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Phase anti-synchronization dynamics between mechanical oscillator and atomic ensemble within a Fabry–Perot cavity

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Abstract

Phase synchronization refers to a kind of collective phenomenon that the phase difference between two or more systems is locked, and it has widely been investigated between systems with the identical physical properties, such as the synchronization between mechanical oscillators and the synchronization between atomic ensembles. Here, we investigate the synchronization behavior between the mechanical oscillator and the atomic ensemble, the systems with different physical properties, and observe a novel synchronization phenomenon, i.e., the phase sum, instead of phase difference, is locked. We refer to this distinct synchronization as the phase anti-synchronization and show that the phase anti-synchronization can be achieved in both the classical and quantum level, which means this novel collective behavior can be tested in experiment. Also, some interesting connections between phase anti-synchronization and quantum correlation are found.

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References

  1. Acebrn, J.A., Bonilla, L.L., Prez Vicente, C.J., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137 (2005)

    Article  ADS  Google Scholar 

  2. Kapitaniak, M., Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Synchronization of clocks. Phys. Rep. 517, 1 (2012)

    Article  ADS  MATH  Google Scholar 

  3. Maianti, M., Pagliara, S., Galimberti, G., Parmigiani, F.: Mechanics of two pendulums coupled by a stressed spring. Am. J. Phys. 77, 834 (2009)

    Article  ADS  Google Scholar 

  4. Pantaleone, J.: Synchronization of metronomes. Am. J. Phys. 70, 992 (2002)

    Article  ADS  Google Scholar 

  5. Aguiar, M., Ashwin, P., Dias, A., Field, M.: Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation. J. Nonlinear Sci. 21, 271 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 354 (1999)

    Article  ADS  Google Scholar 

  7. Porat-Shliom, N., Chen, Y., Tora, M., Shitara, A., Masedunskas, A., Weigert, R.: In vivo tissue-wide synchronization of mitochondrial metabolic oscillations. Cell Rep. 9, 514 (2014)

    Article  Google Scholar 

  8. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Strogatz, S.H., Stewart, I.: Coupled oscillators and biological synchronization. Sci. Am. 269, 102 (1993)

    Article  Google Scholar 

  10. Angelini, L., Lattanzi, G., Maestri, R., Marinazzo, D., Nardulli, G., Nitti, L., Pellicoro, M., Pinna, G.D., Stramaglia, S.: Phase shifts of synchronized oscillators and the systolic-diastolic blood pressure relation. Phys. Rev. E 69, 061923 (2004)

    Article  ADS  Google Scholar 

  11. Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469, 93 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  12. Padmanaban, E., Boccaletti, S., Dana, S.K.: Emergent hybrid synchronization in coupled chaotic systems. Phys. Rev. E 91, 022920 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  13. Mari, A., Farace, A., Didier, N., Giovannetti, V., Fazio, R.: Measures of quantum synchronization in continuous variable systems. Phys. Rev. Lett. 111, 103605 (2013)

    Article  ADS  Google Scholar 

  14. Bemani, F., Motazedifard, A., Roknizadeh, R., Naderi, M.H., Vitali, D.: Synchronization dynamics of two nanomechanical membranes within a Fabry–Perot cavity. Phys. Rev. A 96, 023805 (2017)

    Article  ADS  Google Scholar 

  15. Militello, B., Nakazato, H., Napoli, A.: Synchronizing quantum harmonic oscillators through two-level systems. Phys. Rev. A 96, 023862 (2017)

    Article  ADS  Google Scholar 

  16. Ying, L., Lai, Y.C., Grebogi, C.: Quantum manifestation of a synchronization transition in optomechanical systems. Phys. Rev. A 90, 053810 (2014)

    Article  ADS  Google Scholar 

  17. Xu, M., Tieri, D.A., Fine, E.C., Thompson, J.K., Holland, M.J.: Synchronization of two ensembles of atoms. Phys. Rev. Lett. 113, 154101 (2014)

    Article  ADS  Google Scholar 

  18. Liao, C.G., Chen, R.X., Xie, H., He, M.Y., Lin, X.M.: Quantum synchronization and correlations of two mechanical resonators in a dissipative optomechanical system. Phys. Rev. A 99, 033818 (2019)

    Article  ADS  Google Scholar 

  19. Davis-Tilley, C., Teoh, C.K., Armour, A.D.: Dynamics of many-body quantum synchronisation. New J. Phys. 20, 113002 (2018)

