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

Abstract

Vortices and quasiparticles in 2D systems are studied for the last decades in relation to the development of technologies in quantum electromagnetics, optics, and quantum computation. Hamiltonians are a fundamental part for studying quasiparticles and vortices, and provide models to calculate the eigenvalues and eigenfunctions that describe a real physical state. By devising supersymmetry, a Hamiltonian for the description of vortices is developed forming in an 2D electron gas and study the numerical solutions under the effects of an alternating electromagnetic field. The numerical analysis shows that vorticity is formed spontaneously without symmetry-breaking and vortices arise from the boundaries and converge toward the center of the system in a similar fashion to natural hydrodynamic phenomena in water or plasma. Additionally, the equation under damping conditions is studied, a homogenous magnetic field and under the absence of an electromagnetic field. The results and the study of the parameters indicate that the supersymmetic wave equation (SWE) may be a good model equation to describe vorticity for quantum electromagnetics, hydrodynamics, and other physical phenomena in the realm of physical and quantum physical sciences. levels of n. The model is similar to the Laughlin model which describes the formation of quasiparticles in a quantum hall, [2] however it acts as an extension in describing vorticity and quantum rotation. These vortices are in essence Bose-Einstein condensates [3] which arise at ultra-low temperatures within a gas such as Helium, also known as superfluids or quantum fluids. This phenomenon gives rise to properties such as superfluidity and superconductivity in gases and metals, where a common quantum state arises for holes with the same vorticity, and hence lead to a uniform band for superconductivity and superfluidity. This phenomenon is of particular appeal to quantum computing and quantum optics. [4][5][6][7] Additionally, quantum computing and optical technologies can be developed further also by constructing systems with alternating magnetic fields, which provide an auxiliary effect on the vortices and generates quantum wave dynamics that trigger exotic properties of appeal to the scientific community. For instance, systems of particles subjected to impulse (staggered) magnetic fields have been studied in ultra-cold atoms recently. [8] The study shows that electronic band structures formed under the effects of the variable magnetic field affect the shape of the Dirac cones formed in the conduction band, hence leading to changes in conductivity. Interestingly, the same study shows that the effect of a staggered magnetic field on bosons generates superfluid phases [8] leading to zero viscosity and therefore zero resistance in the optical band. Furthermore, a second study, [9] shows that staggered electromagnetic fields lead to the formation of uniform superfluids and Mott insulating phases [10] and a novel kind of finite-momentum superfluid phase, directly tunable by the quantized stagg...