Quantum Electrodynamics is Crucial for Plasmonic Resonance of Metallic Nanostructures

Plasmonic resonance of a metallic nanostructure results from coherent motion of its conduction electrons driven by incident light. At the resonance, the induced dipole in the nanostructure is proportional to the number of the conduction electrons, hence $10^{7}$ times larger than that in an atom. The interaction energy between the induced dipole and fluctuating virtual field of the incident light can reach a few tenths of an eV. Therefore, the classical electromagnetism dominating the field becomes inadequate. We argue that quantum electrodynamics (QED) should be used instead as the fundamental theory to describe the virtual field and its interaction with the electrons. Based on QED, we derive analytic expressions for the plasmonic resonant frequency, which depends on three easily accessible material parameters. The analytic theory reproduces very well the experimental data, and can be used for rational design of materials.