Superfluidity of fermionic pairs in a harmonic trap. Comparative studies: Local Density Approximation and Bogoliubov-de Gennes solutions

Abstract

Experiments with ultracold gases on the lattice give the opportunity to realize superfluid fermionic mixtures in a trapping potential. The external trap modifies the chemical potential locally. Moreover, this trap also introduces non-homogeneity in the superconducting order parameter. There are, among other approaches, two methods which can be used to describe the system of two-component mixtures loaded into an optical lattice: the Local Density Approximation (LDA) and the self-consistent Bogoliubov-de Gennes equations. Here, we compare results obtained within these two methods. We conclude that the results can be distinguishable only in the case of a small value of the pairing interaction.Recent progress in experimental techniques in ultracold quantum gases gives ground to believe that they are good candidates for realization of quantum simulators [1][2][3][4][5]. This progress gives the opportunity to study different types of quantum models, well established already in many-body physics [5][6][7].Parameters such as the depth of the periodic trapping potential or the lattice geometry can be tuned with a high level of control in experiments. Moreover, the interaction strengths can also be fully controlled via so-called Feshbach resonances [8], giving the possibility of realization of repulsively and attractively interacting systems [9]. The latter allows for an experimental study of superfluid states [10][11][12].Typically, a system of ultracold atoms is located in an external potential, for instance in a harmonic trap. This potential introduces some effective site-dependent chemical potential and non-homogeneous distribution of the particles in the system. As a consequence, non-uniform superconducting order parameter is expected. The system can be studied theoretically via several methods. In this paper, we compare the results obtained within the Local Density Approximation (LDA) and the self-consistent Bogoliubov-de Gennes (BdG) equations, for a system of an ultracold attractive fermionic gas loaded into a one-dimensional (1D) optical lattice, put in a harmonic trap.The paper is organized as follows: first, we shortly introduce the model as well as the above mentioned methods (section 2). In section 3, we present and discuss numerical results. Finally, we summarize in section 4. Model and techniquesWe investigate a fermionic gas loaded onto a one-dimensional lattice, in a harmonic trap, which is schematically illustrated in figure 1. The system under consideration can be described by the Hamiltonian