Open Access
Add to BibSonomy
Abstract
We propose a method to achieve a strong indirect interaction between two distant whispering-gallery-mode (WGM) resonators in a hybrid quantum system at room temperature, even when the distance between them exceeds 40 wavelengths. By exploiting the quantum critical point, we can greatly enhance both the effective damping rate and the coupling strengths between a WGM resonator and a low-frequency polariton. We introduce a large effective frequency detuning to suppress the effective damping rate while maintaining the enhanced coupling strength. The strong indirect interaction between separated WGM resonators is mediated by a far-off-resonant low-frequency polariton through virtual excitations in a process similar to Raman process. This proposal provides a viable approach to building a quantum network based on strongly coupled WGM resonators.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The physical system of coupled resonators has generated significant interest in a wide range of research fields, including fundamental physics [1–5] and quantum information processing [6–8]. This is due to the unique properties of oscillators, such as individual component addressability and interactions with various gain and nonlinear materials. A crucial requirement for these applications is the establishment of strong interactions among resonators, where energy exchange between two resonators occurs faster than energy dissipation from either resonator. The coupling strength decreases exponentially with the distance between separated resonators, as the resonant modes are tightly confined within each resonator [9–13]. In practical systems of coupled ultrahigh-Q resonators, achieving exact resonant conditions for WGM eigenstates is unlikely due to inevitable cavity size variations. The interaction between whispering-gallery mode (WGM) resonators and other physical systems has been extensively investigated for potential applications of WGM resonators in various fields [9,10,14–17]. Efficient coupling between detuned cavities is also difficult, as the spatial and spectral overlaps of eigenstates in adjacent resonators are expected to diminish [18–21]. Recent efforts have made progress towards achieving strong interaction between resonators under specific conditions [11,13,22–35]. For example, the hybridization of two mechanical modes in an optomechanical system requires a cryogenic temperature of approximately $\sim 100$ mk [25]. Low temperature is consistently necessary for effective interaction involving superconducting resonators [26,28,30]. Additionally, experimental realization of strong indirect coupling between distant nanocavities has been achieved by significantly modifying waveguide modes.
In an optomechanical cavity, the interaction between light and mechanical motion via radiation pressure has been extensively studied. Some interesting phenomena that have been explored include optomechanically induced transparency [36–38], cooling of a mechanical resonator to its ground state [39–41], and its applications in gravitational-wave detectors [42]. Apart from the commonly used Fabry-Perot resonators, WGM (whispering-gallery mode) resonators have also shown optomechanical coupling to mechanical modes of the structure [43–47]. To achieve strong coupling between light and mechanical motion, a strong driving field is typically employed [31,48]. This strong driving field not only enhances the linear optomechanical coupling but also leads to the emergence of quantum criticality with appropriate parameters [49]. Quantum criticality offers unique features that have been utilized in various applications, such as mass sensing [50], enhancing nonlinearity [49,51–55], and increasing coupling strength [52–55]. In this work, we demonstrate that the interaction between a WGM optical resonator and a low-frequency polariton mode of an optomechanical WGM resonator is significantly enhanced by quantum criticality. This enhancement can be attributed to the squeezing effect resulting from counter rotating terms [54,56]. Moreover, we show that a strong indirect interaction can be established between two well-separated WGM resonators under the condition of large frequency detuning at room temperature. This relaxes the traditional restrictions on the distance between coupled WGM resonators. As depicted in Fig. (1), greater spacing between the two WGM optical resonators is allowed since only an initial weak coupling is required between adjacent resonators. The two WGM optical resonators are positioned on both sides of the central WGM optomechanical resonator, and the strong indirect coupling is mediated by the low-frequency polariton of the central resonator. As a result, the well-separated distance between the two WGM optical resonators can exceed the diameter of the central resonator. Our scheme has potential application in optical quantum computing [57,58], and quantum sensing [59,60].
Fig. 1. Experimental Setup Simplified Schematic. A laser beam is precisely targeted at the edge of a deformed silica microsphere to stimulate its clockwise optical mode. In this configuration, the silica microsphere undergoes uniform expansion and contraction, akin to a specialized mechanical oscillator interacting with optical modes. Together, they form a unique type of WGM optomechanical resonator. The two WGM optical resonators are located on either side of the central WGM optomechanical resonator. Furthermore, the counter-clockwise optical modes of the other two WGM optical resonators are weakly coupled to the clockwise optical modes of the WGM optomechanical resonator. The intensity of the laser field driving the experiment can be detected and measured using a detector (D0).
