Based on a system-reservoir model and an appropriate choice of nonlinear coupling, we have explored the microscopic quantum generalization of classical Liénard systems. Making use of oscillator coherent states and canonical thermal distributions of the associated $c$ numbers, we have derived the quantum Langevin equation of the reduced system which admits single or multiple limit cycles. It has been shown that detailed balance in the form of the fluctuation-dissipation relation preserves the dynamical stability of the attractors even in the case of vacuum excitation. The quantum versions of Rayleigh, van der Pol, and several other variants of Liénard oscillators are derived as special cases in our theoretical scheme within a mean-field description.

中文翻译：

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经典Liénard振荡器的微观量子广义化

基于一个系统的储层模型和非线性耦合的适当的选择，我们已经探索古典Liénard型系统的微观量子概括。利用振荡器的相干态和相关的规范热分布$C$数字，我们导出了简化系统的量子Langevin方程，该方程允许单个或多个极限环。已经表明，即使在真空激励的情况下，以波动-耗散关系的形式的详细平衡也保持了吸引器的动态稳定性。Rayleigh，van der Pol和Liénard振荡器的其他几种变体的量子形式是在我们的理论方案中通过均值场描述得出的特殊情况。