    Article  ADS  Google Scholar 

  20. Ameri, V., Eghbali-Arani, M., Mari, A., Farace, A., Kheirandish, F., Giovannetti, V., Fazio, R.: Mutual information as an order parameter for quantum synchronization. Phys. Rev. A 91, 012301 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  21. Hush, M.R., Li, W.B., Genway, S., Lesanovsky, I., Armour, A.D.: Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers. Phys. Rev. A 91, 061401(R) (2015)

    Article  ADS  Google Scholar 

  22. Roulet, A., Bruder, C.: Quantum synchronization and entanglement generation. Phys. Rev. Lett. 121, 063601 (2018)

    Article  ADS  Google Scholar 

  23. Davis-Tilley, C., Armour, A.D.: Synchronization of micromasers. Phys. Rev. A 94, 063819 (2016)

    Article  ADS  Google Scholar 

  24. Schrödinger, E.: Sitzungsberichte der Preussischen Akademie der Wissenschaften. Phys. Math. Klasse 14, 296 (1930)

    Google Scholar 

  25. Dodonov, V.V.: Variance uncertainty relations without covariances for three and four observables. Phys. Rev. A 97, 022105 (2018)

    Article  ADS  Google Scholar 

  26. Xu, M., Holland, M.J.: Conditional ramsey spectroscopy with synchronized atoms. Phys. Rev. Lett. 114, 103601 (2015)

    Article  ADS  Google Scholar 

  27. Qiu, H., Zambrini, R., Polls, A., Martorell, J., Julia-Diaz, B.: Hybrid synchronization in coupled ultracold atomic gases. Phys. Rev. A 92, 043619 (2015)

    Article  ADS  Google Scholar 

  28. Aspelmeyer, M., Kippenberg, T.J., Marquardt, F.: Cavity optomechanics. Rev. Mod. Phys. 86, 1391 (2014)

    Article  ADS  Google Scholar 

  29. Motazedifard, A., Bemani, F., Naderi, M.H., Roknizadeh, R., Vitali, D.: Force sensing based on coherent quantum noise cancellation in a hybrid optomechanical cavity with squeezed-vacuum injection. New J. Phys. 18, 073040 (2016)

    Article  ADS  Google Scholar 

  30. Wimmer, M.H., Steinmeyer, D., Hammerer, K., Heurs, M.: Coherent cancellation of backaction noise in optomechanical force measurements. Phys. Rev. A 89, 053836 (2014)

    Article  ADS  Google Scholar 

  31. Bariani, F., Seok, H., Singh, S., Vengalattore, M., Meystre, P.: Atom-based coherent quantum-noise cancellation in optomechanics. Phys. Rev. A 92, 043817 (2015)

    Article  ADS  Google Scholar 

  32. Bemani, F., Roknizadeh, R., Motazedifard, A., Naderi, M.H., Vitali, D.: Quantum correlations in optomechanical crystals. Phys. Rev. A 99, 063814 (2019)

    Article  ADS  Google Scholar 

  33. Hauer, B.D., Clark, T.J., Kim, P.H., Doolin, C., Davis, J.P.: Dueling dynamical backaction in a cryogenic optomechanical cavity. Phys. Rev. A 99, 053803 (2019)

    Article  ADS  Google Scholar 

  34. Huang, G.F., Deng, W.W., Tan, H.T., Cheng, G.L.: Generation of squeezed states and single-phonon states via homodyne detection and photon subtraction on the filtered output of an optomechanical cavity. Phys. Rev. A 99, 043819 (2019)

    Article  ADS  Google Scholar 

  35. Hammerer, K., Sorensen, A.S., Polzik, E.S.: Quantum interface between light and atomic ensembles. Rev. Mod. Phys. 82, 1041 (2010)

    Article  ADS  Google Scholar 

  36. Giovannetti, V., Vitali, D.: Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion. Phys. Rev. A 63, 023812 (2001)

    Article  ADS  Google Scholar 

  37. Benguria, R., Kac, M.: Quantum langevin equation. Phys. Rev. Lett. 46, 1 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  38. Lee, T.E., Sadeghpour, H.R.: Quantum synchronization of quantum van der Pol oscillators with trapped ions. Phys. Rev. Lett. 111, 234101 (2013)

    Article  ADS  Google Scholar 

  39. Dantan, A., Genes, C., Vitali, D., Pinard, M.: Self-cooling of a movable mirror to the ground state using radiation pressure. Phys. Rev. A 77, 011804(R) (2008)

