1. Introduction

Quantum entanglement [1,2] attracts considerable interest due to the verification of the fundamental quantum theory [3] and its potential application in quantum information processing [4] including computation [5,6], secure communication [7,8], and metrology [9,10]. The generation of macroscopic entanglement is an important subject [11–23]. Especially, the mechanical entanglement [16–23] is of wide interest due to the long decoherence time. However, few works investigate the tripartite entanglement among multiple macroscopic mechanical oscillators, except that Hao et al. study the dynamical tripartite entanglement of the mechanical oscillator in an optomechanical array [19].

On the other hand, EPR steering [24,25] is a stronger type of nonlocal correlation than entanglement and receives extensive attention due to the asymmetry [26–30], which indicates the different steerabilities in two opposite directions or even one-way steerability representing that the local measurement of one particle unidirectionally affects the state of the other. With the development of the quantum network [31,32], the EPR steering in the multipartite system has drawn extensive attention, such as the demonstrations of monogamy [27,33–36] and shareability of EPR steering [37]. These works reveal more configurations of multipartite EPR steering and can provide more potential applications in various quantum information protocols [38–40]. However, little work has been done on the study of EPR steering among multiple macroscopic objects. This motivates us to explore multipartite EPR steering among macroscopic mechanical oscillators in a multi-mode optomechanical system.

In this paper, we study the bipartite and tripartite quantum correlations among three distinct mechanical oscillators in an optomechanical system where the cavity is driven by a multi-tone laser. The three mechanical oscillators with no direct interaction commonly interact with the cavity whose frequency components are regarded as Floquet modes. Mediated by the Floquet cavity modes, the quantum correlation of the mechanical oscillator is constructed and manipulated by the optomechanical coupling strength. Not only all bipartite entanglement but also genuine tripartite entanglement can be generated among the three mechanical oscillators. Compared with the scheme mediated by the real cavity in an optomechanical array [19], the stronger entanglement can be obtained in the current method. We show a significant predominance of manipulating the arbitrary bipartite and tripartite entanglement. In contrast to this proposal where the entangled tripartite particles are coupled to each other, the two-tone driving widely employed [41–43] is not enough to generate tripartite entanglement of the uncoupled mechanical oscillators. In addition, the monogamous relation of EPR steering of the mechanical oscillator can be demonstrated. We also show that asymmetric and even one-way EPR steering can be obtained for the three mechanical oscillators possessing identical damping rates. The direction of one-way steering can be manipulated by modulating the ratio of effective optomechanical coupling strength.

2. Model and dynamics

As depicted in Fig. 1(a), we consider that three unconnected mechanical oscillators $b_j\; (j=1,2,3)$ with frequency $\omega _j$ and damping rate $\gamma _j$ are optomechanically coupled to a common cavity with frequency $\omega _c$ and decay rate $\kappa$. It is convenient to change the frame by applying the unitary transformation $\hat {U}=\mathtt {Exp}(iH_0t)$, where $H_0=\omega _La^{\dagger }a+\sum _{j=1}^{3}\omega _jb_j^{\dagger }b_j$. Then the new Hamiltonian $H=\hat {U}H_{\mathtt {old}}\hat {U}^{\dagger }-i\hat {U}\partial \hat {U}^{\dagger }/\partial t$ of the form ($\hbar =1$)

The dynamic of the system is described by the Langevin equations

 

Fig. 1. (a) Schematic diagram of the optomechanical system, where the cavity mode $a$ driven by a four-tone laser interacts with three near-degenerate mechanical oscillators $b_j$ ($j=1,2,3$). (b) Frequency-domain illustration of the optomechanical control scheme. The horizontal axis is the detuning with respect to the cavity resonance. The black curve shows the cavity lineshape. The coloured arrows are control lasers with detuning $-\Delta _j$ ($j=1,2,3,4$). The black dashed curves are the Stokes (blue) and anti-Stokes (red) sidebands that resonate with the mechanical oscillators. (c) Scheme of the mechanical quantum correlation mediated by the Floquet cavity modes $a_\omega, a_{\omega \pm 2\Delta }$ resulting from the frequency components of the cavity mode in the frequency domain. The red dashed and black solid arrows represent the beam-splitter and parametric-type interaction, respectively.