2. Theoretical model
We consider a hybrid quantum system comprising a silica WGM optomechanical resonator $a$ (as proposed in [45–47]) and two WGM optical resonators $c_i(i=1,2)$, as illustrated in Fig. 1. The WGM optomechanical resonator is a silica microsphere with a $36$ µm diameter. In the central WGM optomechanical resonator, the optical modes interact with its mechanical breathing modes under weak optomechanical coupling conditions. In order to achieve a strong linear optomechanical coupling, we employ a high-power laser field at frequency $\omega _0$ to drive the clockwise optical modes. By doing so, we can safely neglect the weak nonlinear optomechanical coupling between the phonon mode and the counter-clockwise optical mode. Additionally, we establish weak couplings with coupling strength $g_i$ between the other two counter-clockwise optical modes of the WGM resonators and the clockwise optical mode of the central resonator. Throughout the rest of this article, modes $a$ and $c_i$ represent the clockwise and counter-clockwise optical modes of the WGM resonators, respectively. The Hamiltonian of the hybrid quantum system can be written as (setting $\hbar =1$) $H_t=H_C+H_{OM}+H_I+H_D$, the first term $H_C=\sum _{i=1}^2\omega _{ci} c^{\dagger} _i c_i$ represents the energy of the two WGM optical resonators with frequency $\omega _{ci}$, and $c_i(c_i^{\dagger} )$ is the corresponding annihilation (creation) operators. The second term $H_{OM}=\omega _a a^{\dagger} a+ \omega _b b^{\dagger} b- g_0 a^{\dagger} a(b^{\dagger} +b)$ is the Hamiltonian of an optomechanical resonator, where $a(a^{\dagger} )$ and $b(b^{\dagger} )$ are the annihilation (creation) operators of the optical mode with frequency $\omega _a$ and the mechanical mode with frequency $\omega _b$, respectively. The coupling rate $g_0$ describes the nonlinear optomechanical interaction between the optical modes and the mechanical breathing modes. The third term $H_I=\sum _{i=1}^2g_i(a^{\dagger} c_i+c^{\dagger} _i a)$ means the interaction between two WGM optical resonators and the optical mode of the central resonator with coupling strength $g_i$. The last term $H_D=\varepsilon (a^{\dagger} e^{-i\omega _0t} + ae^{i\omega _0t})$ shows that the optical mode of the central resonator is driven by a strong external laser field with amplitude $\varepsilon$, where $\varepsilon$ is related to the laser input power $P$ and the optical mode decay rate $\kappa _a$ by $\varepsilon =\sqrt {2P\kappa _a/\hbar \omega _0}$.
In a typical optomechanical system, the significant difference in the resonator frequencies hampers the coupling between the optical and mechanical modes. However, by exciting the optical mode with a powerful external laser field, the frequency discrepancy can be substantially reduced, enabling strong optomechanical coupling. This strong coupling generates a linear coupling effect, giving rise to quantum criticality attributed to the squeezing effect resulting from counterrotating terms. In a frame rotating with frequency $\omega _0$, we employ the displacement $(a\rightarrow \langle a\rangle +a, b\rightarrow \langle b\rangle +b, c_i\rightarrow \langle c_{i}\rangle +c_i)$ to linearize the system [38,61] (see the appendix for details) and derive the steady-state value $\langle a\rangle =\varepsilon /(i\kappa _a-\Delta _a)$ using the mean field approximation, considering the effective frequency detuning $\Delta _a = \omega _a-\omega _0 - \frac {2g^2_0}{\omega _b}\langle a^{\dagger} a\rangle$. Clearly, both the steady-state value $\langle a\rangle$ and the frequency detuning $\Delta _a$ can be regulated by adjusting the frequency $\omega _0$ and amplitude m of the external driving field. Additionally, the mean resonator photon number $\langle a^{\dagger} a\rangle$ can be significantly enhanced. Upon linearizing the system, the transformed Hamiltonian can be obtained as [62–65]:
