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Abstract
The dissipative squeezing mechanism is an effective method to generate the strong squeezing, which is important in the precision metrology. Here, we propose a practical method to achieve arbitrary bosonic squeezing via introducing frequency modulation into the coupled harmonic resonator model. We analyze the effect of frequency modulation and give the analytical and numerical squeezing results, respectively. To measure the accurate dynamic squeezing in our proposal, we give a more general defination of the relative squeezing degree. Finally, the proposed method is extended to generate the strong mechanical squeezing (>3 dB) in a practical optomechanical system consisting of a graphene mechanical oscillator coupled to a superconducting microwave cavity. The result indicates that the strong mechanical squeezing can be effectively achieved even when the mechanical oscillator is not initially in its ground state. The proposed method expands the study on nonclassical state and does not need the bichromatic microwave driving technology.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Squeezing is a famous nonclassical phenomenon in quantum mechanics [1], where the uncertainty of a quadrature component is less than the zero-point level whereas the orthogonal other one is larger to obey the Heisenberg uncertainty principle. Due to the property of less uncertainty for a measurable component, squeezing can be applied to the ultrahigh precision metrology [2,3], e.g., the current gravitational-wave detection [4]. Besides, in the area of continuous variable quantum-information science, squeezing is also an important resource [5]. So how to generate the strong squeezing is always the subject of numerous researchers discussing.
Currently, most schemes to generate squeezing are mainly divided into two categories: various nonlinearities [6–8] and coherent parametric drivings [9–15]. The nonlinear schemes include the nonlinear medium [16], nonlinear coupling [17,18], intrinsic nonlinearity [19], etc. Due to those nonlinearities being weak, the strong squeezing is usually difficult to generate via the nonlinear schemes. On the other hand, kinds of coherent parametric driving schemes have been proposed to generate squeezing, and some of them can be simplified into the modulation of the spring constant via analyzing. When the frequency of the modulated spring constant is twice the eigenfrequency [9,20,21], the squeezing is achieved. Furthermore, there is another coherent parametric driving mechanism to generate squeezing, i.e., bichromatic microwave driving [22–24], which is called as the dissipative squeezing mechanism or reservoir engineering scheme, and can generate the ultrastrong squeezing via dissipation [25–28]. In this mechanism, the bichromatic microwave driving induces some different sidebands, which eventually results in the anisotropic Rabi-type interaction. Accordingly, the steady-state strong squeezing is generated near the boundary of the stable region. Here, what we consider is that whether there is another way to generate the strong squeezing without relying on the bichromatic microwave driving technology. Fortunately, we find that, if a periodic frequency modulation is introduced, the system will produce different Floquet sidebands, whose coupling strengths are also independently tunable [29]. This seems to meet the requirements of generating squeezing via the dissipative mechanism [30]. Over the last several years, some interesting researches related to the frequency modulation have been proposed whether in theory [31–38] or in experiment [39–42], such as quantum state transfer [43], generating the Schrödinger-cat states [44,45], optomechanical cooling [46], entanglement [47], etc. Among them, Ref. [38] has reported the realization of mechanical squeezing with frequency modulation by utilizing the inherent Duffing nonlinearity, which does not belong to the dissipative squeezing mechanism.
In this paper, we focus on exploring an alternative way to induce the anisotropic Rabi-type interaction [48] that is the key to generate the bosonic squeezing and discussing the effect of the modulated parameters on the squeezing degree. Different from the previous dissipative squeezing schemes, the proposed method does not require the bichromatic microwave driving and additional laser control. We first derive the anisotropic Rabi-type interaction under the rotating wave approximation (RWA) and give the analytical results of the bosonic squeezing. Then, we numerically calculate the exact results without any approximations to verify our analyses. Comparing the above results with and without RWAs, we find that the RWA method erases the important information of the bosonic squeezing direction. Since the traditional relative squeezing degree (RSD) is inconvenient to evaluate the squeezing case in our proposal, we redefine a more general one, which is coincident with the traditional definition but independent of the squeezing direction, to measure the accurate dynamic squeezing. In addition, we discuss the effect of the modulation parameters on the bosonic squeezing and analyze the reasons in detail. Resorting to the natural advantages of the optomechanical oscillator in ultrahigh precision metrology, we apply the proposed method to generating the strong mechanical squeezing ($>3\,\mathrm{dB}$) with the experimentally realizable parameters in a practical optomechanical system, where the mechanical oscillator does not need to be initially cooled to its ground state. Compared with the bichromatic microwave driving method, our proposal does not need the additional laser control and would be easier to implement in experiment.
