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AIP conference proceedings

Multicomponent Nonlinear Systems of Bose-Fermi Fields: Exact Solutions

Nikolay A. Kostov, Vladimir Stefanov Gerdjikov, Tihomir Valchev, Kouei Sekigawa, Vladimir Stefanov Gerdjikov, Yasuo Matsushita, Ivaïlo M. Mladenov

Published Jan 1, 2011• 0 citations
Publisher

Abstract

We present families of stationary solutions for a multicomponent nonlinear system of two boson and Nf fermion fields in terms of elliptic functions of modulus k. This system is an extention of models, describing Bose‐Fermi mixtures in the mean field approximation. We also single out the particular cases when the quasiperiodic solutions become periodic ones. In the limit of sinusoidal external potential (k→0) our solutions model periodic waves trapped in an optical lattice. The other limit k→1 provides solutions expressed by hyperbolic functions (vector solitons). Thus we demonstrate that our system describes quasi‐periodic and periodic waves, as well as solitons.

DOI
10.1063/1.3582738

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We present families of stationary solutions for a multicomponent nonlinear system of two boson and Nf fermion fields in terms of elliptic functions of modulus k. This system is an extention of models, describing Bose‐Fermi mixtures in the mean field approximation. We also single out the particular cases when the quasiperiodic solutions become periodic ones. In the limit of sinusoidal external potential (k→0) our solutions model periodic waves trapped in an optical lattice. The other limit k→1 provides solutions expressed by hyperbolic functions (vector solitons). Thus we demonstrate that our system describes quasi‐periodic and periodic waves, as well as solitons.
We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus k. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential (k → 0) our solutions model a quasi-one dimensional quantum degenerate Bose- Fermi mixture trapped in optical lattice. In the limit k → 1 the solutions are expressed by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons. The precise conditions for existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.
We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus k.We also discuss particular cases when the quasiperiodic solutions become periodic ones.In the limit of a sinusoidal potential (k → 0) our solutions model a quasi-one dimensional quantum degenerate Bose-Fermi mixture trapped in optical lattice.In the limit k → 1 the solutions are expressed by hyperbolic function solutions (vector solitons).Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons.The precise conditions for existence of every class of solutions are derived.There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.
Coupled nonlinear Schrodinger equations (CNLS) with an external elliptic function potential model a quasi one--dimensional interacting two-component Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions of the CNLS with a periodic potential are presented and their stability studied by direct numerical simulations. Some of these solutions allow reduction to Manakov system. From a physical point of view these solutions can be interpreted as exact Bloch states at the edge of the Brillouin zone. Some of them are stable while others are found to be unstable against modulations of long wavelength. The solutions which are modulationally unstable are shown to lead to the formation of localized ground states of the coupled BEC system.
The cubic vector nonlinear Schrödinger equation with an external trigonometric potential models a quasi-one-dimensional multi-component Bose–Einstein condensate trapped in a standing light wave. We construct families of exact stationary solutions for the more general case of an elliptic function potential. Some of these solutions degenerate to zero as the effect of the external potential disappears, whereas others limit to solutions of the free vector nonlinear Schrödinger equation. The stability of these solutions is examined both analytically and numerically. The stability results depend on the nature of the atomic interactions not only within the components but also between components. As in the scalar case (one component) with repulsive interaction, all linearly stable solutions are deformations of the ground state of the linear Schrödinger equation. Unlike the scalar case with attractive interaction, no solutions are found to be stable if there is any attractive interaction present.
Using coupled equations for the bosonic and fermionic order parameters, we construct families of gap solitons (GSs) in a nearly one-dimensional Bose-Fermi mixture trapped in a periodic optical-lattice (OL) potential, the boson and fermion components being in the states of the Bose-Einstein condensation and Bardeen-Cooper-Schrieffer superfluid, respectively. Fundamental GSs are compact states trapped, essentially, in a single cell of the lattice. Full families of such solutions are constructed in the first two band gaps of the OL-induced spectrum, by means of variational and numerical methods, which are found to be in good agreement. The families include both intragap and intergap solitons, with the chemical potentials of the boson and fermion components falling in the same or different band gaps, respectively. Nonfundamental states, extended over several lattice cells, are constructed too. The GSs are stable against strong perturbations.