López-Saldívar Julio A, Man'ko Margarita A, Man'ko Vladimir I
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apdo. Postal 70-543, Ciudad de México 04510, Mexico.
Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, 141700 Moscow Region, Russia.
Entropy (Basel). 2020 May 23;22(5):586. doi: 10.3390/e22050586.
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.
在阐述的微分方法中，我们研究高斯、混合、连续变量密度矩阵参数的演化，其动力学由表示为位置和动量算符或正交分量二次形式的厄米哈密顿量给出。具体而言，我们以一般形式得到了多体高斯态的协方差矩阵、平均值和密度矩阵参数的微分方程，该多体高斯态根据哈密顿量(\hat{H})进行幺正演化。我们还给出了描述子系统非幺正演化的相应微分方程。所得的非线性方程用于求解系统的动力学，而非薛定谔方程。所阐述的形式体系使我们能够定义新的特定不变态和准不变态，以及具有不变协方差矩阵的态，即仅平均值根据经典哈密顿方程演化的态。通过使用位置表象和断层概率表象中的密度矩阵，我们研究了这些性质的示例。作为示例，我们给出了双模频率转换器的新型不变态和二分参量放大器的准不变态。