Download Full Size | PDF

The cavity is driven by the strong classical field and evolves to a large mean value triggering the large amplitude of the mechanical oscillators, then the standard linearization technique can be applied with $a\rightarrow \alpha + \delta a$ and $b_j\rightarrow \beta _j+ \delta b_j$. ($\alpha,\;\beta _j$) and ($\delta a,\;\delta b_j$) denote the coherent amplitudes and the fluctuations respectively. In the following, we neglect $\delta$ in $\delta a$ and $\delta b_j$ for simplify. Therefore, Eq. (2) can be separated into two sets of dynamical equations about the coherent amplitude

$\Delta _c^{\prime }=\Delta _c+\sum _{j=1}^{3}g_j(\beta _j^{*}e^{i\omega _jt}+\beta _je^{-i\omega _jt})$ is the cavity detuning modified by the optomechanical interaction. For the classical amplitude, we assume $\alpha =\sum _{k=1}^{4}\alpha _k e^{i(\Delta _kt+\phi _k)}$ and $\beta _j=\tilde {\beta }_j e^{i\omega _jt}$. From Eq. (3), we have $\alpha _k=E_k/(\kappa +i\Delta _c^{\prime }+i\Delta _k)$ and $\tilde {\beta }_j=-ig_j\sum _{k=1}^{4}|\alpha _k|^2/(\gamma _j+i\omega _j)$. In the weak coupling condition, i.e., $|g_j\beta _j| \ll \{\Delta _c,\Delta _k\}$, then $\Delta _c^{\prime }\approx \Delta _c, \alpha _k\approx E_k/(\kappa +i\Delta _c+i\Delta _k)$. When $\Delta _j=-\omega _j-\Delta$ and $\Delta _4=-\Delta _3\,(j=1,2,3)$ are chosen with the frequency difference between mechanical oscillators $\Delta =\frac {1}{2}(\omega _l-\omega _{l+1}) (l=1,2)$, the rotating-wave approximation (RWA) can be satisfied. Then the linearized Hamiltonian corresponding to the Eq. (4) is written as

$\mathcal {G}_j^{(\mathtt {p})}(t)=g_j\sum _{k=1}^{3}\alpha _k e^{i[(\Delta _k+\omega _j)t+\phi _k]}$ is the effective coupling strength of the parametric-type interaction between the $j$th mechanical oscillator and the cavity, which corresponds to the Stokes scattering of the control laser. $\mathcal {G}_j^{(\mathtt {b})}(t)=g_j\alpha _4e^{i[(\Delta _4-\omega _j)t+\phi _4]}$ implies the beam-splitter-type optomechanical interaction resulting from the anti-Stokes scattering of the control laser. We define the time-independent optomechanical coupling strength $G_k=g|\alpha _k|$ with $g_j=g$. Furthermore, rotating the linearized Hamiltonian Eq. (5) in the frame with respect to the cavity detuning $\Delta _c^\prime =\Delta$, we can obtain the Hamiltonian in the interaction picture

Here $\mathtt {arg}(\alpha _1)+\phi _1=\pi,\;\mathtt {arg}(\alpha _{2,3,4})+\phi _{2,3,4}=0$ are chosen. It can be seen that there are five different frequency components, and $\tilde {\mathcal {G}}_l^{(\mathtt {p})}(t)\; (l=1,2,3)$ is determined by $G_{1,2,3}, \;G_4$ contributes to beam-splitter-type interaction $\tilde {\mathcal {G}}_l^{(\mathtt {b})}(t)$. To clarify the effects of these frequency components on the mechanical quantum correlation, we introduce the Fourier transform of the operator $o_{\omega }=\frac {1}{\sqrt {2\pi }}\int \mathtt {d}t o(t)e^{i\omega t}$, the Hamiltonian in the frequency domain can be given by

