Modeling electromagnetic field-matter interaction plays an essential role in technology and
science. Solving classical Maxwell’s equations with bulk electromagnetic parameters
including permittivity and permeability is a commonly-used tool to understand physical
effects and optimize engineering designs. At the quantum regime, when the size of
quantum particles (atoms, molecules, quantum dots, and superconducting circuits) is
pretty small compared to wavelength (typically smaller than 10 nm at optical frequencies),
the “homogenized” bulk permittivity and permeability of the classical Maxwell equation is
invalid or meaningless to describe electromagnetic responses of the quantum particles. In
this situation, quantum effects become significant and therefore the particle system needs
to be quantized.
If the field intensity is strong or the number of photons is large, semi-classical Maxwell-
Schrödinger system is adopted to simulate the electromagnetic field-particle interaction,
where electromagnetic field is treated as a classical field interacting with the quantum
particles. The coupled Maxwell-Schrödinger system can be solved by a unified
Hamiltonian approach with a symplectic framework. Versatile interesting physical
phenomena involving Rabi-oscillation, radiative decay and shift, electromagnetically
induced transparency, saturable absorption, and high-harmonic generation can be
reproduced by the semi-classical framework.
If the field intensity is very weak or the number of photons is quite small, both particle
system and electromagnetic system must be quantized and classical Maxwell equation
breaks down. Regarding quantization of Maxwell equation in lossy and dispersive media,
the quantized field-matter-bath system is still an energy-conserving Hamiltonian system.
When the matter couples to a heat bath (environment), it loses energy to the bath;
simultaneously, the bath feeds the energy back to the matter system. This obeys the spirits
of thermal equilibrium, fluctuation-dissipation theorem, and detailed balance theory. As a
result, we could introduce the loss to the quantized field-matter system by regarding the
heat bath as a Langevin source or noise current. The use of ubiquitous Green’s function is
still present in the full quantum calculations, which find rich applications in a great amount
of cutting-edge problems, such as spontaneous emission, single photon detection, cavity
quantum electrodynamics, and quantum interference.