Maxwell-Schrodinger方程的统一哈密顿解法：用于建模电磁场-粒子相互作用
A Unified Hamiltonian Solution to Maxwell-Schrodinger Equations for Modeling Electromagnetic Field-Particle Interaction
AI总结：
本文提出统一哈密顿方法求解Maxwell-Schrodinger方程，用于建模电磁场与粒子的相互作用。通过变分原理推导出统一系统，确保能量守恒。通过降阶特征模展开技术，将粒子波函数转化为常微分方程，从而解决多尺度耦合系统中的数值挑战。
摘要：
本文提出了一种新的统一哈密顿方法，用于求解Maxwell-Schrodinger方程，以建模经典电磁场与粒子的相互作用。基于电磁学和量子力学的哈密顿量，通过变分原理推导出统一的Maxwell-Schrodinger系统。该耦合系统是良好设定且辛的，确保了时间演化过程中的能量守恒性质。然而，由于电磁波和电子波波长的差异性，对多尺度耦合系统的有限差分时域（FDTD）方法数值实现极具挑战性。为克服这一困难，首先应用降阶特征模展开技术来表示粒子的波函数。然后推导出一组描述扩展系数时间演化的一般微分方程（ODEs），以替代原始的薛定谔方程。最后，用矢量势表示的麦克斯韦方程（采用库仑规范）与这些ODEs一起进行自洽求解。在数值示例中，研究了电磁场与粒子在闭合、开放和非均匀电磁系统中的相互作用。所提出的方法不仅捕捉到闭合腔中的Rabi振荡现象，还捕捉到开放自由空间中的辐射衰减和位移效应。与现有理论近似模型相比，发现近似模型在某些情况下失效，当需要严格自洽方法时。这项工作对新兴纳米器件或下一代量子电动力学系统的电磁模拟具有帮助。
英文摘要：
A novel unified Hamiltonian approach is proposed to solve Maxwell-Schrodinger equation for modeling the interaction between classical electromagnetic (EM) fields and particles. Based on the Hamiltonian of electromagnetics and quantum mechanics, a unified Maxwell-Schrodinger system is derived by the variational principle. The coupled system is well-posed and symplectic, which ensures energy conserving property during the time evolution. However, due to the disparity of wavelengths of EM waves and that of electron waves, a numerical implementation of the finite-difference time-domain (FDTD) method to the multiscale coupled system is extremely challenging. To overcome this difficulty, a reduced eigenmode expansion technique is first applied to represent the wave function of the particle. Then, a set of ordinary differential equations (ODEs) governing the time evolution of the slowly-varying expansion coefficients are derived to replace the original Schrodinger equation. Finally, Maxwell's equations represented by the vector potential with a Coulomb gauge, together with the ODEs, are solved self-consistently. For numerical examples, the interaction between EM fields and a particle is investigated for both the closed, open and inhomogeneous electromagnetic systems. The proposed approach not only captures the Rabi oscillation phenomenon in the closed cavity but also captures the effects of radiative decay and shift in the open free space. After comparing with the existing theoretical approximate models, it is found that the approximate models break down in certain cases where a rigorous self-consistent approach is needed. This work is helpful for the EM simulation of emerging nanodevices or next-generation quantum electrodynamic systems.
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