最大非线性且非保守的量子电路
Maximally Nonlinear and Nonconservative Quantum Circuits
AI总结：
本文提出一种基于关联矩阵的算法，用于求解最大非线性电路（含约瑟夫森结和量子相位滑移）的保守能量与非保守功率，无需拉格朗日量，通过变换得到哈密顿量，并处理耗散、噪声及辅助元件，实现完整正则坐标。
摘要：
在本文中，我们引入一种算法方法，基于电路有向图的关联矩阵，来求解一大类最大非线性电路（包括约瑟夫森隧道结、相干量子相位滑移和超导环路）的保守能量和非保守功率。我们考虑双端口线性电路，其约束大多为完整约束，由麦克斯韦-基尔霍夫电流规则或麦克斯韦-基尔霍夫电压规则提供。电路的独立变量（通常是自由度的超集）从麦克斯韦-基尔霍夫电流或电压规则的解空间中获得。该方法不需要寻找任何拉格朗日量。相反，电路经典或量子哈密顿量是通过与哈密顿方程互补的变换，从无功（即保守）电路元件的能量中获得的。耗散（损失）通过使用瑞利耗散函数并定义广义泊松括号——泊松-瑞利括号来考虑。涨落（噪声）作为由浴模式表征的电压或电流源加入。非保守元件（例如，有噪声的电阻器）从一开始就使用关联矩阵方法包含在内，无需将它们作为独立元件处理。最后，我们表明，为了形成完整的正则坐标集，需要辅助（在某些情况下可能是寄生）电路元件来寻找具有不完整广义速度集的电路的哈密顿量。特别地，我们引入了两种方法来消除与辅助元件相关的坐标，要么通过哈密顿量约化，要么通过运动方程约化。我们使用辅助电路元件来处理一个同时包含约瑟夫森结和量子相位滑移的最大非线性电路。
英文摘要：
In this article, we introduce an algorithmic method to find the conservative energy and non-conservative power of a large class of maximally nonlinear electric circuits (including Josephson tunnel junctions, coherent quantum phase slips, and superconducting loops), based on the incidence matrix of the circuits' digraph. We consider two-port linear circuits with mostly holonomic constraints provided by either Maxwell-Kirchhoff's current rules or Maxwell-Kirchhoff's voltage rules. The circuit's independent variables, generally a superset of the degrees of freedom, are obtained from the solution space of Maxwell-Kirchhoff's current or voltage rules. The method does not require to find any Lagrangian. Instead, the circuit's classical or quantum Hamiltonian is obtained from the energy of the reactive (i.e., conservative) circuit elements by means of transformations complementary to Hamilton's equations. Dissipation (loss) is accounted for by using the Rayleigh dissipation function and defining generalized Poisson brackets--Poisson-Rayleigh brackets. Fluctuations (noise) are added as voltage or current sources characterized by bath modes. Non-conservative elements (e.g., noisy resistors) are included ab initio using the incidence-matrix method, without needing to treat them as separate elements. Finally, we show that in order to form a complete set of canonical coordinates, auxiliary (which could be parasitic in certain cases) circuit elements are required to find the Hamiltonian of circuits with an incomplete set of generalized velocities. In particular, we introduce two methods to eliminate the coordinates associated with the auxiliary elements by either Hamiltonian reduction or equation-of-motion reduction. We use auxiliary circuit elements to treat a maximally nonlinear circuit comprising simultaneously both a Josephson junction and a quantum phase slip.
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