Confined quantum gases in a toroidal-like potential can be studied by a variational method using the like-Gross-Pitaevskii equations. The variational approach determines the dynamics of the quantum cloud, as the evolution of the aspect ratio during the free expansion of the cloud and the collective modes. We adapted the variational method to parametrize on a curved surface and effective dynamics. We use the thin-layer quantization scheme, focusing on three-dimensional surfaces, thus resorting to the Schrödinger equation for a particle with constrained dynamics to a curved surface. We propose a hybrid Lagrangian density and split the wave function into two parts: the normal and a tangential component to the surface that allow us to study the system through the extension of the variational method. Finally, we obtain the equations of motion that describe the dynamics separately in the normal and tangential parts on the surface.
|Ana Karen Cuervo Montiel
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