The paper is organized as follows: In Sec. 2., we derive the anisotropic Rabi-type interaction from the usual Rabi-type interaction via the frequency modulation. In Sec. 3., we first give the analytical expression of the bosonic squeezing under RWA and numerical simulation, respectively. Then we give a more general definition of the RSD and discuss the effect of the modulation parameters on the bosonic squeezing. In Sec. 4., as an application of the proposed method, we present to generate the strong mechanical squeezing in an optomechanical system. Lastly, a conclusion is given in Sec. 5.
2. Model
We consider a generic coupled harmonic resonator model which describes the interaction between two bosonic modes. The Hamiltonian is written as
In the dissipative squeezing mechanism, the anisotropic Rabi-type interaction, $H_{\mathrm {ani}}=\lambda _{-}(a^{\dagger }b+ab^{\dagger })+\lambda _{+}(a^{\dagger }b^{\dagger }+ab)$ (here $\lambda _{-}\neq \lambda _{+}$), is usually constructed by the bichromatic microwave driving technology [22–28]. To better understand the dissipative squeezing mechanism, we apply the squeezing transformation $S(\zeta )=\exp [\zeta /2(a^2-a^{\dagger 2})]$ to the anisotropic Hamiltonian, $S^{\dagger }(\zeta )H_{\mathrm {ani}}S(\zeta )=\sqrt {\lambda _{-}^{2}-\lambda _{+}^{2}}(a^{\dagger }b+ab^{\dagger })$. Here $\zeta =\tanh ^{-1}(\lambda _{+}/\lambda _{-})$ is the corresponding squeezing parameter and it reaches its maximum when $\lambda _{+}$ is close to $\lambda _{-}$ without the influence of the system environment. The transformed Hamiltonian is the standard beam-splitter interaction, which indicates that the transformed bosonic mode (Bogoliubov mode) can be cooled through the decay of the other bosonic mode. Namely, the original bosonic mode can be prepared in the squeezing-vacuum sate. It is worth noting that the above results is also right in the usual quantum Rabi model.
To construct the anisotropic Rabi-type interaction, we apply the frequency modulation to both the bosonic modes, giving
3. General results
3.1 Analytical solution
According to the constructed Hamiltonian (5) and the dissipative squeezing mechanism, it is easy to give the squeezing parameter in our proposal, $\zeta =\tanh ^{-1}[J_{-1}(\xi _{+})/J_{0}(\xi _{-})]$, which is related to the modulation amplitudes. To study the squeezing of bosonic mode in an actual system which includes decay terms, the quantum Langevin equations [49] of the system are given by the RWA Hamiltonian (5),
Before proceeding to study the generation of bosonic mode squeezing, we first analyze the stability of the system, which can be determined through the Routh-Hurwitz criterion [50] due to the fact that the dynamical equations are linear and time-invariant, with the result being $\lambda _{-}^{2}-\lambda _{+}^{2}\geqslant \kappa \gamma /4$. That is to say, the coupling strength of the rotating-wave term is always bigger than that of the counter-rotating-wave term to ensure that the system is stable and the squeezing-vacuum state of the bosonic mode can be generated. Additionally, since the dynamic equations of the quadrature components are related to the stochastic quantum fluctuation noise, they are unsuited to describe the quantum effects of the system directly. However, we can introduce the correlation matrix $V$ of the quadrature components to calculate the bosinic mode squeezing in our research. The correlation matrix element is defined by $V_{ij}=\langle u_{i}u_{j}+u_{j}u_{i}\rangle /2$. Thus the dynamical equation of the correlation matrix can be given as
where $D=\mathrm {diag}[\kappa /2,\kappa /2,\gamma /2,\gamma /2]$ is the diagonal diffusion matrix, which is related to the autocorrelation functions of noise operations.The direct analytical solution of dynamics in Eq. (8) is too cumbersome to be written here. However, under the stable condition of the system, the steady-state analytical solution of Eq. (8) can be given with a simplify expression. According to the definition of the quadrature components variance, i.e., $\langle \delta X^{2}\rangle =\langle X^{2}\rangle -\langle X\rangle ^{2}$ and $\langle \delta Y^{2}\rangle =\langle Y^{2}\rangle -\langle Y\rangle ^{2}$, the variance can be directly obtained from the steady-state analytical solution of the correlation matrix and is written as
3.2 Numerical simulation
