Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates

By L. Salasnich, Alberto Parola, L. Reatto. 2002

469 citations recorded by the source corpus.

Starting from the three-dimensional (3D) Gross-Pitaevskii equation and using a variational approach, we derive an effective 1D wave equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry. The trapping potential is harmonic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent nonpolynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to model cigar-shaped condensates, whose dynamics is essentially 1D. We show that 1D NPSE gives much more accurate results than all other effective equations recently proposed. By using 1D NPSE we find analytical solutions for bright and dark solitons, which generalize the ones known in the literature. We deduce also an effective 2D nonpolynomial Schr\"odinger equation (2D NPSE) that models disk-shaped Bose condensates confined in an external trap that is harmonic along the axial direction and generic in the transverse direction. In the limiting cases of weak and strong interaction, our approach gives rise to Schr\"odinger-like equations with different polynomial nonlinearities.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2cd0c18325269ba694f57bc847c2727fc42d1b06

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Shortcuts to Adiabaticity (2012, background): “Integrating out the transverse modes, and up to a time-dependent constant which can be gauged away, one obtains the reduced GPE i~ ∂ψ(z, t) ∂t = [ − ~ 2 2m ∂2 ∂z2 +Vex(z) + g1(t)|ψ(z, t)|2 ] ψ(z, t), (42) with the effective coupling g1(t) = g3 ∫∫ dxdy|Φ0(x, y, t)|4.”
Solitons in nonlinear lattices (2010, background): “…of the 1D density, ^ = W1/ = + 24 1 ( )R x qs , where R x , thus leading to the derivation of the 1D equa- tion with the nonpolynomial nonlinearity (Salasnich, Parola, and Reatto, 2002; see also an ^/N aº 2 ( )sa x( ) alternative form of the nonpolynomial nonlinearity, derived by Muñoz Mateo…”
Dynamics of Bose-Einstein condensates in optical lattices (2006, background): “This also reduces the description to one dimension, but one ends up with a nonpolynomial nonlinear Schrödinger equation Salasnich et al., 2002 .”