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Description
Bose Einstein condensation (BEC) is a unique state of matter where bosons coalesce to the same (quantum) wave function. Because of their unique properties, BECs allow the possibility of studying quantum dynamics at the macroscopic level. This leads to many exciting applications in gyroscopes, interferometry, and quantum computing. Exciton-polariton systems represent a novel realization of BEC at relatively high temperatures (in contrast with standard BEC that need to be help at nano Kelvin temperatures). In two dimensions, vortices represent the basic/prevalent BEC solutions which are currently being explored both analytically and experimentally. The goal of our work is to study the dynamics and interactions between vortices in exciton-polariton BECs and in particular to (a) showcase the difference between vortex-vortex interactions in standard and exciton-polariton BECs and (b) to cast reduced particle-like equations for vortex-vortex interactions. Our starting model consists of the standard BEC model consisting of the nonlinear Schrodinger equation (NLS), a partial differential equation ̈ (PDE), with no external potential nor loss/gain sources (the vanilla case). We study the vortex steady state and the interaction between two same-charged vortices using a combination of optimization, finite differencing, and Mod-Squared Dirichlet/Laplacian-zero boundary conditions. The study of the vanilla NLS case serves both as a testing ground for our numerics and a quantitative measurement of how vortices behave in the absence of gain/loss or external influences. We then extend the vanilla NLS to model exciton-polariton BECs by incorporating gain and loss terms that are intrinsic to these systems. Due to the presence of gain/loss, vortices inherently develop a spiral phase that is responsible for the spiraling out of vortex-vortex pairs (trait that is absent in the vanilla case where two same-charge vortices just rotate in a circular motion). By phenomenologically fitting the corresponding dynamics for the exciton-polariton case, we are able to obtain a reduced ordinary differential equation (ODE) model that accurately describes the interactions between two (or several) vortices interactions as if they were particle-like entities.