    Article  ADS  Google Scholar 

  40. Mari, A., Eisert, J.: Gently modulating optomechanical systems. Phys. Rev. Lett. 103, 213603 (2009)

    Article  ADS  Google Scholar 

  41. Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)

    Article  ADS  MATH  Google Scholar 

  42. Gardiner, C.W., Zoller, P.: Quantum Noise. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  43. Genes, C., Vitali, D., Tombesi, P.: Emergence of atom-light-mirror entanglement inside an optical cavity. Phys. Rev. A 77, 050307(R) (2008)

    Article  ADS  Google Scholar 

  44. Walter, S., Nunnenkamp, A., Bruder, C.: Quantum synchronization of a driven self-sustained oscillator. Phys. Rev. Lett. 112, 094102 (2014)

    Article  ADS  Google Scholar 

  45. Ludwig, M., Marquardt, F.: Quantum many-body dynamics in optomechanical arrays. Phys. Rev. Lett. 111, 073603 (2013)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11574022).

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Correspondence to Guo-Feng Zhang.

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Appendices

Appendix A

For the ensemble of N effective two-level atoms, the total spin is N/2, and then, one can introduce an effective atomic bosonic annihilation operator \(\hat{d}\), which satisfies [29]:

$$\begin{aligned} \hat{S}_{z}&=\frac{N}{2}-\hat{d}^{\dagger } \hat{d},\nonumber \\ \hat{S}_{+}&=\sqrt{N} \hat{d} \sqrt{\left( 1-\hat{d}^{\dagger } \hat{d}\right) }, \nonumber \\ \quad \hat{S}_{-}&=\sqrt{N} \hat{d}^{\dagger } \sqrt{\left( 1-\hat{d}^{\dagger } \hat{d}\right) }. \end{aligned}$$
(A1)

Under the condition that the atomic ensemble is fully inverted and N is large, we have \(\hat{d}^{\dagger } \hat{d} / N \ll 1\). Then, one can obtain that \(\hat{S}_{+} \simeq \sqrt{N} \hat{d}\) and \(\hat{S}_{-} \simeq \sqrt{N} \hat{d}^{\dagger }\), and the Hamiltonian (2) can be rewritten as [29, 31]:

$$\begin{aligned} \hat{H}_{a t}=- \hbar \omega _{\sigma } \hat{d}^{\dagger } \hat{d} +\hbar G \cos \left( \omega _{\mathrm{G}}t\right) \left( \hat{c}^{\dagger } +\hat{c}\right) \left( \hat{d}^{\dagger }+\hat{d}\right) , \end{aligned}$$
(A2)

where \(G=G_{0} \sqrt{N}\). Taking the resonance condition \(\omega _G=\omega _L\), and applying the rotating wave approximation, one can obtain (in the interaction picture with respect to \(\hbar \omega _{L} \hat{c}^{\dagger } \hat{c}\)) [29]:

$$\begin{aligned} \hat{H}_{a t}=-\hbar \omega _{\sigma } \hat{d}^{\dagger } \hat{d} +\hbar \frac{G}{2}\left( \hat{c}^{\dagger }+\hat{c}\right) \left( \hat{d}^{\dagger }+\hat{d}\right) . \end{aligned}$$
(A3)

Appendix B

The variance \(\Delta \left( \hat{\varphi }_{m}+\hat{\varphi }_{d} \right) ^{2}\) can be calculated by the covariance matrix V(t), with the elements \(V(t)[i, j]=[\langle u(i) u(j)+u(i) u(j)\rangle ] / 2\) and \(u=\left( \delta \hat{q}_{m},\ \delta \hat{p}_{m},\, \delta \hat{q}_{d},\, \delta \hat{p}_{d}, \, \delta \hat{q}_{c},\delta \hat{p}_{c}\right) ^{\mathrm{T}}\). Based on HL equation (5), we can deduce that the time evolution of V(t) is governed by [40, 44, 45]:

