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## The invariant eigen-operator method in matrix form and the eigenfrequency of several mesoscopic circuits

Wu Ze, Fan Hong-Yi
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• #### 摘要

本文把不变本征算符方法(invariant eigen-operator, IEO方法)推广到了基于拉格朗日量的矩阵形式, 将以往计算的思路和过程用简约的矩阵形式表示出来, 这对大规模复杂多回路的介观电路的计算有着重要的意义. 此外用该方法计算了三个LC介观电路的本征频率, 包括存在互感和不存在互感的两种情形. 通过计算结果得出了这些电路的相关性质, 说明了本征频率只与介观电路本身的元件性质有关.

#### Abstract

The Invariant Eigen-operator (IEO) method is widely used in solving the eigenfrequencies of the coulped quantum mesoscopic circuits. The previous IEO method is complicated but stylized, we always wasted much time in this boring processes. Here we extended the IEO method to the matrix form based on Lagrangian of the complex mesoscopic circuits, and express the ideas and processes of the previous calculations of the IEO method in a very simple matrix form. The mathematical methods we used is the indicator representation of the matrix, and we got a very simple and convenient matrix form of the IEO method. This form has important significance for the calculation of large-scale complex multi-loop mesoscopic circuits. Moreover, the matrix form of the IEO method is very friendly to the programming implementation of the complex quantum mesoscopic LC circuits, it is probably a most optimal algorithm for calculating the eigenfrequencies of the quantum mesoscopic LC circuits. In addition, with some help of computer programs, we used this method to calculate the eigenfrequencies of three LC mesoscopic circuits, including two cases with and without mutual inductance. We revealed some relevant properties of these circuits by calculating results, indicating that the eigenfrequency is only related to the element properties of the mesoscopic circuit itself. Finally, we found that this method can also be used in other areas like atom-light coupling systems and solid state physics.

#### 作者及机构信息

###### 通信作者: 范洪义, fhym@ustc.edu.cn
• 基金项目: 国家自然科学基金(批准号: 11775208)资助的课题

#### Authors and contacts

###### Corresponding author: Fan Hong-Yi, fhym@ustc.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11775208)

#### 施引文献

• 图 1  两回路$L\text{-}C$介观电路

Fig. 1.  Two-loop $L\text{-}C$ mesoscopic circuit

图 2  无互感的$L\text-C$介观电路

Fig. 2.  $L\text{-}C$ mesoscopic circuit without mutual inductance

图 3  带互感的$L\text-C$介观电路

Fig. 3.  $L\text-C$ mesoscopic circuit with mutual inductance 2m.