1. Introduction

Quantum entanglement [1–4], as a key resource for quantum technologies such as quantum communication, quantum computing, and quantum sensing [5–7], has been demonstrated by using photons [8], atoms [9] or ions [10], and more recently various cavity optomechanical (COM) systems [11–14]. COM systems with coherent light-motion coupling [15] have enabled many important applications such as ground-state motion cooling [16–18], long-lived quantum memory [19–22], blockade or anti-bunching effects [23–28], ultra-sensitive COM sensing [29–31], and COM squeezing [32–36] or entanglement [37–45], to name only a few. In recent experiments, COM entanglement or correlations have been demonstrated via direct radiation-pressure coupling between optical field and mechanical oscillators [37–41] or even a laser and a $40\,\mathrm {kg}$ mirror [46]. Also, entangled states based on COM devices between propagating optical fields [47] or between massive mechanical elements [12,48] have been achieved. In order to protect and enhance the fragile quantum states of COM devices, many effective schemes have been proposed, especially with an optical parametric amplifier (OPA) [49–56] or similarly, using a mechanical parametric amplification (MPA) [57–66]. We note that, as feasible and effective ways to tune quantum systems, OPA or MPA was also used to realize strong mechanical cooling [67] and enhanced sensors [64,68–70] or enhanced nonlinear effects [43,71–76]. Hu et al. proved that parametric driving can exhibit rich classical and quantum dynamical behaviors in a standard COM system [77]. However, as far as we know, the possibility of combining these powerful means to further enhance COM entanglement, especially the interlay of OPA and MPA in engineering quantum COM dynamics, has remained unexplored till now.

Here, we fill this gap by studying the synergetic effect of both optical and mechanical parametric modulations in enhancing COM entanglement and protecting it against thermal noises. Specifically, we consider a single-cavity COM system under three-field driving, i.e., besides the standard optical pump laser driving the cavity, we also consider an intra-cavity OPA driven by another laser and a mechanical parametric driving. We find that COM entanglement can be significantly enhanced by tuning the parameters of both OPA and MPA, compared to the standard single-driven case (without any OPA or MPA) or the double-driven cases (with the OPA or the MPA only). We find that by tuning the relative strength and the frequency mismatch of optical and mechanical driving fields, constructive interference can emerge and lead to significant improvement of both the strength of COM entanglement and its robustness to thermal noises. Thereby, our work sheds a new light on preparing and protecting quantum states with multi-field driven COM systems for diverse applications based on COM entanglement.

This paper is organized as follows. In Sec. 2, we first describe the theoretical model of a COM system driven by an OPA and a MPA in detail, and the linearized dynamical equations of the classical mean values and quantum fluctuations are presented. In Sec. 3, compared to the no-modulation case, the synergistic effects of modulating an OPA and a MPA on the classical and quantum dynamics are studied in the long-time limit. Finally, conclusions are given in Sec. 4.

2. Theoretical model

As shown in Fig. 1(a), we consider a nonlinear $\chi ^{(2)}$ medium placed inside a Fabry-P$\acute {\mathrm {e}}$rot (FP) cavity that consists of a fixed mirror and a movable mirror. The FP cavity supports one single optical mode with resonance frequency $\omega _{c}$ and decay rate $\kappa = \pi c/(2FL)$, where $F$ and $L$ denote the fineness and length of the FP cavity, respectively. The movable mirror supports a mechanical mode with effective mass $m$, fundamental frequency $\omega _m$, and mechanical damping rate $\gamma _m$. Hereafter, we consider three kinds of driving fields applied to this hybrid COM system. The nonlinear $\chi ^{(2)}$ medium that serves as an OPA is pumped by a parametric laser field with frequency $2\omega _{p}$, amplitude $G_c$ and phase $\theta _c$. Meanwhile, the mechanical mode is electrically driven by a periodic pump field that modulates the mechanical spring constant in time and generates the process of MPA under two-phonon resonance with frequency $2\omega _s$, amplitude $G_m$ and phase $\theta _m$ [80]. Also, the cavity mode is assumed to be driven by an external laser field with frequency $\omega _l$ and amplitude $E=\sqrt {2\kappa P/\hbar \omega _l}$, where $P$ denotes the input laser power. In the frame rotating with respect to the driving laser frequency $\omega _l$, the total Hamiltonian of the system is given by