Although the dynamical solution of Eq. (8) can be obtained via numerical simulation, it is still an approximate solution because RWA is used in the computational procedure. To study the exact dynamical behavior of the system, we need to simulate the dynamical equation of the initial Hamiltonian $H_{\mathrm {full}}=H+H_{M}$. Based on this, we recalculate the dynamical equation of the correlation matrix based on the initial Hamiltonian $H_{\mathrm {full}}$ without performing RWA through similar calculations. The form of the dynamical equation is the same to Eq. (8), and the only difference is that the matrix $M$ needs to be changed to
Fig. 1. The dynamical evolutions of the quadrature component variances for the full Hamiltonian or the RWA Hamiltonian. The parameters of system are chosen as $\omega _{a}=\omega _{b}$, $\kappa =10^{-6}\omega _{a}$, $\gamma =10^{-2}\omega _{a}$, $\lambda =10^{-2}\omega _{a}$, $\xi _{1}=-0.5$, $\xi _{2}=-0.5$, and $\nu =2\omega _{a}$.
In order to clearly show the change of the squeezing direction, we utilize the Wigner function in the phase space, as shown in Fig. 2, where Fig. 2(a-d) respectively correspond to the points $A$, $B$, $C$, and $D$ at different times in Fig. 1. The squeezing direction $\theta$ corresponding to these four points is $\pi /2$, $\pi /4$, $0$, and $-\pi /4$, respectively. The result indicates that, for the full Hamiltonian, the squeezing direction changes periodically with a clockwise rotation following the time evolution of the system. However, for the RWA Hamiltonian, the bosonic mode squeezing is fixed on a single direction of the quadrature component $X$ for the given parameters, which can be seen form the red dotted line in Fig. 1. That means the information of the squeezing direction will be erased when the RWA method is applied.
Fig. 2. The Wigner function for the bosonic mode at different times, which respectively correspond to the points $A$, $B$, $C$, and $D$ in Fig. 1. The parameters of system are the same as given in Fig. 1.
4. Discussion
4.1 Relative squeezing degree
In the above discussion, we already know that the RWA erases the information of the squeezing direction, but the squeezing degree is almost unchanged, which can be roughly seen from Fig. 1. In most of those researches related to squeezing, the squeezing degree is usually measured by the RSD, which is defined by $r=-10\log _{10}(\langle \delta O^{2}\rangle /\langle \delta O^{2}\rangle _{\mathrm {vac}})$, with $\langle \delta O^{2}\rangle _{\mathrm {vac}}$ being the variance of quadrature component $O$ when the system is in the vacuum state [51]. The RSD has a so-called $r=3\,\mathrm {dB}$ limit, i.e., the variance of quadrature component belows half of the zero-point level, which is to satisfy the requirements of ultrahigh-precision measurement.
However, since the squeezing direction is constantly changing, the definition of the above RSD is no longer applicative in our work. This is because it is difficult to determine the squeezing direction timely, i.e., to determine the squeezed quadrature component. In order to measure the squeezing degree in our research, therefore, we need to redefine the RSD, where the key is to determine the physical quantities independent of the squeezing direction. According to the correlation matrix $V$, the information of the bosonic mode is only related to the $2\times 2$ block matrix $V_{b}$, which is located in the upper left corner of $V$ and written as
We can find that the eigenvector of the matrix $V_{b}$, which corresponds to the minimum eigenvalue, is the squeezed quadrature component of the bosonic mode and the minimum eigenvalue is the variance of the squeezed quadrature component. Whereas the eigenvector corresponding to the maximum eigenvalue is the other component which is orthogonal to the squeezed one. Furthermore, since the vacuum variance of any quadrature component is $1/2$, we thus redefine the RSD as where $\min [\mathrm {eig}(V_{b})]$ represents the minimum eigenvalue of the matrix $V_{b}$. For convenience, we call $r$ in Eq. (13) as RRSD hereafter. In Fig. 3, we plot the dynamical behavior of the conventional RSD definition for the RWA Hamiltonian (see the blue dashed line). Meanwhile, the dynamical result of the RRSD for the full Hamiltonian is also given in Fig. 3 (see the red solid line), which agrees very well with the result of the RWA Hamiltonian. It is worth noting that the slight difference between both the results comes from the RWA, not the definition of the different RSDs. Now, we have given the formula of the RRSD, which is convenient to measure the squeezing degree when the squeezing direction is unfixed. Next, we mainly utilize the RRSD to discuss our work with the initial full Hamiltonian, which does not have any approximations.Fig. 3. The dynamical evolutions of the RSD and RRSD for the RWA Hamiltonian and the full Hamiltonian, respectively. The parameters of system are the same as given in Fig. 1.