The first three lines describe the interactions between three mechanical oscillators and Floquet cavity modes $a_{-\omega }^\dagger, a_{-\omega +2\Delta }^\dagger, a_{-\omega -2\Delta }^\dagger$ corresponding to the cavity fields with detecting detuning $0, \;-2\Delta, \;2\Delta$, as shown in Fig. 1 (c). The last line represents the parametric-type optomechanical interaction between the mechanical oscillator $b_1\;(b_3)$ and Floquet cavity mode $a_{-\omega +4\Delta }^\dagger (a_{-\omega -4\Delta }^\dagger )$ with strength $G_3\;(G_1)$. These Floquet modes $a_{-\omega }^\dagger,\; a_{-\omega \pm 2\Delta }^\dagger$, and $a_{-\omega \pm 4\Delta }^\dagger$ are mutually independent and separately mediate the three mechanical oscillators. Mediated by the same Floquet cavity mode, the Hamiltonian implies that: (i) the phonon conversion between different mechanical oscillators can be formed; (ii) the two-mode squeezed state consisting of arbitrary two mechanical oscillators can be generated; (iii) the degenerate parametric amplification of the mechanical oscillator $b_2$ can be created (see the term $a_{-\omega }^{\dagger }(G_2b_{2,-\omega }^{\dagger }+G_4b_{2,\omega })$ in the brackets of the first line). That is to say, the indirect interactions between the three mechanical oscillators can be constructed mediating by Floquet cavity modes, and its strength can be tunable by modulating the effective optomechanical coupling $G_k\; (k=1,2,3,4)$. Consequently, the generation and manipulation of mechanical quantum entanglement can be expected.

We define the quadrature operators of the bosonic modes $o=a, b_j, a_{in}, b_{j,in}$ as $X_o=(o+o^\dagger )/\sqrt {2}$, $P_o=(o-o^\dagger )/\sqrt {2}i$ and introduce the column vectors of quadrature and noise operators $U=[X_{a}, P_{a}, X_{b_1}, P_{b_1}, X_{b_2}, P_{b_2}, X_{b_3}, P_{b_3}]^{T}$, $N=[X_{a_{in}}, P_{a_{in}}, X_{b_{1,in}}, P_{b_{1,in}}, X_{b_{2,in}}, P_{b_{2,in}}, X_{b_{3,in}}, P_{b_{3,in}}]^{T}$. Then the dynamical equations corresponding to the linearized Hamiltonian Eq. (5) can be written in a compact form

Here, $\mathtt {Re}[\cdot ]$ and $\mathtt {Im}[\cdot ]$ are the real and imaginary parts, respectively. Note that the general stable condition of the linear differential equation (Eq. (9)) is determined by the corresponding homogeneous equation $\dot {U}=M(t)U$ which has a principal matrix solution $\Pi (t)$. Based on the Floquet’s theorem [44], the solution of Eq. (9) is stable if all Floquet multipliers satisfy $|\lambda _j|<1$, where $\lambda _j$ is the $j$th eigenvalue of $\Pi ^{-1}(0)\Pi (T)$ and $T$ is a cycle length of $\Pi (t)$. The chosen parameters satisfy the stability condition in the following simulations.

From Eq. (9), we can derive a linear differential equation

3. Quantum entanglement and EPR steering

3.1 Measure of quantum entanglement and EPR steering

For the continuous-variable Gaussian state, logarithmic negativity [45,46] is convenient to quantify the bipartite entanglement between the three mechanical oscillators. The bipartite entanglement between modes $b_i$ and $b_j$ is defined as

The genuine tripartite entanglement can be quantified by the minimum residual contangle $\mathcal {R_{\mathtt {min}}}$ [47] defined as

To measure the Gaussian steering, we employ the intuitive and computable quantification recently put forward in [48], which is a necessary and sufficient criterion for arbitrary bipartite Gaussian states under Gaussian measurements. The steering in the direction from mode $b_j$ to mode $b_i$ is given by

