Analytical Solutions for a Supersymmetric Wave‐Equation for Quasiparticles in a Quantum System

Abstract

Vortices and quasiparticles have been studied for 2D systems over the past decades in relation to the development of technologies in quantum electromagnetics, optics, and quantum computation. Deriving new equations for quasiparticles in quantum systems is therefore a critical part of quantum physics. Recently, a supersymmetric wave equation has been developed which describes quasiparticles and vorticity under the influence of electromagnetic fields which has been studied numerically. The analytical solutions of the supersymmetric wave‐equation are presented by increasing levels of energy. It is shown that the derived linear and nonlinear models of the supersymmetric wave‐equation can represent respectively symmetric and asymmetric vorticity formed in a confined rotating torus. The models described are suitable to represent vorticity formed in 2D and 3D electron gas and can be of significant relevance to future research in quantum electromagnetics.

Supersymmetric Hamiltonian and Vortex Formation Model in a Quantum Nonlinear System in an Inhomogeneous Electromagnetic Field

“…Figure shows furthermore the numerical solution of the system given in under a homogeneous magnetic field (where the periodical initial condition in is absent). The vorticity formed under the homogeneous magnetic field (Figure , 3rd row) is more pronounced than under the absence of a magnetic field (Figure , 2nd row) and shares similarities to our previous work . The model suggests therefore that vorticity arises both under the presence and also under absence of a magnetic field, which is well in agreement with literature of forming vorticity without magnetic fields (i.e., hydrodynamic vorticity or in turbulent fluids).…”

“…In regular Gross–Pitaevskii models the β parameter is related both to the number of particles and their interaction with one another, however in our SWE it defines rather the vortex (quasiparticle) interaction and hence the preferred symmetry the vortices adapt. It follows thus from the data that the β parameter does not affect the number of vortices (see Figure ), but rather affects the amplitude of the wave function, as we have previously established in our work . Most importantly, the β parameter allows vortices to be formed spontaneously, without the use of auxiliary functions or perturbations.…”

“…In a recent work we developed a Hamiltonian for the formation of quasiparticles in a rotating system, subjected to an electromagnetic field . The Hamiltonian .…”

“…There are however no models to our knowledge that describe the effects on vortices in a 2D system subjected to an alternating magnetic field, tuned to oscillate harmonically. We have therefore selected this particular feature by the study of a supersymmetric variant of the Hamiltonian for 2D electron gas systems, as studied and developed to generate wavefunctions for 2D systems by Laughlin …”

“…As mentioned above, in ref. [], we used supersymmetry rules to derive a supersymmetric form of . This form was derived by considering both the negative as well as the positive sign for the two factorial components present in Equations and : yielding the superpair of the two factors: …”