \times \,\left.\left.\left(1-{{\rm{e}}}^{-4{\left|{\alpha }_{1}^{^{\prime} }\right|}^{2}}\right)\left(1-{{\rm{e}}}^{-4{\left|{\alpha }_{2}^{^{\prime} }\right|}^{2}}\right)\right]\right\}.\end{array}\end{eqnarray}$
The first two terms of C(φ1, φ2) are respectively contributed by the components ${\left|-\alpha \right\rangle }_{1}{\left|-\alpha \right\rangle }_{2}$ and ${\left|\alpha \right\rangle }_{1}{\left|\alpha \right\rangle }_{2}$, while the terms inside the brackets arise from their quantum coherence, which is responsible for the entanglement between the two bosonic modes. When $4{\left|{\alpha }_{1}^{{\prime} }\right|}^{2},4{\left|{\alpha }_{2}^{{\prime} }\right|}^{2}\gg 1$, C(φ1, φ2) approximates $\begin{eqnarray}C({\phi }_{1},{\phi }_{2})\simeq \cos {\phi }_{1}\cos {\phi }_{2}-R\sin {\phi }_{1}\sin {\phi }_{2}.\end{eqnarray}$
We note that, with the encoding $\left\{{\left|{0}_{L}\right\rangle }_{j}={\left|{\alpha }_{j}^{^{\prime} }\right\rangle }_{j},{\left|{1}_{L}\right\rangle }_{j}={\left|-{\alpha }_{j}^{^{\prime} }\right\rangle }_{j}\right\}$, the concurrence of the two coherent-state-encoded logic qubits is R, which implies that the second term of equation ( $\begin{eqnarray}\begin{array}{c}{{ \mathcal B }}_{\mathrm{ODM}}=\left|C({\phi }_{1},{\phi }_{2})+C({\phi }_{1},{\phi }_{2}^{^{\prime} })\right.\\ \quad +\,\left.C({\phi }_{1}^{^{\prime} },{\phi }_{2})-C({\phi }_{1}^{^{\prime} },{\phi }_{2}^{^{\prime} })\right|\leqslant 2.\end{array}\end{eqnarray}$
For a classical mixture of ${\left|{\alpha }_{1}^{{\prime} }\right\rangle }_{1}$ ${\left|{\alpha }_{2}^{{\prime} }\right\rangle }_{2}$ and ${\left|-{\alpha }_{1}^{^{\prime} }\right\rangle }_{1}{\left|-{\alpha }_{2}^{^{\prime} }\right\rangle }_{2}$ (R = 0), C(φ1, φ2) can approach 1 and the Bell signal $\left|{{ \mathcal B }}_{\mathrm{ODM}}\right|$ can approximate to 2. This indicates that, for R > 0, $\left|{{ \mathcal B }}_{\mathrm{ODM}}\right|$ can exceed the classical upper bound by optimizing the parameters φ1 , ${\phi }_{1}^{{\prime} }$, φ2, and ${\phi }_{2}^{{\prime} }$ as the bosonic modes exhibit stronger correlations when there is quantum entanglement between them.3. Numerical simulations of Bell signals
The quantum nonlocality is characterized by the violation of the Bell inequality. To maximize the Bell signal, we set2 ) at each time. With these optimized parameters, we perform numerical simulations of the Bell signal evolution. The dash–dot black line of figure 1 shows the maximized Bell signal ${{ \mathcal B }}_{\mathrm{ODM}}$ calculated with the present formalism as a function of γt. For simplicity, we here set α1 = α2 = 3 and γ1 = γ2 = γ. As expected, during the process of the two oscillators evolving from an maximally entangled coherent state into a classical mixture of two pair of mesoscopic components, the Bell signal gradually decreases from $2\sqrt{2}$ to 2; as long as 18 e−γt ≫ 1 so that the coherent states ${\left|3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}$ and ${\left|-3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}$ are quasiorthogonal, the Bell signal remains to be 2 even when the entanglement is completely lost. To explain this point clearly, we approximately express the mixed entangled coherent states of equation (2 ) as a two-qubit mixed entangled state
$\begin{eqnarray}\frac{\partial {{ \mathcal B }}_{\mathrm{ODM}}}{\partial {\phi }_{1}}=\frac{\partial {{ \mathcal B }}_{\mathrm{ODM}}}{\partial {\phi }_{1}^{^{\prime} }}=\frac{\partial {{ \mathcal B }}_{\mathrm{ODM}}}{\partial {\phi }_{2}}=\frac{\partial {{ \mathcal B }}_{\mathrm{ODM}}}{\partial {\phi }_{2}^{^{\prime} }}=0.\end{eqnarray}$
Solving this set of equations, we can obtain a set of optimal parameters $\left\{{\phi }_{1},{\phi }_{1}^{^{\prime} },{\phi }_{2},{\phi }_{2}^{^{\prime} }\right\}$ for the entangled coherent state of equation ( $\begin{eqnarray}\begin{array}{rcl}{\rho }_{c} & = & \displaystyle \frac{1}{2}[| {0}_{L}{\rangle }_{1}\langle {0}_{L}| \otimes | {0}_{L}{\rangle }_{2}\langle {0}_{L}| +| {1}_{L}{\rangle }_{1}\langle {1}_{L}| \\ & & \otimes | {1}_{L}{\rangle }_{1}\langle {1}_{L}| \\ & & +R(| {0}_{L}{\rangle }_{1}\langle {1}_{L}| \otimes | {0}_{L}{\rangle }_{2}\langle {1}_{L}| +| {1}_{L}{\rangle }_{1}\langle {0}_{L}| \\ & & \otimes | {1}_{L}{\rangle }_{2}\langle {0}_{L}| )],\end{array}\end{eqnarray}$
