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arXiv:0804.1277 · Laser Physics

Hydrodynamics of Bose and Fermi superfluids at zero temperature: the superfluid nonlinear Schrödinger equation

Luca Salasnich

Published Apr 1, 2009• 37 citations
arXivHTMLPDF

Abstract

We discuss the zero-temperature hydrodynamics equations of bosonic and fermionic superfluids and their connection with generalized Gross-Pitaevskii and Ginzburg-Landau equations through a single superfluid nonlinear Schrödinger equation.
DOI
10.1134/s1054660x09040185

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We investigate the zero-temperature properties of a superfluid Bose-Fermi mixture by introducing a set of coupled Galilei-invariant nonlinear Schr\"odinger equations valid from weak coupling to unitarity. The Bose dynamics is described by a Gross-Pitaevskii-type equation including beyond-mean-field corrections possessing the correct weak-coupling and unitarity limits. The dynamics of the two-component Fermi superfluid is described by a density-functional equation including beyond-mean-field terms with correct weak-coupling and unitarity limits. The present set of equations is equivalent to the equations of generalized superfluid hydrodynamics, which take into account also surface effects. The equations describe the mixture properly as the Bose-Bose repulsive (positive) and Fermi-Fermi attractive (negative) scattering lengths are varied from zero to infinity in the presence of a Bose-Fermi interaction. The present model is tested numerically as the Bose-Bose and Fermi-Fermi scattering lengths are varied over wide ranges covering the weak-coupling to unitarity transition.
We introduce a nonlinear Schr\"odinger equation to describe the dynamics of a superfluid Bose gas in the crossover from the weak-coupling regime, where $a{n}^{1/3}\ensuremath{\ll}1$ with aaa the interatomic s\mathit{s}s-wave scattering length and nnn the bosonic density, to the unitarity limit, where $a\ensuremath{\rightarrow}+\ensuremath{\infty}$. We call this equation the unitarity Schr\"odinger equation (USE). The zero-temperature bulk equation of state of this USE is parametrized by the Lee-Yang-Huang low-density expansion and Jastrow calculations at unitarity. With the help of the USE we study the profiles of quantized vortices and vortex-core radius in a uniform Bose gas. We also consider quantized vortices in a Bose gas under cylindrically symmetric harmonic confinement and study their profile and chemical potential using the USE and compare the results with those obtained from the Gross-Pitaevskii-type equations valid in the weak-coupling limit. Finally, the USE is applied to calculate the breathing modes of the confined Bose gas as a function of the scattering length.
The Gross-Pitaevskii equation, or more generally the nonlinear Schrödinger equation, models the Bose-Einstein condensates in a macroscopic gaseous superfluid wave-matter state in ultra-cold temperature. We provide analytical study of the NLS with L2 initial data in order to understand propagation of the defocusing and focusing waves for the BEC mechanism in the presence of electromagnetic fields. Numerical simulations are performed for the two-dimensional GPE with anisotropic quadratic potentials.