Residual gauge theory for quanta of surface plasmons

TL;DR

This work develops a gauge-theoretical framework for the quanta of surface plasmons by quantizing Maxwell theory in the Coulomb gauge with explicit boundary counterterms, revealing a residual gauge symmetry that imposes a physical constraint tied to Joule heating. It constructs TM and TE plasmon quanta as entangled states of evanescent photons and surface currents, with dispersion relations determined by the dynamical conductivities ${oldsymbol extsigma}(oldsymbol{ angle},oldsymbol{ angle}$ via $ olinebreak 2i\,oldsymbol{ extomega}oldsymbol{ extxi}oldsymbol{ extepsilon}_0-oldsymbol{ extsigma}_{xx}(oldsymbol{ angle})=0$ and $2/(ioldsymbol{ extomega}oldsymbol{ extxi}oldsymbol{ extmu}_0)-oldsymbol{ extsigma}_{yy}(oldsymbol{ angle})=0$, and a Joule-heating operator $oldsymbol{Q}$ encoding photon–matter entanglement and dissipation. The framework yields concrete predictions for Drude and Lorentz media, magnetoplasmons, edge magnetoplasmons, and quantum Hall states, including TM/TE hybridization at boundaries and chiral edge modes with lifetimes modified by magnetic fields; it also connects residual gauge structure to topological aspects via analogies to Chern–Simons theory and edge states. These results clarify how dissipative boundaries and gauge constraints give rise to physical plasmon quanta that can retain light–matter entanglement, with potential implications for nanoscale quantum information processing and boundary-based spectroscopies.

Abstract

We develop a gauge-theoretical framework to investigate the quanta of surface plasmons. Our formulation, based on quantum electrodynamics, highlights the importance of residual gauge symmetry. We emphasize that residual gauge symmetry, which imposes constraint equations on physical states, is fundamentally linked to Joule heating. This framework is applied to metals, semiconductors, and quantum Hall states, suggesting the presence of a latent transverse electric mode and that the quanta have the ability to maintain light-matter entanglement.

Figures (2)

• Figure 1: (a) The transverse magnetic (TM) mode is defined as a surface wave with an electric field ${\bf E}_m=(E_x,0,E_z)$ and a magnetic field ${\bf B}_m=(0,B_y,0)$. The electric fields induce positive and negative charge densities, which are associated with an electric current ${\bf j}=(j_x,0,0)$. According to Ampère's circuital law, this current produces a discontinuity in $B_y$ at the plane $z=0$, as indicated by the symbols $\otimes$ and $\odot$. (b) The transverse electric (TE) mode is defined as a surface wave with ${\bf E}_e=(0,E_y,0)$ and ${\bf B}_e=(B_x,0,B_z)$. According to Ampère's circuital law, loops of the magnetic field induce electric current along the $y$-axis, ${\bf j}=(0,j_y,0)$, as indicated by the symbols $\otimes$ and $\odot$. We note that the electric (gauge) and magnetic fields are everywhere perpendicular both for the TM and TE modes, ${\bf E}\cdot {\bf B}=0$ (${\bf A}\cdot {\bf B}=0$). Ranada1989
• Figure 2: (a) Drude model: The dispersion of the TM mode $\omega_p(k_x)$ is plotted as a function of $k_x$. The intersection between the TM mode and the light dispersion $ck_x$ occurs at $k_m^*$, where $\xi=\infty$. Stable localized waves can exist for $k_x> k_m^*$. The TE mode appears as a flat band along the $k_x$-axis. (b) Lorentz model: The dispersion of the TM mode appears as a nearly linear band with a constant group velocity. The TE mode exists below the gap and remains immobile. Although anticrossing is expected near the intersection of the TM (TE) and Radiation modes, we omit it here because our focus is on $k_x> k_m^*$.