3.2 Manipulation of bipartite and tripartite entanglement

In Fig. 2 (a), we plot the time evolution of bipartite entanglement $E_{12}, E_{13}, E_{23}$ and the tripartite entanglement $\mathcal {R}_{\mathtt {min}}$ among the three mechanical oscillators. It can be seen that $\mathcal {R}_{\mathtt {min}}$ appears first even if bipartite entanglement is zero, which is similar to the behavior of tripartite GHZ state $\frac {1}{\sqrt {2}}(|111\rangle +|000\rangle )$. With the evolution of time, $E_{ij}$ and $\mathcal {R}_{\mathtt {min}}$ eventually reach their oscillating steady values. At the moment $t=80\pi /\Delta$, the dependence of the mechanical entanglement on optomechanical couplings is plotted in Fig. 2 (b) and (c). For the tripartite entanglement, $\mathcal {R}_{\mathtt {min}}$ shows a non-monotonous dependence of $G_1/G_4$ regardless of the presence of $G_3$, but the amount of $\mathcal {R}_{\mathtt {min}}$ does relate with $G_3$, i.e., the maximum value of $\mathcal {R}_{\mathtt {min}}$ in (b) is larger than that in (c). This result implies that four-tone driving is better than three-tone driving for enhancing tripartite entanglement. It is worth pointing out that $\mathcal {R}_{\mathtt {min}}$ is zero for two-tone driving (when $G_1=G_3=0$ in Fig. 2 (c)). That is to say, two-tone driving is not enough to generate the tripartite entanglement in this system due to the absence of direct interaction of the mechanical oscillator, while two-tone driving would take effect for the generation of tripartite entanglement among three particles with direct coupling [41–43].

For the bipartite entanglement, $E_{12}$, $E_{13}$ and $E_{23}$ can exist simultaneously when the cavity is driven by the four-tone laser, as shown in Fig. 2 (b). It also can be seen that $E_{12}$ is non-monotonously dependent on $G_1$. From the Hamiltonian (8), the generation of $E_{12}$ results from the mediation of the Floquet modes $a_{-\omega }^{\dagger }$ and $a_{-\omega -2\Delta }^{\dagger }$, and then the increase of $G_1$ is beneficial to a larger value of $E_{12}$. However, the increase of $\mathcal {G}_j^{(p)}(t)$ would result in an overlarge phonon number and push the mechanical oscillators into the classical regime. These two contradictory effects make the non-monotonic dependence of $E_{12}$ on $G_1$. When $G_1=0$, $E_{12}$ does not exist, and $E_{13}\neq E_{23}$ due to the different effects of optomechanical coupling on the mechanical oscillator $b_1$ and $b_2$, which can be answered from the Hamiltonian (8). Furthermore, in the case of two-tone driving, e.g., $G_1=G_3=0$, there is only the entanglement between mechanical oscillators $b_1$ and $b_3$, as shown in Fig. 2 (c), and the maximum values of $E_{12}$ and $E_{13}$ in Fig. 2 (c) are larger than those in Fig. 2 (b). Therefore, reducing the number of driving sidebands would bring fewer pairs of bipartite entanglement but be beneficial to stronger bipartite entanglement.

 

Fig. 2. (a) Time evolution of entanglement among three mechanical oscillators with parameters $(\frac {G_1}{G_4},\frac {G_2}{G_4},\frac {G_3}{G_4})=(0.28,0.3,0.32)$. At the moment $t=80\pi /\Delta$, (b-c) the entanglement as a function of $G_1/G_4$ with $G_2/G_4=0.3$, and $G_3/G_2=1$ and $0$. (d) When $G_4/\omega _m=0.02$, the tripartite entanglement $\mathcal {R_{\mathtt {min}}}$ as a function of bath temperature $T$ for $G_1/G_4=0.35$ and $G_1=G_2=G_3$; the bipartite entanglement is plotted with $(\frac {G_1}{G_4},\frac {G_2}{G_4},\frac {G_3}{G_4})=(0.28,0.3,0.32)$. The other parameters are $\omega _m/2\pi =10.34\mathtt {MHz}$, $\omega _1=\omega _m$, $T=1\mathtt {mK}$, $\gamma _{1,2,3}/\omega _m=10^{-5}$, $\kappa /\omega _m=0.08$, $\Delta /\omega _m=0.01$, $\Delta _c^\prime /\Delta =1$, and $G_4/\omega _m=0.01$.