where R ≃ e−36γt. This analogy is valid when the coherent states ${\left|3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}\ $ and ${\left|-3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}$ are quasiorthogonal. The quantum entanglement between the two coherent-state-encoded qubits can be characterized by the concurrence, which is equal to R. The two logic qubits show perfect classical correlations along the z axis. In our formalism the Bell signal is combined by four correlations on the yz plane, each of which can approach 1 even for R → 0 and thus the maximized Bell signal tends to 2. When there remains some quantum entanglement, the optimal Bell signal is stronger than that for a classical mixture so that the inequality is violated. As a result, the revealed nonlocality dynamics agrees well with the entanglement evolution. When the coherent states ${\left|3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}$and ${\left|-3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}$ are not quasiorthogonal, they cannot be considered two basis states of a logic qubit and the oscillators do not show perfect on–off correlations after the displacement ${D}_{j}({\alpha }_{j}^{{\prime} })$. Consequently, the optimal Bell signal drops below 2. When 18e−γt → 0, both the components ${\left|3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}\ $ and ${\left|-3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}$ tend to the vacuum state ${\left|0\right\rangle }_{j}$, which is an eigenstate of the on–off operator Mj with eigenvalue −1. Therefore, the correlation $\left\langle {M}_{1}{M}_{2}\right\rangle $ tends to 1 and the optimal Bell signal is approximately 2. Numerical simulation reveals that the Bell signal is larger than 2 when γt < 0.382, corresponding to R > 1.088 × 10−5.
Figure 1. Evolution of Bell signals for two entangled mesoscopic fields under decoherence. The amplitudes of the coherent state components for both fields are set to be initially &agr;1 = &agr;2 = 3, and the decaying rates are γ1 = γ2 = γ. The dash–dot black line denotes the evolution of the optimal Bell signal obtained by rotated and displaced on–off correlations. The solid blue line represents the result based on displaced parity correlations. The dashed red line denotes the result based on rotated parity correlations. |
The solid blue line represents the evolution of the maximized Bell signal based on joint displaced parity correlations [9], which is defined as
$\begin{eqnarray}\begin{array}{c}{{ \mathcal S }}_{\mathrm{DP}}=\left|{\rm{\Pi }}({\beta }_{1},{\beta }_{2})+{\rm{\Pi }}({\beta }_{1},{\beta }_{2}^{^{\prime} })\right.\\ \quad +\,\left.{\rm{\Pi }}({\beta }_{1}^{^{\prime} },{\beta }_{2})-{\rm{\Pi }}({\beta }_{1}^{^{\prime} },{\beta }_{2}^{^{\prime} })\right|,\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\rm{\Pi }}({\beta }_{1},{\beta }_{2})=\left\langle {D}_{1}({\beta }_{1}){P}_{1}{D}_{1}^{\dagger }({\beta }_{1})\otimes {D}_{2}({\beta }_{2}){P}_{2}{D}_{2}^{\dagger }(\beta )\right\rangle \end{eqnarray}$
is the unnormalized joint Wigner function at the point (β1, β2) in phase space, with ${P}_{j}={\left(-1\right)}^{{a}_{j}^{\dagger }{a}_{j}}$ denoting the parity operator for the jth field mode. We note that each of the components ${\left|{\alpha }_{1}^{{\prime} }\right\rangle }_{1}{\left|{\alpha }_{2}^{{\prime} }\right\rangle }_{2}$ and ${\left|-{\alpha }_{1}^{^{\prime} }\right\rangle }_{1}{\left|-{\alpha }_{2}^{^{\prime} }\right\rangle }_{2}$ shows no parities correlation when ${\left|3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}\ $ and ${\left|-3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}$ are quasiorthogonal. With the choice $\left\{{\beta }_{1}={\alpha }_{1}^{^{\prime} },{\beta }_{2}={\alpha }_{2}^{^{\prime} }\right\}$, the component ${\left|{\alpha }_{1}^{{\prime} }\right\rangle }_{1}{\left|{\alpha }_{2}^{{\prime} }\right\rangle }_{2}$ is transformed to ${\left|0\right\rangle }_{1}{\left|0\right\rangle }_{2}$, showing a perfect parities correlation, while ${\left|-{\alpha }_{1}^{^{\prime} }\right\rangle }_{1}{\left|-{\alpha }_{2}^{^{\prime} }\right\rangle }_{2}$ evolves to ${\left|-2{\alpha }_{1}^{^{\prime} }\right\rangle }_{1}{\left|-2{\alpha }_{2}^{^{\prime} }\right\rangle }_{2}$, exhibiting no parities correlation. For the choice $\left\{{\beta }_{1}=-{\alpha }_{1}^{^{\prime} },{\beta }_{2}=-{\alpha }_{2}^{^{\prime} }\right\}$, the component ${\left|-{\alpha }_{1}^{^{\prime} }\right\rangle }_{1}{\left|-{\alpha }_{2}^{^{\prime} }\right\rangle }_{2}$ has a perfect displaced parities correlation, but the other component shows no correlation. Therefore, for an equal classical mixture of ${\left|{\alpha }_{1}^{{\prime} }\right\rangle }_{1}{\left|{\alpha }_{2}^{{\prime} }\right\rangle }_{2}$ and ${\left|-{\alpha }_{1}^{^{\prime} }\right\rangle }_{1}{\left|-{\alpha }_{2}^{^{\prime} }\right\rangle }_{2}$ the maximized value of Π(β1, β2) is 1/2, so that the optimal Bell signal is 1, which is in distinct contrast with the result based on the present formalism, with which both components show perfect on–off correlations under the joint displacement ${D}_{1}({\alpha }_{1}^{{\prime} }){D}_{2}({\alpha }_{2}^{{\prime} })$ or ${D}_{1}(-{\alpha }_{1}^{{\prime} }){D}_{2}(-{\alpha }_{2}^{{\prime} })$. Numerical simulation shows the Bell inequality is violated only when γt < 0.009, which corresponds to R > 0.724. When both ${\left|3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}\ $ and ${\left|-3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}$ tend to the vacuum state ${\left|0\right\rangle }_{j}$, the correlation $\left\langle {P}_{1}{P}_{2}\right\rangle $ tends to 1 and the optimal Bell signal is approximately 2.The dashed red line denotes the evolution of the Bell signal based on rotated parities correlations, defined as [32]
$\begin{eqnarray}\begin{array}{c}{{ \mathcal S }}_{{RP}}=\left|E({\phi }_{1},{\phi }_{2})+E({\phi }_{1},{\phi }_{2}^{^{\prime} })\right.\\ \quad +\left.E({\phi }_{1}^{^{\prime} },{\phi }_{2})-E({\phi }_{1}^{^{\prime} },{\phi }_{2}^{^{\prime} })\right|,\end{array}\end{eqnarray}$
where $\begin{eqnarray}E({\phi }_{1},{\phi }_{2})=\left\langle {P}_{1,z}({\phi }_{1})\otimes {P}_{2,z}({\phi }_{2})\right\rangle ,\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{ccl}{P}_{j,z}({\phi }_{j}) & = & {R}_{j,z}({\phi }_{j}){P}_{j}{R}_{j,z}^{\dagger }({\phi }_{j}),\\ {R}_{j,z}({\phi }_{j}) & = & {D}_{j}^{\dagger }({\alpha }_{j}^{{\prime} }){G}_{j}({\phi }_{j}){D}_{j}({\alpha }_{j}^{{\prime} }),\\ {G}_{j}({\phi }_{j}) & = & {\left|0\right\rangle }_{j}\left\langle 0\right|{{\rm{e}}}^{{\rm{i}}{\phi }_{j}}+\displaystyle \sum _{n\,=\,1}^{\infty }{\left|n\right\rangle }_{j}\left\langle n\right|.\end{array}\end{eqnarray}$
When ${\left|3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}\ $ and ${\left|-3{{\rm{e}}}^{-\gamma t/2}\right\rangle }_{j}$ are quasiorthogonal, the rotated parity operator Pj,z(φj) corresponds to the spin operator on the xy plane with an angle φj to the x axis. Neither ∣0L⟩1∣0L⟩2 nor ∣1L⟩1∣1L⟩2 exhibits spin correlation on the xy plane, so that E(φ1, φ2) = 0 for a classical mixture of these two components and the maximized Bell signal is about $2\sqrt{2}R$. Thus, the Bell inequality violation requires the concurrence to be higher than $1/\sqrt{2}$, corresponding to $\gamma t\lt (\mathrm{ln}2)/72$.4. Conclusions
We have suggested a new formalism to explore the decoherence process of the quantum nonlocality of two harmonic oscillators initially in a mesoscopic entangled state. We show that the obtained Bell signal progressively decays under decoherence, but it can keep above the classical upper bound even only a small amount of entanglement is remained. Consequently, the Bell signal is suitable for being used as a measure for the remaining nonlocal correlations. This is in distinct contrast with the results obtained with previous formalisms, which drop below the classical bound even when there remains a considerable amount of entanglement. Though the formalism is studied with an initial two-mode entangled cat state, it can be generalized to the case with a two-mode squeezed vacuum state [33]. The potential experimental implementation of our scheme could be reached with the recently demonstrated superconducting circuit techniques [26–31], or sooner to be developed optomechanical techniques [34], and so on.