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d} t} V(t)=A V(t)+V(t) A^{\mathrm{T}}+D, \end{aligned}$$
(B1)

where \(D=\mathrm{diag}\{0, \gamma (2 \overline{n}+1), \Gamma , \Gamma , \kappa , \kappa \}\) is the diffusion matrix. The drift matrix A reads:

$$\begin{aligned} A=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {0} &{} {\omega _{m}} &{} {0} &{} {0} &{} {0} &{} {0} \\ {-\,\omega _{m}} &{} {-\,\gamma } &{} {0} &{} {0} &{} {-\,g_{m} q_{c}} &{} {-\,g_{m} p_{c}} \\ {0} &{} {0} &{} {-\,\Gamma } &{} {-\,\omega _{\sigma }} &{} {0} &{} {0} \\ {0} &{} {0} &{} {\omega _{\sigma }} &{} {-\,\Gamma } &{} {-\,\sqrt{2} g_{m}} &{} {0} \\ {g_{m} p_{c}} &{} {0} &{} {0} &{} {0} &{} {-\,\kappa } &{} {g_{m} q_{m}+\Delta } \\ {-\,g_{m} q_{c}} &{} {0} &{} {-\,\sqrt{2} g_{m}} &{} {0} &{} {-\,\Delta -g_{m} q_{m}} &{} {-\,\kappa } \end{array}\right) . \end{aligned}$$
(B2)

Assume that the mechanical oscillator is initially prepared in a thermal state corresponding to the temperature T, the atomic ensemble and the cavity mode fluctuations are initially in the vacuum state. Then, one can obtain the initial condition \(V(0)=\mathrm{diag}[\overline{n}+1 / 2, \overline{n}+1 / 2,1 / 2,1 / 2,1 / 2,1 / 2]\). Taking advantage of Eq. (B1), one can obtain the numerical result of the variance \(\Delta \left( \hat{\varphi }_{m}+\hat{\varphi }_{d}\right) ^{2}\).

Appendix C

In fact, the covariance matrix V(t) can be written as:

$$\begin{aligned} V=\left[ \begin{array}{ll} {V_{A}} &{}\quad {V_{C}} \\ {V_{C}^{\mathrm{T}}} &{}\quad {V_{B}} \end{array}\right] . \end{aligned}$$
(C1)

where \(V_A\), \(V_B\) and \(V_C\) are \(2\times 2\) matrics, \(V_A\) and \(V_B\) account for the local properties of modes A and B, respectively, and \(V_C\) describes the correlation between them. As we known, the Gaussian quantum discord is an asymmetric, and we here mainly focus on the Gaussian quantum A-discord, which is given by:

$$\begin{aligned} D_G=f(\sqrt{\beta })-f\left( v_{-}\right) -f\left( v_{+}\right) -f(\sqrt{\varepsilon }). \end{aligned}$$
(C2)

where \(f(x)=\left( \frac{x+1}{2}\right) \log \left( \frac{x+1}{2}\right) -\left( \frac{x-1}{2}\right) \log \left( \frac{x-1}{2}\right) \), \(v_{\pm }=\sqrt{\frac{\Sigma _{+} \pm \sqrt{\Sigma _{+}^{2}-4 \mathrm{det} V}}{2}}\), and \(\Sigma _{\pm }=\mathrm{det} V_{A}+\mathrm{det} V_{B} \pm 2 \mathrm{det} V_{C}\). \(\varepsilon \) is given by:

$$\begin{aligned} \varepsilon =\left\{ \begin{array}{lr} {\frac{2 \gamma ^{2}+(\beta -1)(\delta -\alpha )+2|\gamma | \sqrt{\gamma ^{2} +(\beta -1)(\delta -\alpha )}}{(\beta -1)^{2}},} &{} {\frac{(\delta -\alpha \beta )^{2}}{(\beta +1) \gamma ^{2}(\alpha +\delta )} \leqslant 1} \\ {\frac{\alpha \beta -\gamma ^{2}+\delta -\sqrt{\gamma ^{4}+(\delta -\alpha \beta )^{2} -2 \gamma ^{2}(\delta +\alpha \beta )}}{2 \beta },} &{} \text {otherwise} \end{array}\right. . \end{aligned}$$
(C3)

where \(\alpha =\mathrm{det} V_{A}\), \(\beta =\mathrm{det} V_{B}\), \(\gamma =\mathrm{det} V_{C}\), and \(\delta =\mathrm{det} V\). Then, using Eqs. (B1) and (C2), one can obtain the Gaussian quantum discord between the mechanical oscillator and the atomic ensemble.

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Ma, SQ., Zheng, X. & Zhang, GF. Phase anti-synchronization dynamics between mechanical oscillator and atomic ensemble within a Fabry–Perot cavity. Quantum Inf Process 19, 152 (2020). https://doi.org/10.1007/s11128-020-02646-0

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