Quantizing the discrete self-trapping equation

Terrence L. Reese, Bruce N. Miller

V. Z. Enolski, Mario Salerno, Nikolay A. Kostov +1 more

In this paper we study the classical and quantum two freedoms Discrete Self-Trapping system by the Inverse Scattering Method. We develop the algebraic approach in terms of the classical r -matrix and quantum R -matrix and give an interpretation of the commutation relations in terms of Sklyanin's quadratic algebra. The quantization is then developed in detail and the first energy levels are obtained. The results are compared with those obtained from the quantization scheme of Scott and Eilbeck.

J. I. Cirac, Luis Lorenzo Sánchez-Soto

Sergio De Filippo, M. Fusco Girard, Mario Salerno

Miguel Orszag, Ricardo Ramı́rez

Goran Gligorić, Aleksandra Maluckov, Luca Salasnich +2 more

The Bose-Einstein condensate (BEC), confined in a combination of the cigar-shaped trap and axial optical lattice, is studied in the framework of two models described by two versions of the one-dimensional (1D) discrete nonpolynomial Schrodinger equation (NPSE). Both models are derived from the three-dimensional Gross-Pitaevskii equation (3D GPE). To produce "model 1" (which was derived in recent works), the 3D GPE is first reduced to the 1D continual NPSE, which is subsequently discretized. "Model 2," which was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction in the dimension. The two models seem very different; in particular, model 1 is represented by a single discrete equation for the 1D wave function, while model 2 includes an additional equation for the transverse width. Nevertheless, numerical analyses show similar behaviors of fundamental unstaggered solitons in both systems, as concerns their existence region and stability limits. Both models admit the collapse of the localized modes, reproducing the fundamental property of the self-attractive BEC confined in tight traps. Thus, we conclude that the fundamental properties of discrete solitons predicted for the strongly trapped self-attracting BEC are reliable, as the two distinct models produce them in a nearly identical form. However, a difference between the models is found too, as strongly pinned (very narrow) discrete solitons, which were previously found in model 1, are not generated by model 2-in fact, in agreement with the continual 1D NPSE, which does not have such solutions either. In that respect, the newly derived model provides for a more accurate approximation for the trapped BEC.