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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This work is supported by the National Natural Science Foundation of China (Grant No. 11574022).
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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]:
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]:
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]:
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]:
where \(D=\mathrm{diag}\{0, \gamma (2 \overline{n}+1), \Gamma , \Gamma , \kappa , \kappa \}\) is the diffusion matrix. The drift matrix A reads:
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:
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:
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:
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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DOI: https://doi.org/10.1007/s11128-020-02646-0