Qubit Lattice Algorithm Simulations of the Scattering of a Bounded Two Dimensional Electromagnetic Pulse from an Infinite Planar Dielectric Interface

TL;DR

This paper extends the quantum lattice algorithm (QLA) approach to Maxwell equations to handle a finite, two-dimensional electromagnetic pulse incident on a planar dielectric interface. By employing a 2D bounded pulse and a collision–streaming scheme with carefully designed potential operators, the authors capture total internal reflection, evanescent-energy transfer, and a Goos-Hanchen-type lateral shift, while preserving energy to seven significant figures. For normal incidence, the study demonstrates that Fresnel coefficients for the electric and magnetic fields can be recovered by averaging the quasi-1D field components, validating the method against analytic expectations. The work also discusses the challenges of achieving a fully unitary evolution (via Dyson maps and linear-combination-of-units) and confirms energy conservation as a key metric, highlighting the potential for quantum hardware implementations of Maxwell solvers in complex dielectric geometries.

Abstract

Qubit lattice algorithm (QLA) simulations are performed for a two-dimensional (2D) spatially bounded pulse propagating onto a plane interface between two dielectric slabs. QLA is an initial value scheme that consists of a sequence of unitary collision and streaming operators, with appropriate potential operators, that recover Maxwell equations in inhomogeneous dielectric media to second order in the lattice discreteness. For the case of total internal reflection, there is transient energy transfer into the second medium due to the evanescent fields as the Poynting unit vector of the pulse is rotated from its incident to reflected direction. Because of the finite spatial extent of the pulse, a self-consistent Goos-Hanchen-type displacement along the interface is found without imposing any explicit interface boundary conditions on the fields. For normal incidence. the standard Fresnel coefficients are recovered for appropriately averaged QLA fields. Energy is conserved at all times to seven significant figures.

Figures (6)

• Figure 1: Initial 2D magnetic field amplitude $H_z$ that is bounded in both $\zeta$ and $\chi$. The initial pulse propagates in the $\zeta$-direction in dielectric $n_1$, with dispersion relation $\omega / k = c/n_1$. The initial electric field amplitude $E_{\chi} = Z_1 H_z$. The square lattice has a spatial grid of side $L = 1024$.
• Figure 2: The time evolution of the normalized energy in dielectric $n_1$, $\epsilon_1(t) = \mathcal{E}_1(t) / [\mathcal{E}_1(0) + \mathcal{E}_2(0) ]$, and in dielectric $n_2$, $\epsilon_2(t) = \mathcal{E}_2(t) / [\mathcal{E}_1(0) + \mathcal{E}_2(0) ]$ for $\theta = 25^o < \theta_c$ - - - dashed curves, and for $\theta=35^o>\theta_c$ - solid curves. At every time output, the total energy, $\mathcal{E}_1(t) + \mathcal{E}_2(t) = const.$ to the 7th significant figure. In these simulations $L=1024$. Thus the $\epsilon_1(35)$-plot is the time evolution of the normalized energy in the refractive index medium $n_1=2$ for angle of incidence $\theta = 35^o$.
• Figure 3: Evolution of the magnetic field $H_z(x,y) > 0$ for $\theta = 35^o >\theta_c$. Red: region $n_1=2, y < L/2$. Grey: region $n_2=1 , y \ge L/2$.
• Figure 4: Evolution of the magnetic field $H_z(x,y) > 0$ for $\theta = 35^o >\theta_c$ at later times. Red: region $n_1=2, y < L/2$. Grey: region $n_2=1 , y \ge L/2$. The Goos-Hanchen [9] longitudinal boundary shift is clearly seen on comparing Fig 3b and 4c.
• Figure 5: (a) The initial magnetic field $H_z(x,y)$ at $t = 0$ as the pulse propagates to the plane dielectric boundary, and (b) the reflected $H_z(x,y)$ at $t = 72k$ as the pulse moves from the dielectric interface. The color coding for the $H_z$-profile is shown in the horizontal strip: blue-green for $H_z< 0$ and yellow-red for $H_z > 0$.