The application of a strong external driving field leads to normal-mode splitting, indicating the presence of the strong coupling regime [66]. In order to examine the influence of quantum criticality on the interaction between distinct resonant modes, we diagonalize the Hamiltonian (1) using the Bogoliubov transformation $[a,a^{\dagger},b,b^{\dagger} ]^T=M[\mathcal {A}_-,\mathcal {A}_-^{\dagger},\mathcal {A}_+,\mathcal {A}_+^{\dagger} ]^T$. The resulting transformation matrix $M$ and the diagonalized Hamiltonian are
The parameter $\vartheta$ is determined by the relation $\tan (2\vartheta )=4G\sqrt {\Delta _a\omega _b}/(\Delta _a^2-\omega _b^2)$. The coupling strengths $\zeta _{\pm }(\zeta =g,\mu )$, describing the interaction between the WGM optical resonators $c_i$ and the polariton modes $\mathcal {A}_{\pm }$, are shown as
From Eq. (4), it is observed that a sufficiently strong linear optomechanical coupling g can result in a purely imaginary frequency $\omega _-$ and give rise to the quantum criticality of the subsystem [49,67], as depicted in Fig. 2. We further show the influence of different parameters on the quantum criticality in Figs. 2(c)–2(d). Due to the strong linear optomechanical coupling $G$, the counterrotating terms $(a^{\dagger} b^{\dagger} +ab)$ need to be taken into account. In fact, the remarkable characteristics of the quantum criticality predominantly arise from the squeezing effect associated with the counter-rotating terms [52,56]. As the linear optomechanical coupling $G$ approaches the critical value $G_{QC}$, the squeezing effect becomes significant, leading to the emergence of polaritons with exceptional properties within the linearized optomechanical subsystem.
Fig. 2. The frequencies of normal-mode versus the linear coupling $G/\omega _b$ with the effective frequency detuning (a) $\Delta _a/\omega _b=1$ and (c) $\Delta _a/\omega _b=1,5,10,15$. The frequencies of normal-mode versus the effective frequency detuning $\Delta _a/\omega _b$ with (b) $G/\omega _b=0.2$ and (d) $G/\omega _b=0.2,0.4,0.6,0.8$. In (c) and (d), solid lines represent the real part of the frequency $\omega _-$, while dashed lines represent the imaginary part of the frequency $\omega _-$. $G_{QC}=\frac {1}{2}\sqrt {\Delta _a\omega _b}$ and $\Delta _{QC}=4G^2/\omega _b$ mean the quantum criticality.
However, excessive driving can result in system instability. By applying the Routh-Hurwitz criterion [68], we can derive the stability condition for this system as follows:
As the coupling strength $G$ approaches the quantum critical point $G_{QC}$, there is a possibility of a significant interaction between the WGM optical resonators $c_i$ and the low-frequency polariton mode $\mathcal {A}_-$. Additionally, the effective damping rate may experience a substantial increase. The effective damping rates can be obtained using a Lindblad approach, and the system’s master equation reads [69,70]
To achieve strong interaction, it is crucial to effectively suppress the effective damping rate $\kappa _-$ while significantly enhancing the coupling strength $g_-$. However, near the quantum critical point, the effective damping rate $\kappa _-$ increases rapidly along with the coupling strength $\zeta _- (\zeta =g,\mu )$. By carefully analyzing Eqs. (5) and (9), we have discovered that the effective damping rate can be effectively suppressed while keeping the coupling strengths relatively unchanged by increasing the ratio of the effective frequency detuning $\Delta _a$ to the mechanical mode frequency $\omega _b$ around the quantum criticality. Our numerical simulation in Fig. (3) demonstrates that a higher ratio $\Delta _a/\omega _b$ ensures the establishment of a strong coupling regime. Figures 3(a)-3(b) illustrate that a higher ratio $\Delta _a/\omega _b$ leads to a much stronger interaction between the WGM optical resonators and the low-frequency polariton mode $\mathcal {A}_-$, while the WGM optical resonators become completely decoupled from the high-frequency polariton mode $\mathcal {A}_+$ simultaneously. Figure 3(c) shows a significant reduction in the effective damping rate $\mathcal {A}_-$ due to the high ratio $\Delta _a/\omega _b$. When maintaining a high ratio $\Delta _a/\omega _b>3$, the system enters the strong-coupling regime, denoted by $g_->\kappa _-,\kappa _c$ as presented in Fig. 3(d).