4.2 Effect of modulation parameters
In the previous discussion, the modulation parameters, mainly the normalized modulation amplitudes $\xi _{1}$ and $\xi _{2}$, are chosen as a set of fixed values. Here, we study the effect of variable modulation parameters on the bosonic mode squeezing and discuss the condition of generating the bosonic mode squeezing.
Firstly, according to the previous calculation, we can show the squeezing parameter $\zeta$ changing with the normalized modulation amplitudes $\xi _{1}$ and $\xi _{2}$ in Fig. 4(a). We can see that the squeezing parameter value can be positive (see the red region) or negative (see the blue region), which corresponds to the bosonic mode squeezing of the quadrature component $X$ or $Y$, respectively. That is because the squeezing parameter comes from the RWA Hamiltonian and the squeezing direction is fixed on quadrature component either $X$ or $Y$. Moreover, the diagonal white region shows that the squeezing parameter is close to 0 and the bosonic mode squeezing disappears. The reason is that these diagonal region corresponds to the zero point of $J_{-1}(\xi _{+})$ and the counter-rotating-wave interaction disappears. Although the squeezing parameter can also reveal the squeezing degree, it is still not as intuitive as the RSD. Thus, we calculate the RRSD with the full Hamiltonian and show the relationship with regard to the normalized modulation amplitudes $\xi _{1}$ and $\xi _{2}$ in Fig. 4(b). We can see that the RRSD increases with a pair of larger modulation amplitudes, which is consistent with the previous analytical results. But the squeezing result is still not ideal due to it is hard to implement the excessively large modulation frequency in experiments. Next, we are committed to study the strong mechanical squeezing ($>3\mathrm {dB}$) based on the frequency modulation method in a general optomechanical system, which does not has the difficulty to achieve in experiments.
Fig. 4. (a) The squeezing parameter $\zeta$ versus both the normalized modulation amplitudes $\xi _{1}$ and $\xi _{2}$. (b) The RRSD $r$ versus both the normalized modulation amplitudes $\xi _{1}$ and $\xi _{2}$.
4.3 Application in optomechanics
In the above sections, we have studied how to construct the anisotropic Rabi-type interaction Hamiltonian via the frequency modulation method and discussed the effect of modulation parameters on the bosonic mode squeezing in detail. We now turn to study the applications in the actual system. As we all know, due to the property of decreasing and suppressing the fluctuation efficiently, squeezing has an important application in the area of the precision metrology. On the other hand, the nanomechanical oscillator with high quality is important and helpful to ultrahigh precision metrology, such as the detection of weak force or weak signal in the actual experiment. To this end, here we study the mechanical squeezing in an optomechanical system, which consists of a nanomechanical oscillator coupled to a microwave optical mode [52–57]. As shown in Fig. 5, the mechanical mode is acted by the graphene mechanical oscillator, where the modulation frequency can be realized by tuning the voltage of the gate electrode for the graphene [40–42]. And the optical mode is acted by the superconducting microwave cavity, which couples to the graphene [58]. Here the frequency of superconducting microwave cavity can be modulated by tuning the magnetic flux.
Fig. 5. Schematic of the superconducting optomechanical system. The PMMA is the polymethyl methacrylate.