Download Full Size | PDF

Tripartite entanglement and bipartite entanglement altered by bath temperature of mechanical oscillators are displayed in Fig. 2 (d). $\mathcal {R}_{\mathtt {min}}$ can only exist in the thermal bath with temperature below 32mK, shown by black-solid curve. Pairwise entanglement can exist when the temperature of the thermal bath is below 10mK. However, there is no bipartite entanglement for the bath temperature high 20mK. In all, both bipartite and tripartite entanglement require a low-temperature experimental environment.

3.3 Manipulation of EPR steering

Given that entanglement can be achieved in this system, EPR steering among the three mechanical oscillators can be expected. Meanwhile, the complementary relation between tripartite entanglement and the maximum steering inequality violation for an arbitrary three-qubit pure state has been demonstrated [49,50], which inspires us to explore the connection between tripartite entanglement and EPR steering in this system.

Figure 3 (a) displays the dependence of EPR steering $\mathcal {S}_{b_i\leftrightarrow b_j}^{\mathtt {max}}$ ($i\neq j$) and $\mathcal {R}_{\mathtt {min}}$ on the ratio $G_1/G_2\; (G_2/G_3)$. $\mathcal {S}_{b_i\leftrightarrow b_j}^{\mathtt {max}}$ is the maximum value in $\mathcal {S}_{b_i\rightarrow b_j}$ and $\mathcal {S}_{b_j\rightarrow b_i}$. $\mathcal {S}_{b_i\leftrightarrow b_j}^{\mathtt {max}}>0$ indicates the presence of EPR steering between mechanical oscillators $b_i$ and $b_j$, but the asymmetry and directionality of steering are missing. It can be seen that tripartite entanglement and EPR steering can be present simultaneously. However, the parameter satisfying the optimal tripartite entanglement is not beneficial to the generation of steering, and $\mathcal {R}_{\mathtt {min}}$ is very small with the parameter used to optimize steering. This result shows a complementary relation between the optimal tripartite entanglement and the optimal steering in this system.

 

Fig. 3. (a) Tripartite entanglement and the maximum value of EPR steering $\mathcal {S}_{b_i\leftrightarrow b_j}^{\mathtt {max}}$ ($i\neq j$) for two mechanical oscillators $b_i,b_j$ as a function of $G_1/G_2$ with $G_1=G_2=G_3$ and $G_2/G_4=0.3$. When $G_2/G_4=0.35$ and $G_3=0$, (b) EPR steering as a function of $G_1/G_4$, and (c) the maximum value of the three sets of EPR steering $\mathcal {S}_{b_1b_2b_3}^{\mathtt {max}}$ as functions of bath temperature $T$ and $G_1/G_4$, where the white solid line is the boundary line of whether EPR steering exists or not. The other parameters are same as those in Fig. 2.

Download Full Size | PDF

On the other hand, we are also interested in the configuration of the EPR steering shared among the three mechanical oscillators. From Fig. 3 (a), we can find that $\mathcal {S}_{b_1\leftrightarrow b_2}^{\mathtt {max}}$ ($\mathcal {S}_{b_2\leftrightarrow b_3}^{\mathtt {max}}$) is present in the range of $1<G_1/G_2<2.5$ ($0.5<G_1/G_2<1$) which indicates $G_1>G_2>G_3$ ($G_1<G_2<G_3$). And $\mathcal {S}_{b_1\leftrightarrow b_3}^{\mathtt {max}}$ is absent because the parameters do not satisfy $(G_1, \; G_3)<G_2$. It also shows the difficulty of obtaining more than one set of EPR steering among the three mechanical oscillators in the system driven by a four-tone laser. Thus, we choose $G_3=0$ and plot the dependence of $\mathcal {S}_{b_i\rightarrow b_j}$ on $G_1/G_4$ in Fig. 3 (b). Obviously, when $G_1/G_4\in (0.15,\;0.29)$, the one-way steering from $b_1$ to $b_2$ and asymmetric steering between $b_1$ and $b_3$ can be present simultaneously, which demonstrates the monogamous relation of EPR steering, i.e., the mechanical oscillator $b_1$ cannot be simultaneously steered by mechanical oscillators $b_2$ and $b_3$. Moreover, the asymmetry of steering is dependent on coupling strength $G_{1,2,3,4}$ because optomechanical interactions $\tilde {\mathcal {G}}_j^{(p)}(t), \;\tilde {\mathcal {G}}_j^{(b)}(t)$ alter the effective decoherence property of the mechanical oscillator, though the three mechanical oscillators possess identical intrinsic damping rates. It has a clear advantage over unequal losses of subsystems resulting in asymmetric steering, manifested by the avoidance of reducing steerability.