Fig. 3. The effective coupling strength versus the dimensionless parameter $(G_{QC}-G)/\omega _b$ with ratio (a) $\Delta _a/\omega _b=10$ and (b) $\Delta _a/\omega _b=50$; (c) The effective damping rate $\kappa _-$ versus the dimensionless parameter $(G_{QC}-G)/\omega _b$; (d) The effective coupling strength $g_-$ and damping rate $\kappa _-$ versus the effective frequency detuning $\Delta _a/\omega _b$ with $\omega _-/\omega _b=5\times 10^{-7}$.
3. Strong indirect coupling between two WGM optical resonators
In the previous analysis, we can safely neglect the high-frequency polariton mode $\mathcal {A}_+$ due to its weak interaction with the WGM optical resonators $c_i$. The interaction between the low-frequency polariton mode $\mathcal {A}_-$ and the WGM optical resonators $c_i$ is in the strong coupling regime, and the rotating-wave approximation remains valid. As a result, the counterrotating terms are omitted, and Hamiltonian (Eq. (3)) can be simplified to the following form:
For the simple case of $\Delta _1=\Delta _2=\Delta$, we analytically solve for the eigenvalues of the normal modes as
and the corresponding eigenvectors areThis indicates that, when there is a large detuning, only two of the three polariton modes undergo significant overlap, effectively isolating the third polariton mode $\mathcal {A}_-$. As a result, the third normal mode corresponding to $\mathcal {A}_-$ can be simplified and considered off-resonance from the other two normal modes. In a similar manner as the upper level in atomic Raman-type processes [71–73], we can adiabatically eliminate the normal mode $\mathcal {A}_-$. Consequently, our focus narrows down to the remaining two normal modes $\{\beta,\beta _+\}$, and the dynamic equations of modes $c_1$ and $c_2$ are determined by the derived matrix $M_{J2}$, as follows:
The two WGM optical modes $c_i(i=1,2)$ are both shifted by $g_-^2/\Delta$ and the effective coupling strength is also $g_-^2/\Delta$.
Numerical simulations indicate that the adiabatic approximation is valid with current experimental parameters. When considering decoherence, the dynamics of the three modes described by Hamiltonian (10) can be governed by a master equation:
Fig. 4. Rabi oscillations between two distant optical cavities $c_i(i=1,2)$. Normalized intensity of modes $c_i$ and $\mathcal {A}_-$ as functions of the evolution time with (a) $\Delta _2/g_-=10.5$, (b) $\Delta _2/g_-=10.1$, and (c) $\Delta _2/g_-=10$ for Hamiltonian (10); with (d) $\Delta _2/g_-=10$ for Hamiltonian (18). Other parameters are $\Delta _1/g_-=10$, $\kappa _-/g_-=0.1$, and $\kappa _{c1}/g_-=\kappa _{c2}/g_-=10^{-2}$.
Fig. 5. Hybridization between three modes $\mathcal {A}_-$ and $c_i(i=1,2)$. A distinct avoided level crossing between modes $c_i$ (red and blue lines)is observed. The green line corresponds to the mode $\mathcal {A}_-$. Other parameters are the same in Fig. (4).
Before concluding this section, let’s discuss some important points regarding our proposal. Firstly, our scheme is feasible in current experiments. For the coupled-WGM-resonator system, we can consider the parameters, $\omega _b\sim 2\pi \times 150$ MHz, $\gamma _b\sim 2\pi \times 60$ KHz, $\kappa _a\sim \kappa _c\sim 2\pi \times 15$ MHz, and the weak coupling strength $g\sim 12$ MHz, which are all experimentally achievable at room temperature [43–45,47]. Moreover, even in a $100-$µm diameter spherical WGM resonator, mechanical modes have been successfully excited at rates as high as X-band ($\sim 11$ GHz) rates [74]. Secondly, we can achieve strong coupling between distant WGM optical resonators, which are indirectly connected through a low-frequency polariton. The separation between the two WGM resonators exceeds the diameter ($\sim 36$ µm) of the central WGM optomechanical resonator. Despite this large distance exceeding $40$ wavelengths, strong coupling can still be achieved by utilizing the wavelength of approximately $800$ nm for the WGM optical mode.