Before proceeding to study the generation of the mechanical squeezing, we first analyze the characteristics of the proposed optomechanical system. In most experiments [40–42], the frequency of a graphene oscillator is about megahertz, which results in a high thermal distributed phonon number even in low temperature, e.g., the thermal phonon number $n_{\mathrm {th}}\simeq 10^{2}$ when the mechanical frequency $\omega _{m}=2\pi \times 30\,\mathrm {MHz}$ and the temperature $T=15\,\mathrm {mK}$. However, the frequency of microwave cavity is almost gigahertz ($\omega _{a}\sim \mathrm {GHz}$). To generate the mechanical squeezing, therefore, the nanomechanical oscillator should be high quality ($\gamma _{m}=\omega _{m}/10^{6}$) and first be cooled. Moreover, the coupling between the optical and mechanical modes is usually too weak to manipulate the nanomechanical oscillator. So the microwave optical cavity needs to be driven by a strong pump field in the actual experiment. According to the standard linearization process, we can derive the linearized Hamiltonian, $H_{L}=\Delta _{c}a^{\dagger }a+\Omega _{m}b^{\dagger }b-G\left (a^{\dagger }+a\right )\left (b^{\dagger }+b\right )$, where $\Delta _{c}=\delta _{c}+\xi _{1}\nu \cos (\nu t)/2$ ($\delta _{c}$ is the cavity pump field detuning), $\Omega _{m}=\omega _{m}+\xi _{m}\nu \cos (\nu t)/2$, and $G$ is the pump enhanced optomechanical coupling strength. Here, for the dynamical results, we have assumed that the $\langle b^{\dagger }b\rangle (t=0)=n_{\mathrm {th}}$, which corresponds to the RRSD $r(t=0)\simeq -23\,\mathrm {dB}$ and the autocorrelation function of the mechanical noise operator is $\langle b_{\mathrm {in}}(t)b_{\mathrm {in}}^{\dagger }(t^{\prime })\rangle =(n_{\mathrm {th}}+1)\delta (t-t^{\prime })$. The dynamical evolutions of the quadrature component variances and the RRSD for the nanomechanical oscillator are given via numerical simulation, as shown in Fig. 6, where the quadrature component variances are respectively defined as $q=(b^{\dagger }+b)/\sqrt {2}$ and $p=i(b^{\dagger }-b)/\sqrt {2}$. We can see that, after a long time of evolution, the quadrature component variances can be greatly reduced and the RRSD up to $8.12\,\mathrm {dB}$. It is worth noting that the squeezing direction of the mechanical oscillator is also constantly changing according the previous analysis, see the insert in Fig. 6.
Fig. 6. The dynamical evolutions of the quadrature component variances and the RRSD for the nanomechanical oscillator. The parameters of system are chosen as $\omega _{m}=2\pi \times 30\,\mathrm {MHz}$, $T=15\,\mathrm {mk}$, $\delta _{c}=\omega _{m}$, $\kappa =2\pi \times 0.3\,\mathrm {MHz}$, $\gamma _{m}=2\pi \times 30\,\mathrm {Hz}$, $G=2\pi \times 0.3\,\mathrm {MHz}$, $\xi _{1}=-2.8$, $\xi _{m}=-0.5$, and $\nu =2\pi \times 60\,\mathrm {MHz}$.
As the previous analysis and the display in Fig. 7, we also study the RRSD of the mechanical oscillator changing with the normalized modulation amplitudes $\xi _{1}$ and $\xi _{m}$. The white regions show that the mechanical squeezing disappears with $r=0\,\mathrm {dB}$. That is because these regions correspond to the zero point of $J_{-1}(\xi _{1}/2+\xi _{m}/2)$ and the counter-rotating-wave interaction disappears. It is worth noting that the large cavity modulation amplitude ($|\xi _{1}|\sim 10$) is not really too big for the microwave cavity because the frequency of the optical cavity is usually thousands of times the mechanical oscillator in a general optomechanical system and $\delta _{c}=\omega _{m}$. On the other hand, the mechanical frequency modulation is enough weak compared to its eigenfrequency in the numerically simulation. Moreover, we find that, even if we cut off the frequency modulation of the mechanical oscillator ($\xi _{m}=0$), the strong mechanical squeezing can still be achieved at the same temperature ($T=15\,\mathrm {mK}$), for example, when setting $\xi _{1}=-2.6$ and $\xi _{m}=0$, the RRSD $r=7.46\,\mathrm {dB}$. It will be more feasible in experiment due to the fact that the mechanical oscillator does not need any modulation and the modulation amplitude of the microwave cavity is small enough to its eigenfrequency.
