Unitary quantum lattice simulations for Maxwell equations in vacuum and in dielectric media

Abstract

Utilizing the similarity between the spinor representation of the Dirac and the Maxwell equations that has been recognized since the early days of relativistic quantum mechanics, a quantum lattice algorithm (QLA) representation of unitary collision-stream operators of Maxwell's equations is derived for both homogeneous and inhomogeneous media. A second-order accurate 4-spinor scheme is developed and tested successfully for two-dimensional (2-D) propagation of a Gaussian pulse in a uniform medium whereas for normal (1-D) incidence of an electromagnetic Gaussian wave packet onto a dielectric interface requires 8-component spinors because of the coupling between the two electromagnetic polarizations. In particular, the well-known phase change, field amplitudes and profile widths are recovered by the QLA asymptotic profiles without the imposition of electromagnetic boundary conditions at the interface. The QLA simulations yield the time-dependent electromagnetic fields as the wave packet enters and straddles the dielectric boundary. QLA involves unitary interleaved non-commuting collision and streaming operators that can be coded onto a quantum computer: the non-commutation being the very reason why one perturbatively recovers the Maxwell equations.

Quantum Lattice Representation for the Curl Equations of Maxwell Equations

“…The pulse propagates from a region of refractive index n 1 that is joined to another dielectric of refractive index n 2 by a thin boundary layer. We recover exactly the same physics as with the QLA for the full Maxwell equations [1,2]: in particular the transmitted to initial field amplitude scales as 2…”

“…With that choice we now proceed to determine the required form of the collision, scattering and potential operators that will recover the two curl equations of Maxwell in the continuum limit. As details are presented in our earlier papers [1][2][3][4][5], we just summarize the results here. Since we are considering 1D propagation in the x-direction, we automatically have E x = 0 = B x .…”

“…Gauss' law can also be problematic and not so easily discarded. Here we shall consider one dimensional (1D) propagation of an electromagnetic pulse, governed by the two curl Maxwell equations of Eq, (1). In a scalar dielectric medium these 1D pulse are transverse so that automatically ∇.…”

“…Recently [1][2][3][4][5] we have been developing quantum lattice algorithms (QLA) for the solution of the full Maxwell equations in 1D, 2D and 3D. In this note we wish to develop a 1D QLA for the solution of the two curl Maxwell equations.…”

“…The QLA [1][2][3][4][5][11][12] consists of a sequence of non-commuting collide-stream operators and some potential collision operators so chosen that in the continuum limit one recovers the Khan representation. In Sec 3 we present initial value 1D QLA simulations for a Gaussian pulse propagating in the x-direction and consider the two polarizations: one with non-zero field components E y and B z and the other with non-zero −E z and B y .…”