圆柱波导的规范量子化：一种基于规范的方法
Canonical Quantization of Cylindrical Waveguides: A Gauge-Based Approach
AI总结：
本文提出了一种基于规范的方法，用于圆柱波导中电磁波模的规范量子化，统一了圆柱形和直角引导模式的处理，推导了电容、电感及电压、电流的表达式。
摘要：
我们提出了圆柱波导中电磁波模的规范量子化，扩展了之前为直角几何开发的规范形式化方法[1]。通过引入TEM（横电磁）模式的两个场四元 $X,Y$，也包括TM（横磁）和TE（横电）传播模式，我们为每种模式识别出一个特征一维标量场（一种广义通量 $φ$），其受类似克莱因-戈登方程的支配。关联的哈密顿量明确从麦克斯韦方程导出，允许构造玻色子阶梯算符。广义通量直接从电磁势 $A,V$ 通过适当的规范选择推导出来，扩展了Devoret的方法[2]。我们的分析在一致且通用的框架下统一处理圆柱形和直角引导模式，确保了理论洞察和实验相关性。我们从场分布中推导出特定模式的电容和电感，并将电压和电流用规范场变量表示。因此，可测量的量因此从模式量子算符中正确定义，特别是对于非平凡的TM和TE模式。该形式化将在未来的工作中扩展到其他类型的波导，特别是与量子技术特别相关的芯片上共面几何。
英文摘要：
We present a canonical quantization of electromagnetic modes in cylindrical waveguides, extending a gauge-based formalism previously developed for Cartesian geometries [1]. By introducing the two field quadratures $X,Y$ of TEM (transverse electric-magnetic), but also of TM (transverse magnetic) and TE (transverse electric) traveling modes, we identify for each a characteristic one-dimensional scalar field (a generalized flux $φ$) governed by a Klein-Gordon type equation. The associated Hamiltonian is derived explicitly from Maxwell's equations, allowing the construction of bosonic ladder operators. The generalized flux is directly deduced from the electromagnetic potentials $A,V$ by a proper gauge choice, generalizing Devoret's approach [2]. Our analysis unifies the treatment of cylindrical and Cartesian guided modes under a consistent and generic framework, ensuring both theoretical insight and experimental relevance. Especially, we find out that TE$_{n=0}$ modes break a fundamental gauge symmetry that is observed by all other families. We derive mode-specific capacitance and inductance from the field profiles and express voltage and current in terms of the canonical field variables. Measurable quantities are therefore properly defined from the mode quantum operators, especially for the non-trivial TM and TE ones. The formalism shall extend in future works to any other type of waveguides, especially on-chip coplanar geometries particularly relevant to quantum technologies.
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