Quantum-statistical approach to electromagnetic wave propagation and dissipation inside dielectric media and nanophotonic and plasmonic waveguides

Quantum-statistical effects occur during the propagation of electromagnetic (EM) waves inside the dielectric media or metamaterials, which include a large class of nanophotonic and plasmonic waveguides with dissipation and noise. Exploiting the formal analogy between the Schrödinger equation and the Maxwell equations for dielectric linear media, we rigorously derive the effective Hamiltonian operator which describes such propagation. This operator turns out to be essentially non-Hermitian in general, and pseudo-Hermitian in some special cases. Using the density operator approach for general non-Hermitian Hamiltonians, we derive a master equation that describes the statistical ensembles of EM wave modes. The method also describes the quantum dissipative and decoherence processes which happen during the wave's propagation, and, among other things, it reveals the conditions that are necessary to control the energy and information loss inside the above-mentioned materials.

Figure 1

Propagation of EM wave through a dielectric medium located at $z\ge 0$.

Figure 2

EM wave propagation along the $z$ direction for different kinds of media. Vertical planes represent the generalized waveguide's effective cross-section, at different values of $z$. For a general medium, deviations from uniformity arise due to varying cross-section (wave-shaded areas on the lower panel) or varying permittivity and/or permeability (brick-shaded areas on the lower panel).

Figure 3

Eigenvalues (124) vs frequency. The vertical axis's units are $n{\omega }_0$ (left) or $n\times \text{Hz}$ (right). Solid curves correspond to ${\mathrm{\Lambda }}_{+}$, dashed ones to ${\mathrm{\Lambda }}_{-}$.

Figure 4

Period of oscillations (132) vs frequency. The vertical axis's units are ${\left(n{\omega }_0\right)}^{-1}$ (left) or ${(n\times \text{Hz})}^{-1}$ (right).

Figure 5

Total energy density (135), divided by ${({\mathcal{N}}_0/2)}^2$, vs $z$, at $\varepsilon =\mu +1=2$ and different frequencies: $\omega /{\omega }_u=1/5$ (solid curves), $1/2$ (dashed curves), 1 (dash-dotted curves), 2 (dotted curves), and 5 (dash-double-dotted curves), where ${\omega }_u$ equals to ${\omega }_0$ (left) or 1 Hz (right). The horizontal axis's units are $h_{-}{\omega }_0$ (left) or $h_{-}\times \text{Hz}$ (right).

Figure 6

Total energy density (135), divided by ${({\mathcal{N}}_0/2)}^2$, vs $z$, at $\omega /{\omega }_u=2$, $\mu =1$ and various values of relative permittivity: $\varepsilon =1$ (solid curves), $3/4$ (dashed curves), $3/2$ (dash-dotted curves), 2 (dotted curves), and 3 (dash-double-dotted curves), where ${\omega }_u$ equals to ${\omega }_0$ (left) or 1 Hz (right). The horizontal axis's units are $h_{-}{\omega }_0$ (left) or $h_{-}\times \text{Hz}$ (right).

Figure 7

Total energy density (135), divided by ${({\mathcal{N}}_0/2)}^2$, vs $z$, at $\omega /{\omega }_u=2$, $\varepsilon =3/2$ and various values of relative permeability: $\mu =1$ (solid curves), $3/4$ (dashed curves), $3/2$ (dash-dotted curves), 2 (dotted curves), and 3 (dash-double-dotted curves), where ${\omega }_u$ equals to ${\omega }_0$ (left) or 1 Hz (right). The horizontal axis's units are $h_{-}{\omega }_0$ (left) or $h_{-}\times \text{Hz}$ (right).

Figure 8

Gibbs-von-Neumann entropy (145) vs $z$, at $\varepsilon =\mu +1=2$ and different frequencies: $\omega /{\omega }_u=1/4$ (solid curves), $1/2$ (dashed curves), 0.9 (dash-dotted curves), 1 (dotted curves), and 2 (dash-double-dotted curve, right panel only), where ${\omega }_u$ equals to ${\omega }_0$ (left) or 1 Hz (right). The horizontal axis's units are $h_{-}{\omega }_0$ (left) or $h_{-}\times \text{Hz}$ (right).