Boris Malomed | school of electrical enginnering - Tel Aviv university - Israel

Abstract

The project aims to predict a possibility of the existence of gap solitons (GSs) in degenerate Fermi gases (DFGs) trapped in periodic potentials of the optical-lattice (OL) type. The description will be based on stationary equation(s) of the mean-field-hydrodynamic (MFHD) type for effective real wave function(s) $\psi $ (in fact, $\psi$ is defined as the square root of the local density in the DFG). The derivation of the MFHD equations from the one-, two-, and three-dimensional (1D, 2D, 3D) local (“microscopic") Fermi distributions, and applicability conditions for the equations will be revised. The effective macroscopic geometry (which is determined by anadditional confining field applied to the DFG, in a combination with the OL) may also be one-, two-, and three-dimensional (the macroscopicdimension should be equal to or smaller than its microscopic counterpart). It is expected that some equations derived in this way will be new, and some may coincide with previously known ones. The most fundamental set of MFHD equations, derived from the underlying 3D Fermi distribution, is expected to feature an effective self-repulsion term $\sim \psi ^{7/3}$. A fundamental problem is to construct families of fundamental 1D, 2D, and 3D GS solutions in this equation (this problem was never considered before). A modification of the model for a mixture of two mutually repulsive species of fermion atoms will be considered too; in that case, the equation contains a combination of repulsive nonlinear terms $\psi ^{7/3}$ and $\psi ^{3}$. The analysis will be focused on finding compact (tightly-bound) solitons, and willemploy a combination of a semi-analytical variational approximation (VA) and direct numerical solutions. In addition to the fundamental solitons, it is planned to look for other soliton families, such as subfundamental ones, as well as even and odd bound states of the fundamental solitons. The results and methods to be elaborated in this work may also help to produce new results for GSs in Bose-Einsteincondensates trapped in OL potentials. (AU)