Fig. 7. The RRSD $r$ of the mechanical oscillator versus both the normalized modulation amplitudes $\xi _{1}$ and $\xi _{m}$. The blank region corresponds to the unstable region of the system.
5. Conclusions
In conclusion, we have proposed an alternative method to generate the strong bosonic mode squeezing based on the dissipative mechanism, which is achieved via modulating frequency in a general coupled harmonic resonator model. Firstly, utilizing the frequency modulation method, we construct the anisotropic Rabi-type interaction, which is the key to generate squeezing via dissipation. Then, in the stable region, we respectively calculate the bosonic squeezing through the analytical and numerical methods, in which the calculated results agree well with each other and are also consistent with our analyses. By comparing these results, we find that the RWA erases the information of the change of squeezing direction, which results in that the squeezing direction is fixed to one of a pair of quadrature components, i.e., $X$ or $Y$. Whereas the squeezing direction is constantly changing for the actual system with the initially full Hamiltonian. Since the conventional method of calculating the dynamical RSD is unfeasible, we give a more general definition of the RSD, i.e., RRSD, which is convenient to describe the dynamical squeezing process. Moreover, we study the effect of modulation parameters on the squeezing in the stable region and discuss the physical reason. Finally, as an application of the proposed method, we discuss the squeezing generation of a graphene mechanical oscillator with the experimentally feasible parameters. We also find that the strong mechanical squeezing ($7.46\,\mathrm {dB}$) can be achieved at a high temperature ($15\,\mathrm {mK}$) even when the frequency modulation of the graphene mechanical oscillator is cut off, which will be helpful to improve the measurement accuracy greatly for some weak forces or weak signals in metrology.
Funding
National Natural Science Foundation of China (61822114; 61575055; 11874132).
Disclosures
The authors declare no conflicts of interest.
References
1. D. F. Walls and G. J. Milburn, Quantum optics (Springer Science & Business Media, 2007).
2. C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, “On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. i. issues of principle,” Rev. Mod. Phys. 52(2), 341–392 (1980). [CrossRef]
3. M. F. Bocko and R. Onofrio, “On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress,” Rev. Mod. Phys. 68(3), 755–799 (1996). [CrossRef]
4. LIGO scientific collaboration and Virgo collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116(6), 061102 (2016). [CrossRef]
5. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005). [CrossRef]
6. L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57(20), 2520–2523 (1986). [CrossRef]
7. H. Shi and M. Bhattacharya, “Quantum mechanical study of a generic quadratically coupled optomechanical system,” Phys. Rev. A 87(4), 043829 (2013). [CrossRef]
8. D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. 6(1), 38559 (2016). [CrossRef]
9. A. Mari and J. Eisert, “Gently modulating optomechanical systems,” Phys. Rev. Lett. 103(21), 213603 (2009). [CrossRef]
10. S. L. Ma, P. B. Li, A. P. Fang, S. Y. Gao, and F. L. Li, “Dissipation-assisted generation of steady-state single-mode squeezing of collective excitations in a solid-state spin ensemble,” Phys. Rev. A 88(1), 013837 (2013). [CrossRef]
11. Z. C. Zhang, Y. P. Wang, Y. F. Yu, and Z. M. Zhang, “Quantum squeezing in a modulated optomechanical system,” Opt. Express 26(9), 11915–11927 (2018). [CrossRef]
12. H. Tan, G. Li, and P. Meystre, “Dissipation-driven two-mode mechanical squeezed states in optomechanical systems,” Phys. Rev. A 87(3), 033829 (2013). [CrossRef]
