A Unified Hamiltonian Solution To Maxwell–Schrödinger Equations For Modeling Electromagnetic Field–particle Interaction


A novel unified Hamiltonian approach is proposed to solve Maxwell–Schrödinger equation for modeling the interaction between classical electromagnetic (EM) fields and particles. Based on the Hamiltonian of electromagnetics and quantum mechanics, a unified Maxwell–Schrödinger 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 Schrödinger 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.


Electrical and Computer Engineering


Universiteit Stellenbosch, Grant 61201122

Keywords and Phrases

Finite-difference time-domain; Hamiltonian; Maxwell–Schrödinger equation; Rabi oscillation; Radiative decay; Reduced eigenmode expansion

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Article - Journal

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Publication Date

01 Jun 2017