Finally, the interaction between polaritons and WGM optical resonators plays a crucial role in establishing the indirect connection between the two WGM optical resonators. The Bogoliubov polaritons consist of photons and phonons originating from the central WGM optomechanical resonator. In our scheme, the high-frequency polaritons $\mathcal {A}_+$ are completely independent from the WGM optical resonators. Meanwhile, under the condition of a large frequency difference, the interaction between the low-frequency polaritons and the WGM resonators occurs in the strong coupling regime near the quantum criticality. This difference in coupling can be attributed to the distinct components of these two types of polaritons. From the transformation matrix $M$ (Eq. (2)), we can give the expression of polaritons as
Fig. 6. The relative intensity of the components of (a) the high-frequency polaritons $a_+$ and (b) the low-frequency polaritons $a_-$ versus the effective frequency detuning $\Delta _a/\omega _b$ with $\omega _-/\omega _b=5\times 10^{-7}$. Other parameters are $\kappa _a/\omega _b=0.1, \gamma _b/\omega _b=4\times 10^{-4}$, and $g_i/\omega _b=8\times 10^{-2}$.
4. Conclusion
In summary, our proposal introduces a scheme for achieving strong indirect coupling between distant WGM resonators, even when the distance exceeds $40$ wavelengths. We achieve this through the utilization of a WGM optomechanical resonator, which indirectly connects the two WGM resonators. The optomechanical resonator’s optical mode is excited by a high-powered laser field to greatly enhance the linear optomechanical coupling. Through the use of Bogoliubov transformation, the direct interaction between the WGM optomechanical resonator and the WGM optical resonators is transformed into an interaction between the Bogoliubov polaritons and the optical resonators. Consequently, the indirect interaction between the WGM resonators is mediated by low-frequency Bogoliubov polaritons via virtual excitations under significant frequency detuning. The findings of this study are predicted to serve as a fundamental building block for long-distance quantum information processing.
Appendix
In the interaction picture, the Hamiltonian of this system is ($\hbar =1$)
For the steady state, $\langle \dot {a} \rangle =\dot {\langle b\rangle } =\langle \dot {c_1} \rangle =\langle \dot {c_2} \rangle =0$. In a typical WGM resonator, the resonator frequency is much larger than its damping rate and coupling strength, i.e. $\delta _{ci}\gg \kappa _{ci}>g_i (i=1,2)$. After considering this simplification, we derive the solution of Eqs. (24) as below $\langle a \rangle =\frac {\varepsilon }{i\kappa _a-\Delta _a},\quad \langle b \rangle =-\frac {g_0\langle a^{{\dagger} }a \rangle }{i\gamma _b-\omega _b}, \quad \langle c_1 \rangle =\frac {g_1\langle a \rangle }{i\kappa _{c1}-\delta _{c1}}, \quad \langle c_2 \rangle =\frac {g_2\langle a \rangle }{i\kappa _{c2}-\delta _{c2}}$. According to the standard linearization procedure, the annihilation operator $A$ is decomposed into the sum of its steady-state mean $\langle A \rangle$ and the perturbation value $\delta A$, i.e. $a=\langle a\rangle +\delta a,\quad b=\langle b\rangle + \delta b,\quad c_1 = \langle c_1\rangle + \delta c_1,\quad c_2 = \langle c_2\rangle + \delta c_2$. Performing this substitution process on Eqs. (23), we get
Funding
Hubei Key Laboratory of Intelligent Geo-Information Processing (KLIGIP2022-C02); National Natural Science Foundation of China (41974059, 41830537); Ministry of Education Science and Technology Industry-University Cooperation and Education Project (202102575031).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
