Open Access
Add to BibSonomy
Abstract
We propose a system for observing the spontaneous phase locking of two frequency separate mechanical modes in an anti-parity-time symmetric optomechanical system. In our approach, a common optical cavity mode mediates the coupling between two phonon modes, leading to the phase locking of the coupled mechanical modes to a common frequency in the symmetry unbroken regime. We furthermore observe the change of quantum correlation near the exceptional point. Our results are also directly relevant to numerous other physical platforms, such as atomic ensembles in cavity quantum electrodynamics (QED) systems and spin interaction mediated by collective motional mode in trapped ions.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Hermiticity is required in standard quantum mechanics to ensure real eigenvalues of the energy spectra and conservation of the total probability of a quantum system. However, in practice non-Hermitian Hamiltonians are widely used to describe the open system, i.e. quantum system interacting with the environment bath. A series of pioneering works [1–4] have exploited the relationship between optical gain and loss to simulate the parity-time (PT) symmetry. As one of non-Hermitian systems, the emergence of exceptional point singularities and the associated properties of the PT symmetric systems have the potential to create a variety of new opportunities in optics and photonics. As a counterpart of PT symmetric system, anti-PT [5] symmetry was first experimentally observed in thermal atoms [6], and it has inspired many researchers who turned their attention from PT to anti-PT. For instance, anti-PT symmetry has been experimentally demonstrated in electrical circuit resonators [7] and heat systems [8]. Anti-PT symmetric systems with distinct and useful properties start a new route to exploring the non-Hermitian Hamiltonian for applications in enhanced sensor sensitivity [9] and constructing topological superconductors [10].
The emergence of spontaneous phase locking of coupled system to a common frequency [11], occurs widely in nature, such as in fireflies colonies, pacemaker cells in the heart, nervous systems and circadian cycles, and is of great technological interest in biological [12,13], physical [14,15] and engineering [16] systems. This concept has been extended to quantum systems such as nanomechanical resonantors [17–19], optomechanical arrays [20], atomic clock [21] and interacting quantum dipoles [22]. Although there has been significant progress in the study of spontaneous phase locking in classical systems [23], the understanding of the same phenomena in the quantum realm remains limited.
Here, we propose a coupled multimode optomechanical system. By dissipation engineering, an anti-PT symmetric Hamiltonian is realized in the platform of frequency detuned mechanial modes via cavity photon mediated coupling. Moreover, we observed the spontaneous phase locking of two mechanical modes in the symmetry unbroken phase, and the change of quantum correlation near the exceptional point (EP) [24].
Fig. 1. (a) Light is evanescently coupled into the sphere resonator by means of a tapered optical fibre. The resonator supports the three participants, pump and anti-Stokes optical modes, and the acoustical modes in (b). (c) Schematic of the frequencies of the various tones.
2. Physical model
The proposed scheme can be realized via three-wave interaction [25], for instance Brillouin optomechanics. Brillouin interaction has stronger optical nonlinearity than Kerr and Raman interactions by orders of magnitude [26]. Stimulated Brillouin scattering (SBS) has been realized in optical microresonators, such as silica microspheres [27], disks [28], and crystalline cylinders [29]. In these whispering gallery resonators, optical and acoustic waves circulate along the equatorial surface, forming optical and mechanical whispering-gallery modes (WGMs) with high quality (Q) factor. Here, let us consider the case of two frequency-separate phonon modes in a resonator, as shown in Fig. 1, one with frequency $\Omega _1$ and the other with frequency $\Omega _2$. Pump and anti-Stokes light are both resonant with two discrete cavity modes, their separation matching the mechanical-resonance frequency. The Stokes mode is