In the differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian H^. We also present the corresponding differential equations, which describe the nonunitary evolution of the subsystems. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schrödinger equation. The formalism elaborated allows us to define new specific invariant and quasi-invariant states, as well as states with invariant covariance matrices, i.e., states were only the mean values evolve according to the classical Hamilton equations. By using density matrices in the position and in the tomographic-probability representations, we study examples of these properties. As examples, we present novel invariant states for the two-mode frequency converter and quasi-invariant states for the bipartite parametric amplifier.

中文翻译：

---

高斯态演化的微分参数形式主义：非酉演化和不变态


在阐述的微分方法中，我们研究高斯混合连续变密度矩阵参数的演化，其动力学由厄米哈密顿量给出，表示为位置和动量算子或正交分量的二次形式。具体来说，我们以通用形式获得协方差矩阵、平均值和多部分高斯状态的密度矩阵参数的微分方程，根据哈密顿量 H^ 统一演化。我们还提出了相应的微分方程，它描述了子系统的非酉演化。由此产生的非线性方程代替薛定谔方程来求解系统的动力学。所阐述的形式主义使我们能够定义新的特定不变和准不变状态，以及具有不变协方差矩阵的状态，即状态只是根据经典哈密顿方程演化的平均值。通过在位置和断层扫描概率表示中使用密度矩阵，我们研究了这些属性的示例。作为示例，我们提出了双模频率转换器的新颖不变状态和二分参量放大器的准不变状态。