1. A. Xuereb, C. Genes, A. Dantan, et al., “Strong coupling and long-range collective interactions in optomechanical arrays,” Phys. Rev. Lett. 109(22), 223601 (2012). [CrossRef]
2. M. Ludwig and F. Marquardt, “Quantum many-body dynamics in optomechanical arrays,” Phys. Rev. Lett. 111(7), 073603 (2013). [CrossRef]
3. A. Tomadin, S. Diehl, M. D. Lukin, et al., “Reservoir engineering and dynamical phase transitions in optomechanical arrays,” Phys. Rev. A 86(3), 033821 (2012). [CrossRef]
4. A. D. Greentree, C. Tahan, J. H. Cole, et al., “Quantum phase transitions of light,” Nat. Phys. 2(12), 856–861 (2006). [CrossRef]
5. J. Cho, D. G. Angelakis, S. Bose, et al., “Fractional quantum hall state in coupled cavities,” Phys. Rev. Lett. 101(24), 246809 (2008). [CrossRef]
6. L. Zhou, Z. Gong, Y.-x. Liu, et al., “Controllable scattering of a single photon inside a one-dimensional resonator waveguide,” Phys. Rev. Lett. 101(10), 100501 (2008). [CrossRef]
7. D. Chang, A. H. Safavi-Naeini, M. Hafezi, et al., “Slowing and stopping light using an optomechanical crystal array,” New J. Phys. 13(2), 023003 (2011). [CrossRef]
8. M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, et al., “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]
9. L. Chang, X. Jiang, S. Hua, et al., “Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014). [CrossRef]
10. B. Peng, Ş. K. Özdemir, F. Lei, et al., “Parity–time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014). [CrossRef]
11. A. V. Kanaev, V. N. Astratov, W. Cai, et al., “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88(11), 111111 (2006). [CrossRef]
12. V. Ilchenko, M. Gorodetsky, S. Vyatchanin, et al., “Coupling and tunability of optical whispering-gallery modes: a basis for coordinate meter,” Opt. Commun. 107(1-2), 41–48 (1994). [CrossRef]
13. M. Notomi, E. Kuramochi, T. Tanabe, et al., “Large-scale arrays of ultrahigh-q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008). [CrossRef]
14. X. Zhou, H. Tamura, T.-H. Chang, et al., “Coupling single atoms to a nanophotonic whispering-gallery-mode resonator via optical guiding,” Phys. Rev. Lett. 130(10), 103601 (2023). [CrossRef]
15. M. Cai, O. Painter, K. J. Vahala, et al., “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85(1), 74–77 (2000). [CrossRef]
16. S.-X. Zhang, L. Wang, Z.-Y. Li, et al., “Free-space coupling efficiency in a high-q deformed optical microcavity,” Opt. Lett. 41(19), 4437–4440 (2016). [CrossRef]
17. S. V. Boriskina, “Coupling of whispering-gallery modes in size-mismatched microdisk photonic molecules,” Opt. Lett. 32(11), 1557–1559 (2007). [CrossRef]
18. T. Mukaiyama, K. Takeda, H. Miyazaki, et al., “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82(23), 4623–4626 (1999). [CrossRef]
19. Y. Hara, T. Mukaiyama, K. Takeda, et al., “Photonic molecule lasing,” Opt. Lett. 28(24), 2437–2439 (2003). [CrossRef]
20. K. A. Fuller, “Optical resonances and two-sphere systems,” Appl. Opt. 30(33), 4716–4731 (1991). [CrossRef]
21. H. Miyazaki and Y. Jimba, “Ab initio tight-binding description of morphology-dependent resonance in a bisphere,” Phys. Rev. B 62(12), 7976–7997 (2000). [CrossRef]
22. L. Chen, L. Jiang, C. Li, et al., “Modified dfs-based term weighting scheme for text classification,” Expert Syst. Appl. 168, 114438 (2021). [CrossRef]
23. J. Pan, S. Sandhu, Y. Huo, et al., “Experimental demonstration of an all-optical analogue to the superradiance effect in an on-chip photonic crystal resonator system,” Phys. Rev. B 81(4), 041101 (2010). [CrossRef]
24. J. D. Teufel, D. Li, M. Allman, et al., “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471(7337), 204–208 (2011). [CrossRef]
25. A. Shkarin, N. Flowers-Jacobs, S. Hoch, et al., “Optically mediated hybridization between two mechanical modes,” Phys. Rev. Lett. 112(1), 013602 (2014). [CrossRef]