Lastly, we investigate the robustness of EPR steering on the bath temperature in Fig. 3 (c), where $\mathcal {S}_{b_1b_2b_3}^{\mathtt {max}}\equiv \mathtt {max}[\mathcal {S}_{b_i\leftrightarrow b_j}^{\mathtt {max}}] (i\neq j)$ denotes the presence of EPR steering among the three mechanical oscillators. According to the white-solid boundary line, EPR steering does not exist in the thermal bath with temperature $T>4.2$mK. This is because the thermal noise is harmful to quantum correlation. When the temperature of the thermal bath increases, the thermal noise of the mechanical oscillator increases resulting in the decrease of quantum correlation between mechanical oscillators. Compared with bipartite and tripartite entanglement, the experimental environment should be at a lower temperature to manipulate EPR steering.

4. Discussion and conclusion

We now discuss the experimental feasibility of the current scheme. The superconducting microcircuit [20,21,51,52] provides a promising platform to structure this optomechanical system. The cavity is formed by the lumped element inductors and capacitors. Three movable capacitors are embedded in parallel into the single microwave cavity to form three mechanical oscillators without direct interaction. The motion of the mechanical oscillator results in a shift of capacitance leading to the optomechanical interaction. The resonant frequency of the mechanical oscillator can be controlled by modulating the inductance and magnetic flux [52]. In the recent experiment, the macroscopic entanglement between two mechanical drum-type oscillators has been observed [20,21]. In [21], the cavity is irradiated with two pulses simultaneously which respectively result in the two-mode squeezing and beam-splitter interactions between the cavity and drums. And the decay rate of the cavity is $\kappa /2\pi =800$kHz, two mechanical oscillators are of frequencies $\omega _1=15.898$MHz, $\omega _2=10.865$MHz, and the decay time of mechanical oscillators $1/\gamma _1=5.8$ms, $1/\gamma _1=6.9$ms. The two effective electromechanical couplings are $G_1=82\times 2\pi$ kHz and $G_2=94\times 2\pi$ kHz and the experimental parameters are measured at the temperature $7$mK. The higher frequencies of drum-type mechanical oscillators with $6.692\times 2\pi$MHz and $9.032\times 2\pi$MHz have been reported in [20]. By generalizing the experimental device [20,21] into three mechanical oscillators and applying three or four pulses, the current scheme might be reliable. The bipartite and tripartite entanglement should be realized based on the current level of experiment, but EPR steering should be present in the experimental environment at a lower temperature. Compared to these experiment results [20,21], the current scheme using classical multi-tone driving is more flexible to manipulate bipartite mechanical entanglement and can be used to generate tripartite entanglement among three macroscopic mechanical oscillators.

In conclusion, we have studied quantum correlation among three distinct mechanical oscillators in an optomechanical system. The cavity pumped by multi-tone driving has different frequency components, which serve as Floquet cavity modes and are used to construct channels of quantum correlations for three mechanical oscillators. By modulating the effective optomechanical coupling, all bipartite and tripartite quantum correlations can be manipulated. The genuine tripartite entanglement can be optimized significantly in the system pumped by four-tone driving compared to that pumped by three-tone driving and cannot be obtained with two-tone driving. All bipartite entanglement can be present simultaneously for four-tone driving. As the number of multi-tone pumping reduces, we can only observe less than three pairs of bipartite correlation, but the maximum magnitudes of bipartite entanglement and EPR steering can be strong with the optimal parameters. Besides, asymmetric and even one-way steering can be obtained, though the mechanical oscillators process almost decay rates. And the monogamous relation of EPR steering of the macroscopic mechanical oscillator can be demonstrated. This work utilizes a simple approach to explore the manipulation of bipartite and tripartite quantum correlations among three macroscopic mechanical oscillators and can be expected to extend more macroscopic objects.