Reliability of the Ginzburg–Landau Theory in the BCS-BEC Crossover by Including Gaussian Fluctuations for 3D Attractive Fermions

We calculate the parameters of the Ginzburg–Landau (GL) equation of a three-dimensional attractive Fermi gas around the superfluid critical temperature. We compare different levels of approximation throughout the Bardeen–Cooper–Schrieffer (BCS) to the Bose–Einstein Condensate (BEC) regime. We show that the inclusion of Gaussian fluctuations strongly modifies the values of the Ginzburg–Landau parameters approaching the BEC regime of the crossover. We investigate the reliability of the Ginzburg–Landau theory, with fluctuations, studying the behavior of the coherence length and of the critical rotational frequencies throughout the BCS-BEC crossover. The effect of the Gaussian fluctuations gives qualitative correct trends of the considered physical quantities from the BCS regime up to the unitary limit of the BCS-BEC crossover. Approaching the BEC regime, the Ginzburg–Landau equation with the inclusion of Gaussian fluctuations turns out to be unreliable.

Reliability of the Ginzburg–Landau Theory in the BCS-BEC Crossover by Including Gaussian Fluctuations for 3D Attractive Fermions

We calculate the parameters of the Ginzburg–Landau (GL) equation of a three-dimensional attractive Fermi gas around the superfluid critical temperature. We compare different levels of approximation throughout the Bardeen–Cooper–Schrieffer (BCS) to the Bose–Einstein Condensate (BEC) regime. We show that the inclusion of Gaussian fluctuations strongly modifies the values of the Ginzburg–Landau parameters approaching the BEC regime of the crossover. We investigate the reliability of the Ginzburg–Landau theory, with fluctuations, studying the behavior of the coherence length and of the critical rotational frequencies throughout the BCS-BEC crossover. The effect of the Gaussian fluctuations gives qualitative correct trends of the considered physical quantities from the BCS regime up to the unitary limit of the BCS-BEC crossover. Approaching the BEC regime, the Ginzburg–Landau equation with the inclusion of Gaussian fluctuations turns out to be unreliable.

Numerical stability of solitons waves through splices in quadratic optical media

The propagation of soliton waves is simulated through splices in quadratic optical media, in which fluctuations of dielectric parameters occur. A new numerical scheme was developed to solve the complex system of partial differential equations (PDE) that describes the problem. Our numerical approach to solve the complex problem was based on the mathematical theory of Taylor series of complex functions. In this context, we adapted the Finite Difference Method (FDM) to approximate derivatives of complex functions and resolve the algebraic system, which results from the discretization, implicitly, by means of the relaxation Gauss-Seidel method. The mathematical modeling of local fluctuations of dielectric properties of optical media was performed by Gaussian functions. By simulating soliton wave propagation in optical fibers with Gaussian fluctuations in their dielectric properties, it was observed that the perturbed soliton numerical solution presented higher sensitivity to fluctuations in the dielectric parameter β, a measure of the nonlinearity intensity in the fiber. In order to verify whether the fluctuations of β parameter in the splices of the optical media generate unstable solitons, the propagation of a soliton wave, subject to this perturbation, was simulated for large time intervals. Considering various geometric configurations and intensities of the fluctuations of parameter β, it was found that the perturbed soliton wave stabilizes, i.e., the amplitude of the wave oscillations decreases as the values of propagation distance increases. Therefore, the propagation of perturbed soliton wave presents numerical stability when subjected to local Gaussian fluctuations (perturbations) of the dielectric parameters of the optical media.