26. T. Miyanaga, A. Tomonaga, H. Ito, et al., “Ultrastrong tunable coupler between superconducting l c resonators,” Phys. Rev. Appl. 16(6), 064041 (2021). [CrossRef]
27. F. Gong, X. Wang, L. Jiang, et al., “Fine-grained attribute weighted inverted specific-class distance measure for nominal attributes,” Inf. Sci. 578, 848–869 (2021). [CrossRef]
28. X. Han, C.-L. Zou, H. X. Tang, et al., “Multimode strong coupling in superconducting cavity piezoelectromechanics,” Phys. Rev. Lett. 117(12), 123603 (2016). [CrossRef]
29. Y. Sato, Y. Tanaka, J. Upham, et al., “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2012). [CrossRef]
30. V. Singh, S. Bosman, B. Schneider, et al., “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9(10), 820–824 (2014). [CrossRef]
31. G. Luo, Z.-Z. Zhang, G.-W. Deng, et al., “Strong indirect coupling between graphene-based mechanical resonators via a phonon cavity,” Nat. Commun. 9(1), 383 (2018). [CrossRef]
32. M. M Siskins, E. Sokolovskaya, M. Lee, et al., “Tunable strong coupling of mechanical resonance between spatially separated feps3 nanodrums,” Nano Lett. 22(1), 36–42 (2022). [CrossRef]
33. J. Huang, L. K. Chin, H. Cai, et al., “Dynamic phonon manipulation by optomechanically induced strong coupling between two distinct mechanical resonators,” ACS Photonics 6(8), 1855–1862 (2019). [CrossRef]
34. I. Mahboob and H. Yamaguchi, “Bit storage and bit flip operations in an electromechanical oscillator,” Nat. Nanotechnol. 3(5), 275–279 (2008). [CrossRef]
35. H. Lin, J.-H. Chen, S.-S. Chao, et al., “Strong coupling of different cavity modes in photonic molecules formed by two adjacent microdisk microcavities,” Opt. Express 18(23), 23948–23956 (2010). [CrossRef]
36. S. Weis, R. Rivière, S. Deléglise, et al., “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010). [CrossRef]
37. A. H. Safavi-Naeini, T. M. Alegre, J. Chan, et al., “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011). [CrossRef]
38. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010). [CrossRef]
39. T. W. Hänsch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13(1), 68–69 (1975). [CrossRef]
40. D. J. Wineland, R. E. Drullinger, F. L. Walls, et al., “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40(25), 1639–1642 (1978). [CrossRef]
41. R. Riviere, S. Deleglise, S. Weis, et al., “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83(6), 063835 (2011). [CrossRef]
42. C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45(2), 75–79 (1980). [CrossRef]
43. T. Kippenberg, H. Rokhsari, T. Carmon, et al., “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005). [CrossRef]
44. H. Rokhsari, T. J. Kippenberg, T. Carmon, et al., “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13(14), 5293–5301 (2005). [CrossRef]
45. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20(7), 7672–7684 (2012). [CrossRef]
46. Z. Shen, Y.-L. Zhang, Y. Chen, et al., “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016). [CrossRef]
47. C. Dong, V. Fiore, M. C. Kuzyk, et al., “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012). [CrossRef]
48. J. Kerckhoff, R. W. Andrews, H. Ku, et al., “Tunable coupling to a mechanical oscillator circuit using a coherent feedback network,” Phys. Rev. X 3(2), 021013 (2013). [CrossRef]
49. X.-Y. Lü, W.-M. Zhang, S. Ashhab, et al., “Quantum-criticality-induced strong kerr nonlinearities in optomechanical systems,” Sci. Rep. 3(1), 2943 (2013). [CrossRef]
50. S.-W. Bin, X.-Y. Lü, T.-S. Yin, et al., “Mass sensing by quantum criticality,” Opt. Lett. 44(3), 630–633 (2019). [CrossRef]
51. T.-S. Yin, X.-Y. Lü, L.-L. Wan, et al., “Enhanced photon-phonon cross-kerr nonlinearity with two-photon driving,” Opt. Lett. 43(9), 2050–2053 (2018). [CrossRef]