absent because of the asymmetric optical resonance structure of the resonator. In Brillouin scattering process, the pump field interacts with a mechanical traveling wave and generates a frequency-shift optical field, i.e. anti-Stokes field aforementioned. As two facets of such coupling, the electrostriction, one aspect of radiation pressure, represents the influences on mechanical wave induced by electric fields, whiles photoelasticity illustrates the mechanical wave modifies the light field. The associated configeration has been applied to Brillouin cooling [30] and light storage [31]. The acoustical and the optical modes are both WGM type and circulate in unison with considerable overlap. The pump laser beam at frequency $\omega _1$ drives the three-wave mixing interaction of phonon and optical modes, i.e. electrostrictive Brillouin scattering of light from sound, and the generated anti-Stokes photon at frequencies $\omega _1+\Omega _i$ lying within the cavity linewidth $\kappa$. Given the pump field remains undepleted, the system Hamiltonian can be written as
Under the unitary transformation with $H_0=\hbar (\omega _1+\Omega _0)a^{\dagger }a+\hbar \Omega _0(b_1^{\dagger }b_1+b_2^{\dagger }b_2)$ with $\Omega _0=(\Omega _1+\Omega _2)/2$ and $\Omega =(\Omega _1-\Omega _2)/2$, the effective Hamiltonian is
Thus, the quantum Langevin equations including the coupling between the mechanical and optical bath can be given
3. Spontaneous phase locking of mechanical multimodes in anti-PT optomechanics
In the limit of $\kappa \gg \Gamma _i, G_i$ and working in the resolved sideband regime $\Omega _i\gg \kappa$, one can make the adiabatic approximation of the cavity mode, and write it as $a=-i(G_1b_1+G_2b_2)/\kappa$. Therefore, the equation of motion for phonon modes can be rewritten as
Most strikingly, under the simplifying assumption $G_1=G_2 \in \mathbb {R}$, $H_{\textrm {eff}}$ satisfies $\hat {P}\hat {T}H_{\textrm {eff}}=-H_{\textrm {eff}}$, that is $\left \{ H_{\textrm {eff}},\hat {P}\hat {T} \right \}=0$ with the parity-time (PT) operator $\hat {P}\hat {T}$. In the presence of photon-mediated coupling between the mechanical modes, the non-Hermitian Hamiltonian Eq. (5) becomes anti-PT-symmetric. The eigenvalues of Eq. (5) correspond to the two eigen-supermodes $\omega _{\pm }=-i\Gamma _c\pm \sqrt {\Omega ^2-\Gamma _c^2}$. In such scenario (Fig. 2), anti-PT symmetry breaking occurs at the EP with $\Omega =\Gamma _c$ where the two supermodes perfectly overlap, i.e. $\Delta =(Re[\omega _+]-Re[\omega _-])/2=0$. In the symmetry-unbroken regime ($\Omega <\Gamma _c$) , the two eigen- resonance centres coincide but with different linewidths. Spontaneous phase locking of mechanical multimodes observed in the symmetry unbroken regime occurs when the coupling exceeds the initial frequency difference of the modes. When $\Omega >\Gamma _c$, the driven system enters the symmetry breaking regime, and the resonances bifurcate.
Fig. 2. Anti-PT supermodes in coupled phonon modes. Characteristics of the real part $\textrm {Re}[\omega ]$ (red solid) of the two eigenfrequencies of the coupled supermodes as a function of $\Omega$.
Let’s describe the system more generally. By defining the mechanical response function $\chi _{m_1}(\omega )=\frac {1}{\Omega -\omega -i\Gamma _1}$ and $\chi _{m_2}(\omega )=\frac {1}{-\Omega -\omega -i\Gamma _2}$ and cavity response $\chi _{c}(\omega )=\frac {1}{-\omega -i\kappa }$, one can further obtain the reflection probability amplitude of the cavity as follows,
In the triply resonant forward SBS process as presented in Ref. [31], the condition mentioned above is satisfied with $\Omega _0/2\pi =43.2 \textrm {MHz}\gg \kappa /2\pi =1.75 \textrm {MHz}\gg \Gamma /2\pi =2 \textrm {kHz}$. Also, the required $G$ to cross over the EP is much smaller than $\kappa$ if assuming the half of frequency separation $\Omega /2\pi$ of $\sim 10 \textrm {kHz}$. Figure 3 shows the spectrum of the generated anti-Stokes photon with different values of $\Omega /\Gamma _c$. For a given $\Omega$, the smaller the value, the larger the coupling. In the symmetry breaking regime ($\Omega /\Gamma _c>1$) illustrated in the right column of Fig. 3, two peaks belonging to initially independent mechanical modes gradually merge when approaching the EP ($\Omega /\Gamma _c=1$).