 

Fig. 1. (a) Schematic of the COM system under three-field driving. The FP cavity is driven by an externally driven laser with strength $E$, and an OPA inside the cavity is driven by a laser with amplitude $G_c$ and phase $\theta _c$, while the mechanical oscillator is driven by a laser with amplitude $G_m$ and phase $\theta _m$. The inset table shows the reasonable experimental parameters that we used in our numerical calculation [78,79]. (b) Cartoon diagram of the interaction between the system and the driving fields. The photon and phonon are coupled together via radiation pressure, and the phase difference between the parametric drive of OPA and MPA is $\Delta \theta$. (c), (d) The cavity field strength $\left |\left \langle a(t)\right \rangle \right |$ as a function of time $t/\tau$. The parameters are $\Delta _c/\omega _{m} =1$, $T=0$, $G_{c}/\omega _{m}=0.01$, $\Omega _{c}/\omega _{m}=1.18$, $G_{m}/\omega _{m}=0.01$, and $\Omega _{m}/\omega _{m}=1.18$.

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Taking into account the fluctuations and dissipations of the optical and mechanical modes, the dynamics of the system can be described by the following set of quantum Langevin equations (QLEs):

In order to find the solutions to the QLEs (2), one can implement a standard linearization procedure by expanding each operator as a sum of its steady-state mean value and a small quantum fluctuation around it, i.e., $\hat {O}(t)=\langle \hat {O}(t)\rangle +\delta \hat {O}(t)$ $(\hat {O}=\hat {a},\hat {b})$, which requires the fulfillment of large cavity intensity conditions. Then, by substituting the above assumption into the QLEs (2), we can obtain the dynamics for the steady-state mean values as

By defining the quadrature operators of the optical and mechanical modes as

Here we have grouped together the quadrature fluctuation operators and the input noise operators into the state vectors $\hat {u}(t)$ and $\hat {n}(t)$, respectively, i.e.,

Due to the linearized dynamics and the Gaussian nature of the input noises, the final state of the system can eventually evolve into a bipartite zero-mean Gaussian state independently of any initial condition, which allows us to characterize the quantum property of the system through a $4\times 4$ covariance matrix (CM) $V(t)$, with its matrix elements defined as

Combining Eqs. (8) and (11), the equation of motion of the CM $V(t)$ can be written as follows:

In order to verify the bipartite COM entanglement, we adopt the logarithmic negativity $E_\mathcal {N}$ as a quantitative measure [81], which is defined as

Eq. (13) indicates that COM entanglement emerges if and only if the inequality $\eta < 1/2$, which is equivalent to Simon’s necessary and sufficient entanglement criterion for Gaussian states.

It should be stressed that the entanglement measure $E_\mathcal {N}$ is valid only when the system is stable in the long-time limit. Here, we have confirmed that all real parts of the eigenvalues of $A$ are negative, which, according to the Routh-Hurwitz criterion, ensures the stability condition of our system in the long-time limit. Besides, we note that the trajectory of the solution of Eq. (4) is an asymptotic periodic orbit with a period of $\tau$, which implies that, in the long time limit, we have $A(t +\tau )=A(t)$. On the other hand, since Eq. (12) is a linear differential equation with periodic coefficients, the Floquet theory can still be valid in principle. Therefore, $V(t)$ can persevere the same periodicity in the long time limit, i.e., $V (t+\tau )=V(t)$. Similarly, the logarithmic negativity $E_\mathcal {N}$ will acquire the same periodicity in time as $V(t)$. To see the behavior of COM entanglement in the presence of OPA and MPA, we introduce the following definition for $E_\mathcal {N}$, i.e.,

3. Results and disscussion

As discussed above, we derive the dynamics of the CM of this system, by which we can further evaluate the entanglement measure by solving such dynamical functions. In this section, based on some numerical calculations, we study how to regulate and enhance the COM entanglement by exploiting the interplay effect of OPA and MPA.