52. W. Xiong, D.-Y. Jin, J. Jing, et al., “Controllable coupling between a nanomechanical resonator and a coplanar-waveguide resonator via a superconducting flux qubit,” Phys. Rev. A 92(3), 032318 (2015). [CrossRef]
53. J. Chen, Z. Li, X.-Q. Luo, et al., “Strong single-photon optomechanical coupling in a hybrid quantum system,” Opt. Express 29(20), 32639–32648 (2021). [CrossRef]
54. W. Xiong, J. Chen, B. Fang, et al., “Strong tunable spin-spin interaction in a weakly coupled nitrogen vacancy spin-cavity electromechanical system,” Phys. Rev. B 103(17), 174106 (2021). [CrossRef]
55. W. Xiong, M. Wang, G.-Q. Zhang, et al., “Optomechanical-interface-induced strong spin-magnon coupling,” Phys. Rev. A 107(3), 033516 (2023). [CrossRef]
56. V. Sudhir, M. G. Genoni, J. Lee, et al., “Critical behavior in ultrastrong-coupled oscillators,” Phys. Rev. A 86(1), 012316 (2012). [CrossRef]
57. B. Lu, L. Liu, J.-Y. Song, et al., “Recent progress on coherent computation based on quantum squeezing,” AAPPS Bull. 33(1), 7 (2023). [CrossRef]
58. J. Huang, X. Chen, X. Li, et al., “Chip-based photonic graph states,” AAPPS Bull. 33(1), 14 (2023). [CrossRef]
59. M. Wang and F. Zhang, “Squeezing for cosmic symphony,” AAPPS Bull. 33(1), 5 (2023). [CrossRef]
60. Y.-P. Gao, C. Cao, P.-F. Lu, et al., “Phase-controlled photon blockade in optomechanical systems,” Fundam. Res. 3(1), 30–36 (2023). [CrossRef]
61. A. Pace, M. Collett, D. Walls, et al., “Quantum limits in interferometric detection of gravitational radiation,” Phys. Rev. A 47(4), 3173–3189 (1993). [CrossRef]
62. C. Genes, D. Vitali, P. Tombesi, et al., “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77(3), 033804 (2008). [CrossRef]
63. F. Marquardt, J. P. Chen, A. A. Clerk, et al., “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99(9), 093902 (2007). [CrossRef]
64. S. Liu, W.-X. Yang, Z. Zhu, et al., “Quadrature squeezing of a higher-order sideband spectrum in cavity optomechanics,” Opt. Lett. 43(1), 9–12 (2018). [CrossRef]
65. I. Wilson-Rae, N. Nooshi, W. Zwerger, et al., “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99(9), 093901 (2007). [CrossRef]
66. S. Gröblacher, K. Hammerer, M. R. Vanner, et al., “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009). [CrossRef]
67. F. P. Laussy, E. Del Valle, C. Tejedor, et al., “Luminescence spectra of quantum dots in microcavities. i. bosons,” Phys. Rev. B 79(23), 235325 (2009). [CrossRef]
68. 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]
69. M.-A. Lemonde and A. A. Clerk, “Real photons from vacuum fluctuations in optomechanics: The role of polariton interactions,” Phys. Rev. A 91(3), 033836 (2015). [CrossRef]
70. G.-L. Zhu, X.-Y. Lü, L.-L. Wan, et al., “Controllable nonlinearity in a dual-coupling optomechanical system under a weak-coupling regime,” Phys. Rev. A 97(3), 033830 (2018). [CrossRef]
71. Y. Wu, “Effective raman theory for a three-level atom in the λ configuration,” Phys. Rev. A 54(2), 1586–1592 (1996). [CrossRef]
72. U. Gaubatz, P. Rudecki, S. Schiemann, et al., “Population transfer between molecular vibrational levels by stimulated raman scattering with partially overlapping laser fields. a new concept and experimental results,” J. Chem. Phys. 92(9), 5363–5376 (1990). [CrossRef]
73. C. V. Raman and K. S. Krishnan, “A new type of secondary radiation,” Nature 121(3048), 501–502 (1928). [CrossRef]
74. M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at x-band (11-ghz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009). [CrossRef]
75. Y.-C. Liu, Y.-F. Xiao, Y.-L. Chen, et al., “Parametric down-conversion and polariton pair generation in optomechanical systems,” Phys. Rev. Lett. 111(8), 083601 (2013). [CrossRef]
