A quadratic invariant operator for general time-dependent three coupled nano-optomechanical oscillators is investigated. We show that the invariant operator that we have established satisfies the Liouville-von Neumann equation and coincides with its classical counterpart. To diagonalize the invariant, we carry out a unitary transformation of it at first. From such a transformation, the quantal invariant operator reduces to an equal, but a simple one which corresponds to three coupled oscillators with time-dependent frequencies and unit masses. Finally, we diagonalize the matrix representation of the transformed invariant by using a unitary matrix. The diagonalized invariant is just the same as the Hamiltonian of three simple oscillators. Thanks to such a diagonalization, we can analyze various dynamical properties of the nano-optomechanical system. Quantum characteristics of the system are investigated as an example, by utilizing the diagonalized invariant. We derive not only the eigenfunctions of the invariant operator, but also the wave functions in the Fock state.

中文翻译：

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动态不变量在一般瞬态三耦合纳米光机振荡器上的应用

研究了一般瞬态三耦合纳米光机械振荡器的二次不变算子。我们表明，我们建立的不变算子满足刘维尔-冯诺依曼方程并与其经典对应物一致。为了使不变量对角化，我们首先对其进行幺正变换。从这样的变换中，量子不变算子简化为一个相等但简单的算子，它对应于三个具有时间相关频率和单位质量的耦合振荡器。最后，我们使用酉矩阵对变换后的不变量的矩阵表示进行对角化。对角化的不变量与三个简单振荡器的哈密顿量相同。由于这种对角化，我们可以分析纳米光机械系统的各种动力学特性。以系统的量子特性为例，利用对角化不变量进行研究。我们不仅推导出不变算子的本征函数，而且推导出 Fock 状态下的波函数。