13. J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83(3), 033820 (2011). [CrossRef]
14. C. G. Liao, R. X. Chen, H. Xie, M. Y. He, and X. M. Lin, “Quantum synchronization and correlations of two mechanical resonators in a dissipative optomechanical system,” Phys. Rev. A 99(3), 033818 (2019). [CrossRef]
15. C. H. Bai, D. Y. Wang, S. Zhang, S. Liu, and H. F. Wang, “Engineering of strong mechanical squeezing via the joint effect between duffing nonlinearity and parametric pump driving,” Photonics Res. 7(11), 1229–1239 (2019). [CrossRef]
16. G. S. Agarwal and S. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93(4), 043844 (2016). [CrossRef]
17. A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, “Cooling and squeezing via quadratic optomechanical coupling,” Phys. Rev. A 82(2), 021806 (2010). [CrossRef]
18. M. Asjad, G. S. Agarwal, M. S. Kim, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Robust stationary mechanical squeezing in a kicked quadratic optomechanical system,” Phys. Rev. A 89(2), 023849 (2014). [CrossRef]
19. X. Y. Lü, J. Q. Liao, L. Tian, and F. Nori, “Steady-state mechanical squeezing in an optomechanical system via duffing nonlinearity,” Phys. Rev. A 91(1), 013834 (2015). [CrossRef]
20. W. J. Gu and G. X. Li, “Squeezing of the mirror motion via periodic modulations in a dissipative optomechanical system,” Opt. Express 21(17), 20423–20440 (2013). [CrossRef]
21. C. H. Bai, D. Y. Wang, S. Zhang, S. Liu, and H. F. Wang, “Modulation-based atom-mirror entanglement and mechanical squeezing in an unresolved-sideband optomechanical system,” Ann. Phys. (Berlin, Ger.) 531(7), 1800271 (2019). [CrossRef]
22. A. Kronwald, F. Marquardt, and A. A. Clerk, “Dissipative optomechanical squeezing of light,” New J. Phys. 16(6), 063058 (2014). [CrossRef]
23. A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88(6), 063833 (2013). [CrossRef]
24. X. You, Z. Li, and Y. Li, “Strong quantum squeezing of mechanical resonator via parametric amplification and coherent feedback,” Phys. Rev. A 96(6), 063811 (2017). [CrossRef]
25. D. Porras and J. J. García-Ripoll, “Shaping an itinerant quantum field into a multimode squeezed vacuum by dissipation,” Phys. Rev. Lett. 108(4), 043602 (2012). [CrossRef]
26. J. M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, “Squeezing of quantum noise of motion in a micromechanical resonator,” Phys. Rev. Lett. 115(24), 243601 (2015). [CrossRef]
27. P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70(20), 205304 (2004). [CrossRef]
28. S. L. Ma, X. K. Li, J. K. Xie, and F. L. Li, “Two-mode squeezed states of two separated nitrogen-vacancy-center ensembles coupled via dissipative photons of superconducting resonators,” Phys. Rev. A 99(1), 012325 (2019). [CrossRef]
29. D. Y. Wang, C. H. Bai, S. Liu, S. Zhang, and H. F. Wang, “Optomechanical cooling beyond the quantum backaction limit with frequency modulation,” Phys. Rev. A 98(2), 023816 (2018). [CrossRef]
30. Y. Y. Zhang and X. Y. Chen, “Analytical solutions by squeezing to the anisotropic rabi model in the nonperturbative deep-strong-coupling regime,” Phys. Rev. A 96(6), 063821 (2017). [CrossRef]
31. M. P. Silveri, J. A. Tuorila, E. V. Thuneberg, and G. S. Paraoanu, “Quantum systems under frequency modulation,” Rep. Prog. Phys. 80(5), 056002 (2017). [CrossRef]
32. W. M. Zhang, K. M. Hu, Z. K. Peng, and G. Meng, “Tunable micro- and nanomechanical resonators,” Sensors 15(10), 26478–26566 (2015). [CrossRef]
33. J. Q. Liao, C. K. Law, L. M. Kuang, and F. Nori, “Enhancement of mechanical effects of single photons in modulated two-mode optomechanics,” Phys. Rev. A 92(1), 013822 (2015). [CrossRef]
34. F. Xue, Y. D. Wang, C. P. Sun, H. Okamoto, H. Yamaguchi, and K. Semba, “Controllable coupling between flux qubit and nanomechanical resonator by magnetic field,” New J. Phys. 9(2), 35 (2007). [CrossRef]