Fig. 3. Spectrum of the generated anti-Stokes light for different values of $\Omega /\Gamma _c$. In the forward SBS scheme, $\Omega _0=2\pi \times 43.2 \textrm {MHz}$, $\Gamma _1=\Gamma _2=2\pi \times 2 \textrm {kHz}$, $\kappa =2\pi \times 1.75 \textrm {MHz}$, $\Omega =2\pi \times 10 \textrm {kHz}$ and $\kappa _{ex}/\kappa =0.8$ [31]. Inset: Zoom-out view of the panel.
As demonstrated in Ref. [25], in such a system satisfying degenerate condition $\Omega =0$, collective effect leads to the apperance of a bright superadiant mode $b_+=(b_1+b_2)/\sqrt {2}$ with linewidth $\Gamma _+=\Gamma +2\Gamma _{\textrm {eff}}$ ($\Gamma _{\textrm {eff}}\approx \frac {\left | G \right |^2\delta }{\kappa ^2+\delta ^2}$) and a dark mode $b_-=(b_1-b_2)/\sqrt {2}$ with linewidth $\Gamma _- =\Gamma$, which is effectively decoupled from the cavity. Most strikingly, here in the symmetry unbroken regime appears a dark mode on top of the broadened bright mode when $G$ is large enough, as shown in the inset of the panel.
We now go step further by considering the strong coupling case as observed in the backward SBS [32]. Since the mechanical decay rate is of the order of $\textrm {MHz}$, and the cavity linewidth is of the order of few tens of $\textrm {MHz}$, the coupling strength $G$ is still smaller than $\kappa$ when approaching to the EP given few $\textrm {MHz}$ separation. However, in the deep sysmmetry unbroken phase, i.e. very small value of $\Omega /\Gamma _c$, $G$ is comparable to $\kappa$ and thus, Eq. (5) could fail to describe the system. There are abundant phenomena associated to strong coupling.
Let’s rewrite the response function $M(\omega )$ as
Of particular interest is that in the stong-coupling limit satisfying $2\chi _c(\omega )G_1G_2\gg \left | \tilde {\omega }_1^e-\tilde {\omega }_2^e \right |$, a dark mode positioned at the central frequency of two mechanial modes appears with a damping rate of $(\Gamma _1+\Gamma _2)/2$ even though two modes are non-degenerate. Substituting the mechanical reponse functions and cavity response with frequency dependence, one can reach the frequency of a bright mode, $\tilde {\omega }_{1,2}^c=\frac {1}{4}(i2\kappa \pm \sqrt {(-i2\kappa )^2+8G_1^2+8G_2^2})$.
Figure 4 shows the spectrum of the generated anti-Stokes photon with different values of $\Omega /\Gamma _c$ in a backward SBS system. In constrast to the forward SBS case, there always exists the dark mode as a narrow peak in the symmetry unbroken regime. Addtionally, when $G_1^2+G_2^2>\kappa ^2/2$, the frequency tends to $\omega _1^c\rightarrow \pm \frac {\sqrt {2(G_1^2+G_2^2)-\kappa ^2}}{2}$ with the linewidth $\sim \kappa /2$. Such feature is of significance in the spectrum with $\Omega /\Gamma _c=0.1$ and $0.25$ in the left column. On the other hand, in the symmetry breaking regime ($\Omega /\Gamma _c>1$) illustrated in the right column of Fig. 4, two peaks belonging to initially independent mechanical modes gradually merge when approaching to the exceptional point ($\Omega /\Gamma _c=1$).
Fig. 4. Spectrum of the generated anti-Stokes light for different values of $\Omega /\Gamma _c$. Parameters: $\Gamma _1=\Gamma _2=\Gamma =2\pi \times 0.5 \textrm {MHz}$, $\kappa /\Gamma =12$ and $\kappa _{ex}/\kappa =0.8$.