In Fig. 2, we first investigate the behavior of COM entanglement under the double-driven case (say, with the standard optical pump laser and the OPA or the MPA only). Figure 2(a) and 2(d) show the dependence of the logarithmic negativity $E_{\mathcal {N},max}$ on the scaled optical (mechanical) detuning $\Omega _{c}/\omega _{m}$ ($\Omega _{m}/\omega _{m}$) and the scaled amplitude of OPA (MPA) $G_{c}/\omega _{m}$ ($G_{m}/\omega _{m}$). In the presence of OPA modulation only (i.e., $G_{c}\neq 0$ and $G_{m}=0$), the profile of COM entanglement is characterized by two sharp peaks around $\Omega _{c}/\omega _{m}=0.5$ and $\Omega _{c}/\omega _{m}=1.5$, respectively, which is reminiscent of the results discussed in preceding investigation [77]. In addition, they find that COM entanglement and classical nonlinear dynamics have good quantum-classical correspondence at these two positions. Also, we note that the maximum value of $E_{N,max}$ is about $E_{N,max}=0.27$ with an optimal optical detuning $\Omega _{c}/\omega _{m} =1.18$. In contrast, in the presence of MPA modulation only (i.e., $G_{c}=0$ and $G_{m}\neq 0$), the profile of COM entanglement becomes highly peaked within a finite interval of values of $\Omega _{m}/\omega _{m}$ around $\Omega _{m}/\omega _{m}=2$, whose maximum value is about $E_{N,max}=0.46$ with an optimal mechanical detuning $\Omega _{m}/\omega _{m} =1.9$. In Figs. 2(b) and 2(e), we plot the logarithmic negativity $E_{\mathcal {N}}$ as a function of time $t$ for different values of $\theta _{c}$ ($\theta _{m}$). It is seen that in the case of either $G_{c}\neq 0$ and $G_{m}=0$ or $G_{c}=0$ and $G_{m}\neq 0$, both of their $E_{\mathcal {N}}$ oscillate back and forth periodically over $t$ with a periodicity of $\tau$. Also, when tuning the phase $\theta _{c}$ ($\theta _{m}$) of the OPA (MPA), there is only a phase shift in $t$ for $E_{\mathcal {N}}$, and its maximum value remains unchanged. This result indicates that under the double-driven case, the COM entanglement can be merely modulated by adjusting the amplitude of the OPA or MPA, and it is not sensitive to their phases, which is consistent with the previous discussion of the field amplitude $|\langle a(t)\rangle |$ in Fig. 1(c). Moreover, as shown in Figs. 2(c) and 2(f), we also plot the logarithmic negativity $E_{\mathcal {N},max}$ as a function of the scaled optical detuning $\Delta _{c}/\omega _{m}$ for different values of the field amplitude $G_{c}$ ($G_{m}$). It is found that in comparison with that of a conventional COM system, the COM entanglement can be enhanced by $\textrm {1.16}$ or $\textrm {1.3}$ times when applying OPA or MPA modulation, respectively.

 

Fig. 2. The COM entanglement under separate modulation of OPA or MPA. (a), (d) $E_{\mathcal {N},max}$ versus the scaled optical (mechanical) detuning $\Omega _{c}/\omega _m$ ($\Omega _{m}/\omega _m$) and the scaled amplitude of OPA (MPA) $G_c/\omega _{m}$ ($G_m/\omega _{m}$). (b), (e) $E_{\mathcal {N}}$ as a function of time $t/\tau$ for different phase modulations of OPA or MPA. (c), (f) $E_{\mathcal {N},max}$ versus the optical detuning $\Delta _c/\omega _m$ with or without OPA (MPA) modulation.