35. A. Farace and V. Giovannetti, “Enhancing quantum effects via periodic modulations in optomechanical systems,” Phys. Rev. A 86(1), 013820 (2012). [CrossRef]
36. T. Chen and Z. Y. Xue, “Nonadiabatic geometric quantum computation with parametrically tunable coupling,” Phys. Rev. Appl. 10(5), 054051 (2018). [CrossRef]
37. J. F. Huang, J. Q. Liao, and L. M. Kuang, “Ultrastrong jaynes-cummings model,” Phys. Rev. A 101(4), 043835 (2020). [CrossRef]
38. X. Han, D. Y. Wang, C. H. Bai, W. X. Cui, S. Zhang, and H. F. Wang, “Mechanical squeezing beyond resolved sideband and weak-coupling limits with frequency modulation,” Phys. Rev. A 100(3), 033812 (2019). [CrossRef]
39. M. Agarwal, H. Mehta, R. N. Candler, S. A. Chandorkar, B. Kim, M. A. Hopcroft, R. Melamud, G. Bahl, G. Yama, T. W. Kenny, and B. Murmann, “Scaling of amplitude-frequency-dependence nonlinearities in electrostatically transduced microresonators,” J. Appl. Phys. 102(7), 074903 (2007). [CrossRef]
40. C. Chen, S. Lee, V. V. Deshpande, G. H. Lee, M. Lekas, K. Shepard, and J. Hone, “Graphene mechanical oscillators with tunable frequency,” Nat. Nanotechnol. 8(12), 923–927 (2013). [CrossRef]
41. V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9(10), 820–824 (2014). [CrossRef]
42. P. Weber, J. Güttinger, I. Tsioutsios, D. E. Chang, and A. Bachtold, “Coupling graphene mechanical resonators to superconducting microwave cavities,” Nano Lett. 14(5), 2854–2860 (2014). [CrossRef]
43. X. Li, Y. Ma, J. Han, T. Chen, Y. Xu, W. Cai, H. Wang, Y. P. Song, Z. Y. Xue, Z. Q. Yin, and L. Sun, “Perfect quantum state transfer in a superconducting qubit chain with parametrically tunable couplings,” Phys. Rev. Appl. 10(5), 054009 (2018). [CrossRef]
44. J. Q. Liao, J. F. Huang, and L. Tian, “Generation of macroscopic schrödinger-cat states in qubit-oscillator systems,” Phys. Rev. A 93(3), 033853 (2016). [CrossRef]
45. B. Xiong, X. Li, S. L. Chao, Z. Yang, W. Z. Zhang, and L. Zhou, “Generation of entangled schrödinger cat state of two macroscopic mirrors,” Opt. Express 27(9), 13547–13558 (2019). [CrossRef]
46. M. Bienert and P. Barberis-Blostein, “Optomechanical laser cooling with mechanical modulations,” Phys. Rev. A 91(2), 023818 (2015). [CrossRef]
47. D. Y. Wang, C. H. Bai, S. Liu, S. Zhang, and H. F. Wang, “Manipulation of nanomechanical resonator via shaking optical frequency,” J. Phys. B: At., Mol. Opt. Phys. 52(4), 045502 (2019). [CrossRef]
48. Q. T. Xie, S. Cui, J. P. Cao, L. Amico, and H. Fan, “Anisotropic rabi model,” Phys. Rev. X 4(2), 021046 (2014). [CrossRef]
49. X. R. Xiong, Y. P. Gao, X. F. Liu, C. Cao, T. J. Wang, and C. Wang, “The analysis of high-order sideband signals in optomechanical system,” Sci. China: Phys., Mech. Astron. 61(9), 90322 (2018). [CrossRef]
50. E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987). [CrossRef]
51. C. S. Hu, Z. B. Yang, H. Wu, Y. Li, and S. B. Zheng, “Twofold mechanical squeezing in a cavity optomechanical system,” Phys. Rev. A 98(2), 023807 (2018). [CrossRef]
52. F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009). [CrossRef]
53. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]
54. D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016). [CrossRef]
55. Y. M. Liu, C. H. Bai, D. Y. Wang, T. Wang, M. H. Zheng, H. F. Wang, A. D. Zhu, and S. Zhang, “Ground-state cooling of rotating mirror in double-laguerre-gaussian-cavity with atomic ensemble,” Opt. Express 26(5), 6143–6157 (2018). [CrossRef]
56. J. H. Gan, Y. C. Liu, C. Lu, X. Wang, M. K. Tey, and L. You, “Intracavity-squeezed optomechanical cooling,” Laser Photonics Rev. 13(11), 1900120 (2019). [CrossRef]
57. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007). [CrossRef]
58. X. Song, M. Oksanen, J. Li, P. J. Hakonen, and M. A. Sillanpää, “Graphene optomechanics realized at microwave frequencies,” Phys. Rev. Lett. 113(2), 027404 (2014). [CrossRef]