4. Quantum correlations
Similar to the quantum correlation evaluation in Ref. [34], we adopt Gaussian discord to evaluate the quantum correlation between the two mechanical modes. Quantum discord is a kind of quantum correlation beyond entanglement, and it is evaluated via the reconstruction of bipartite covariance matrix. Specifically, Gaussian discord is defined in the following. In a bipartite system, the total amount of correlations (classical and quantum) is given by the von Neumann mutual information $I(\rho _{AB})=S(\rho _A)+S(\rho _B)-S(\rho _{AB})$, where $S(\rho )$ is the von Neumann entropy and $\rho _{A(B)}$ is the reduced density matrix of the A (B) subsystem. Another measure of mutual information that only quantifies the amount of classical correlations extractable by a Gaussian measurement is $J_A(\rho _{AB})=S(\rho _A)-\textrm {inf}_{\sigma _M}S(\rho _A\mid \sigma _M)$, where $\sigma _M$ is the covariance matrix of the measurement on mode B. As it only captures the classical correlations, the difference, $D_A=I(\rho _{AB})-J_A(\rho _{AB})$, is a measure of Gaussian quantum correlation that is coined Gaussian quantum discord [35,36]. Introducing the amplitude and phase quadratures of light and phonon as $\hat {X}_a=\frac {\hat {a}+\hat {a}^{\dagger }}{\sqrt {2}}$, $\hat {P}_a=\frac {\hat {a}-\hat {a}^{\dagger }}{\sqrt {2}i}$ and $\hat {x}_{b_j}=\frac {\hat {b}_j+\hat {b}_j^{\dagger }}{\sqrt {2}}$, $\hat {p}_{b_j}=\frac {\hat {b}_j-\hat {b}_j^{\dagger }}{\sqrt {2}i}$ with $j=1,2$, we can focus on the Heisenberg-Langevin equations as
where $\Sigma =(X_a,P_a,x_{b_1},p_{b_1},x_{b_2},p_{b_2})^T$ is a vector of mechanical and cavity quadrature operators, and $N=(\sqrt {2\kappa _{ex}}X_{a,in}+\sqrt {2\kappa _{in}}X_{c},\sqrt {2\kappa _{ex}}P_{a,in}+\sqrt {2\kappa _{in}}P_{c},\sqrt {2\Gamma _1}x_{b_1}^{in},\sqrt {2\Gamma _1}p_{b_1}^{in},\sqrt {2\Gamma _2}x_{b_2}^{in},\sqrt {2\Gamma _2}p_{b_2}^{in})^T$. The matrix given in Eq. (9) isAround the EP, we observe apparent changes of the Gaussian discord with respect to $\Omega /\Gamma _c$, as shown in Fig. 5. When $\Gamma _c$ is small enough, the phases of the two mechanical modes are not synchronized, reducing the efficiency of mutual coherence stimulation between the two modes, further reducing the discord. However, when $\Gamma _c$ is larger than $\Omega$, the system is in the phase unbroken regime, and the two mechanical modes’ frequencies are pulled together, giving rise to relatively larger discord generated by photon-mediated coupling operation. Note that backward SBS is considered in Fig. 5 with $\Omega _0=2\pi \times 11 \textrm {GHz}$ at 4K, causing a low thermal occupation with $\bar {n}=7$. It allows for the observation of significant quantum coherence in term of quantum correlation, Gaussian discord. In stark contrast, for the forward SBS, lower mechanical oscillation frequency leads to much higher thermal occupation ($\bar {n}=1960$ at 4K). As we know, quantum states are very fragile under dissipation due to the coupling to their environment, where the induced dissipation tends to destroy and wash out the interesting quantum effects [38]. Therefore, quantum discord in Fig. 6(a) drops much faster than that in Fig. 6(b) where $\bar {n}=49$ at 100 mK when optomechanical coupling reduces. It impedes us to observe the sharp change of quantum correlation near EP as illustrated in Fig. 5.
Fig. 5. Discord vs. coupling strength at T=4K. In the backward SBS scheme, $\Omega _0=2\pi \times 11 \textrm {GHz}$, $\Gamma _1=\Gamma _2=\Gamma =2\pi \times 0.5 \textrm {MHz}$, $\kappa /\Gamma =12$ and $\kappa _{ex}/\kappa =0.8$ with thermal occupation $\bar {n}=7$.