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In Fig. 3, we further explore the behavior of COM entanglement under the three-field driving case (say, with the standard optical pump laser and the OPA as well as the MPA). For this purpose, we plot the dependence of the logarithmic negativity $E_{\mathcal {N},max}$ on the ratio of $\Omega _m/\Omega _c$ and the ratio of $G_m/G_c$ in Fig. 3(a), with $\Delta _{c}/\omega _{m}=1$, $\Omega _c/\omega _m=1.18$, and $G_c/\omega _m=0.01$. It is seen that for the ratio of $G_{m}/G_{c}$, $E_{\mathcal {N},max}$ is a monotonically increasing function, while for the ratio of $\Omega _{m}/\Omega _{c}$, $E_{\mathcal {N},max}$ is characterized by a sharp peak with maximum value $E_{\mathcal {N},max}=0.4$ around the optimal optical detuning $\Omega _{m}/\Omega _{c}=1.8$. Figure 3(b) shows $E_{\mathcal {N},max}$ as a function of the phase difference $\Delta \theta$ in a polar coordinate, with $\Delta _{c}/\omega _{m}=1$, $G_{m}/G_{c}=1$, and $\Omega _m/\Omega _c=0.95$, $1$, and $1.8$. Here the phase difference $\Delta \theta$ is defined by $\Delta \theta \equiv \theta _{m}-\theta _{c}$. It is seen that $E_{\mathcal {N},max}$ becomes tunable with the variation of the phase difference $\Delta \theta$, which is different from that of the double-driven case discussed previously. From these results, one can intuitively find that when simultaneously applying both OPA and MPA modulations to the COM system, the ratio of their modulation parameters and the phase difference start to play an important role in the manipulation of COM entanglement. To clearly see this, we further present the dependence of the logarithmic negativity $E_{\mathcal {N},max}$ on the phase difference $\Delta \theta$ and the ratio of $G_{m}/G_{c}$ in Figs. 3(d) and 3(e). Interestingly, it is seen that when choosing proper parameters of the OPA and MPA, one can achieve an enhanced and controllable COM entanglement. Figure 3(c) shows $E_{\mathcal {N}}$ as a function of time $t$ for $\Omega _m/\Omega _c=0.95$ and $\Omega _m/\Omega _c=1$, respectively. It is found that to achieve such enhancement of COM entanglement, one should also choose a proper ratio for $\Omega _m/\Omega _c$. Finally, we plot the logarithmic negativity $E_{\mathcal {N},max}$ versus the bath temperature $\textrm {T}$ of the mechanical mode with respect to the standard case, double-driven case, and three-field driving case. One can find that for the same value of bath temperature $\textrm {T}$, $E_{\mathcal {N},max}$ of the three-field driving case can reach a much higher value than that of the other cases, indicating that the COM entanglement tends to become more robust against thermal noises by exploiting the interplay of OPA and MPA.

 

Fig. 3. The COM entanglement under simultaneous modulation of OPA and MPA. (a) $E_{\mathcal {N},max}$ versus the detuning frequency ratio $\Omega _m/\Omega _c$ and the amplitude ratio $G_m/G_c$, where $\Omega _c/\omega _m=1.18$ and $G_c/\omega _m=0.01$. (b) $E_{\mathcal {N},max}$ as a function of the phase difference in polar coordinates, with $\Omega _m/\Omega _c= 0.95$, $1$ or $1.8$. (c) $E_\mathcal {N}$ as a function of time $t/\tau$ with $\Omega _m/\Omega _c= 0.95$ or $1$. (d) $E_{\mathcal {N},max}$ versus the optical detuning $\Delta _c/\omega _m$ with or without co-modulated modulation. (e) $E_{\mathcal {N},max}$ versus the amplitude ratio $G_m/G_c$ and the phase difference $\Delta \theta$. (f) $E_{\mathcal {N},max}$ versus temperature $T$ under four modulation conditions. Among them, OMPA represents the simultaneous existence of MPA and OPA.

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4. Conclusions

In summary, by combining the powerful means of OPA and MPA modulation, we have studied how to generate, manipulate, and enhance the COM entanglement in a single COM resonator under three-field driving. Specifically, we show that the interplay of OPA and MPA allows us to regulate the field amplitude of the cavity mode by tuning their phase difference and strength, which thus provides an efficient and flexible way to manipulate the COM entanglement. More interestingly, by choosing the proper parameters for OPA and MPA, a considerable enhancement of the COM entanglement can also be achieved. Besides, we also find that compared to the standard case and the double-driven case, the COM entanglement with respect to the three-field driving case is more robust against the thermal noises. As such, we believe that our work, serving as a powerful tool to coherently manipulate the light-motion interaction through controlling the interplay of OPA and MPA, holds the promise to be useful for the manipulation of a variety of quantum effects based on COM system, such as photon blockade or anti-bunching effect [23–26], optical or mechanical squeezing [32,33], and mechanical ground-state cooling [16–18].