Tightly bound gap solitons in a Fermi gas

Abstract

Abstract. -Within the framework of the mean-field-hydrodynamic model of a degenerate Fermi gas (DFG), we study, by means of numerical methods and variational approximation (VA), the formation of fundamental gap solitons (FGSs) in a DFG (or in a BCS superfluid generated by weak interaction between spin-up and spin-down fermions), which is trapped in a periodic opticallattice (OL) potential. An effectively one-dimensional (1D) configuration is considered, assuming strong transverse confinement; in parallel, a proper 1D model of the DFG (which amounts to the known quintic equation for the Tonks-Girardeau gas in the OL) is considered too. The FGSs found in the first two bandgaps of the OL-induced spectrum ( unless they are very close to edges of the gaps) feature a (tightly-bound) shape, being essentially confined to a single cell of the OL. In the second bandgap, we also find antisymmetric tightly-bound subfundamental solitons (SFSs), with zero at the midpoint. The SFSs are also confined to a single cell of the OL, but, unlike the FGSs, they are unstable. The predicted solitons, consisting of ∼ 10 4 − 10 5 atoms, can be created by available experimental techniques in the DFG of 6 Li atoms.Introduction: Matter-wave solitons were created as localized nonlinear excitations in Bose-Einstein condensates (BECs) of attractively interacting 7 Li [1] and 85 Rb [2] atoms loaded in a cigar-shaped trap. In both cases, the interaction was switched from repulsion to attraction by applying magnetic field near the Feshbach resonance [3,4]). In the absence of axial confinement, the solitons can propagate freely in the axial direction. This was followed by the creation of gap solitons, GSs (formed by a few hundred atoms of 87 Rb) in a self-repulsive BEC loaded in a cigar-shaped trapped combined with an optical lattice (OL) [5], which was created as the interference pattern by counter-propagating laser beams (see also review [6]). Theoretical description of the dilute BEC relies upon the Gross-Pitaevskii equation (GPE), which provides for a remarkably accurate description of various matter-wave patterns, including solitons [7]. In particular, GSs exist at values of the chemical potential that fall in finite gaps of the band spectrum of the linear problem, induced by the periodic OL potential. In that case, the system possesses a negative effective mass, which allows the formation of solitons in balance with the repulsive nonlinearity [8].

Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps

“…The density scale is represented by the intensity of the print. Horizontal stipes represent double well structure and modulation illustrate the lattice latter case (for σ = +1) a known fact is that solutions exist only for ρ > ρ (0) ≡ e 2 /16 ≈ 0.462 (in fact, this constraint predicts, with high accuracy, the edge of the first finite bandgap in the linear spectrum induced by the OL [20]). Results for asymmetric solitons are presented in the next section.…”

“…For symmetric and antisymmetric solitons, Eqs. (14) and (16), with ν = 0, are tantamount to equations that were derived, by means of the VA, for solitons in 1D models with a periodic sinusoidal potential and attractive or repulsive nonlinearity [19,20]. In particular, in the Fig.…”

“…(20) is a small correction to 2, the main correction given by the last term, which demonstrates that theoretically there may be two different bifurcation points. Obviously, expressions (20) are meaningful, i.e., the bifurcation takes place, if…”

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

“…The SNLS equation can be applied to study not only collective modes [1,2,11,22] and free expansion [1,2,26], but also quantized vortices where the characteristic length is the vortex-core size [1,10], and tunneling phenomena where the characteristic length is the tunneling penetration depth [5,27]. In addition, with the SNLS equation one can investigate interesting nonlinear effects, like solitons [25,28], shock waves [19], and also chaos [29]. Finally, we observe that the SNLS equation satisfies the requirements suggested by Greiter, Wilczek and Witten [30] to have a well-founded theory of neutral superconductors: it is Galilei invariant, it mantains the current-momentum algebric identity, and at low-energy it shows a Nambu-Goldstone boson field with linear dispersion relation, i.e.…”

Gap solitons in elongated geometries: The one-dimensional Gross-Pitaevskii equation and beyond

“…In such condensates, the radial ground state satisfying Eq. (12) is, in general, a linear combination of many harmonic-oscillator modes (a similar situation takes place in the derivation of the effective 1D equation for fermionic gases by means of the densityfunctional approach [25]). …”