Fig. 6. Discord vs. coupling strength at 4K (a) and 100 mK (b). In the forward SBS scheme, $\Omega _0=2\pi \times 43.2 \textrm {MHz}$, $\Gamma _1=\Gamma _2=2\pi \times 2 \textrm {kHz}$, $\kappa =2\pi \times 1.75 \textrm {MHz}$, $\Omega =2\pi \times 10 \textrm {kHz}$ and $\kappa _{ex}/\kappa =0.8$. $\bar {n}=1969$ (a) and $\bar {n}=49$ (b).
5. Conclusion
In conclusion, we present a proposal of spontaneous phase locking of two mechanical modes in the platform of anti-PT optomechanics via three-wave interaction, and discuss the potential realization in both forward and backward SBS. Moreover, we study the quantum correlation behavior near the EP. It is possible to realize the proposed scheme in Brillouin optomechanics [30] since strong coupling has been observed in such system [32]. Our results are also directly relevent to numerous other physical platforms, such as atomic ensembles in cavity QED system [39], and spin interaction mediated by collective motional modes in trapped ions [40].
Funding
Royal Society (Internation Fellowship NF170876); National Natural Science Foundation of China (11604069, 61875147); the Program of State Key Laboratory of Quantum Optics and Quantum Optics Devices (KF201802); Fundamental Research Funds for the Central Universities (PA2019GDQT0023).
Acknowledgments
The authors are grateful to Jie Li, Wenlin Li, David Vitali, L. Freisem, G. Enzian, J. Price and M. Vanner for fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
References
1. Ş Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019). [CrossRef]
2. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-hermitian physics and pt symmetry,” Nat. Phys. 14(1), 11–19 (2018). [CrossRef]
3. L. Feng, R. El-Ganainy, and L. Ge, “Non-hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017). [CrossRef]
4. M.-A. Miri and A. Alu, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019). [CrossRef]
5. L. Ge and H. E. Türeci, “Antisymmetric pt-photonic structures with balanced positive-and negative-index materials,” Phys. Rev. A 88(5), 053810 (2013). [CrossRef]
6. P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity–time symmetry with flying atoms,” Nat. Phys. 12(12), 1139–1145 (2016). [CrossRef]
7. Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, “Observation of an anti-pt-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators,” Nat. Commun. 9(1), 2182 (2018). [CrossRef]
8. Y. Li, Y.-G. Peng, L. Han, M.-A. Miri, W. Li, M. Xiao, X.-F. Zhu, J. Zhao, A. Alù, and S. Fan, “Anti–parity-time symmetry in diffusive systems,” Science 364, 170–173 (2019).
9. H.-K. Lau and A. A. Clerk, “Fundamental limits and non-reciprocal approaches in non-hermitian quantum sensing,” Nat. Commun. 9(1), 4320 (2018). [CrossRef]
10. Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, “Topological phases of non-hermitian systems,” Phys. Rev. X 8(3), 031079 (2018). [CrossRef]
11. A. Pikovsky, J. Kurths, M. Rosenblum, and J. Kurths, Synchronization: a universal concept in nonlinear sciences, vol. 12 (Cambridge university press, 2003).
12. S. Yamaguchi, H. Isejima, T. Matsuo, R. Okura, K. Yagita, M. Kobayashi, and H. Okamura, “Synchronization of cellular clocks in the suprachiasmatic nucleus,” Science 302(5649), 1408–1412 (2003). [CrossRef]
13. M. Steriade, “Synchronized activities of coupled oscillators in the cerebral cortex and thalamus at different levels of vigilance,” Cereb. Cortex 7(6), 583–604 (1997). [CrossRef]
14. M. Cross, “Improving the frequency precision of oscillators by synchronization,” Phys. Rev. E 85(4), 046214 (2012). [CrossRef]
15. D. Stefanatos and J.-S. Li, “Antiphase synchronization of phase-reduced oscillators using open-loop control,” Phys. Rev. E 85(3), 037201 (2012). [CrossRef]
16. M. Rohden, A. Sorge, M. Timme, and D. Witthaut, “Self-organized synchronization in decentralized power grids,” Phys. Rev. Lett. 109(6), 064101 (2012). [CrossRef]
17. C. A. Holmes, C. P. Meaney, and G. J. Milburn, “Synchronization of many nanomechanical resonators coupled via a common cavity field,” Phys. Rev. E 85(6), 066203 (2012). [CrossRef]
18. M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. 109(23), 233906 (2012). [CrossRef]
19. M. Bagheri, M. Poot, L. Fan, F. Marquardt, and H. X. Tang, “Photonic cavity synchronization of nanomechanical oscillators,” Phys. Rev. Lett. 111(21), 213902 (2013). [CrossRef]
20. G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, “Collective dynamics in optomechanical arrays,” Phys. Rev. Lett. 107(4), 043603 (2011). [CrossRef]
21. M. Xu, D. A. Tieri, E. Fine, J. K. Thompson, and M. J. Holland, “Synchronization of two ensembles of atoms,” Phys. Rev. Lett. 113(15), 154101 (2014). [CrossRef]
22. B. Zhu, J. Schachenmayer, M. Xu, F. Herrera, J. G. Restrepo, M. J. Holland, and A. M. Rey, “Synchronization of interacting quantum dipoles,” New J. Phys. 17(8), 083063 (2015). [CrossRef]
23. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008). [CrossRef]
24. A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111(10), 103605 (2013). [CrossRef]
25. G. Agarwal and S. S. Jha, “Multimode phonon cooling via three-wave parametric interactions with optical fields,” Phys. Rev. A 88(1), 013815 (2013). [CrossRef]
26. B. J. Eggleton, C. G. Poulton, P. T. Rakich, M. J. Steel, and G. Bahl, “Brillouin integrated photonics,” Nat. Photonics 13(10), 664–677 (2019). [CrossRef]
27. M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at x-band (11-ghz) rates,” Phys. Rev. Lett. 102(11), 113601 (2009). [CrossRef]
28. J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip brillouin oscillator,” Nat. Commun. 4(1), 2097 (2013). [CrossRef]
29. I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a caf2 whispering gallery mode resonator,” Phys. Rev. Lett. 102(4), 043902 (2009). [CrossRef]
30. G. Bahl, M. Tomes, F. Marquardt, and T. Carmon, “Observation of spontaneous brillouin cooling,” Nat. Phys. 8(3), 203–207 (2012). [CrossRef]
31. C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6(1), 6193 (2015). [CrossRef]
32. G. Enzian, M. Szczykulska, J. Silver, L. Del Bino, S. Zhang, I. A. Walmsley, P. Del’Haye, and M. R. Vanner, “Observation of brillouin optomechanical strong coupling with an 11 ghz mechanical mode,” Optica 6(1), 7–14 (2019). [CrossRef]
33. F. Massel, S. U. Cho, J.-M. Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, and M. A. Sillanpää, “Multimode circuit optomechanics near the quantum limit,” Nat. Commun. 3(1), 987 (2012). [CrossRef]
34. D. Stefanatos, “Maximising optomechanical entanglement with optimal control,” Quantum Sci. Technol. 2(1), 014003 (2017). [CrossRef]
35. G. Adesso and A. Datta, “Quantum versus classical correlations in gaussian states,” Phys. Rev. Lett. 105(3), 030501 (2010). [CrossRef]
36. P. Giorda and M. G. Paris, “Gaussian quantum discord,” Phys. Rev. Lett. 105(2), 020503 (2010). [CrossRef]
37. D. Vitali, S. Gigan, A. Ferreira, H. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007). [CrossRef]
38. D. Witthaut, S. Wimberger, R. Burioni, and M. Timme, “Classical synchronization indicates persistent entanglement in isolated quantum systems,” Nat. Commun. 8(1), 14829 (2017). [CrossRef]
39. M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, “Cavity-mediated collective spin-exchange interactions in a strontium superradiant laser,” Science 361(6399), 259–262 (2018). [CrossRef]
40. P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. Lanyon, M. Heyl, R. Blatt, and C. Roos, “Direct observation of dynamical quantum phase transitions in an interacting many-body system,” Phys. Rev. Lett. 119(8), 080501 (2017). [CrossRef]
