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Journal of Experimental and Theoretical Physics Letters

Controlling potential traps for filtering solitons in Bose-Einstein condensates

Renato Fedele, Padma Kant Shukla, Sergio M. De Nicola, Margarita A. Man’ko, Vladimir I. Man’ko, F. S. Cataliotti

Published Oct 1, 2004• 19 citations
Publisher

Abstract

We present a controlling potential method for solving the three-dimensional Gross-Pitaevskii equation (GPE), which governs the nonlinear dynamics of the Bose-Einstein condensates (BECs) in an inhomogeneous potential trap. Our method allows one to construct ground and excited matter wave states whose longitudinal profiles can have bright solitons. This method provides the confining potential that filters and controls localized BECs. Moreover, it is predicted that, while the BEC longitudinal soliton profile is controlled and kept unchanged, the transverse profile may exhibit oscillatory breathers (the unmatched case) or move as a rigid body in the form of either coherent states (performing the Lissajous figures) or a Schrödinger cat state (matched case).

DOI
10.1134/1.1846114

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We present a controlling potential method for solving the three-dimensional Gross-Pitaevskii equation (GPE), which governs the nonlinear dynamics of the Bose-Einstein condensates (BECs) in an inhomogeneous potential trap. Our method allows one to construct ground and excited matter wave states whose longitudinal profiles can have bright solitons. This method provides the confining potential that filters and controls localized BECs. Moreover, it is predicted that, while the BEC longitudinal soliton profile is controlled and kept unchanged, the transverse profile may exhibit oscillatory breathers (the unmatched case) or move as a rigid body in the form of either coherent states (performing the Lissajous figures) or a Schrödinger cat state (matched case).
The results of recently developed investigations, that have been carried out within the framework of the controlling potential method (CPM), are reviewed. This method allows one to decompose a three dimensional (3D) Gross‐Pitaevskii equation (GPE) into the pair of coupled Schrödinger‐type equations. Under suitable mathematical conditions, the solutions of the 3D controlled GPE can be constructed from the solutions of a 2D linear Schrödinger equation (the transverse component of the GPE) coupled with a 1D nonlinear Schrödinger equation (the longitudinal component of the GPE). Such decomposition allows one to cast the solutions in the form of the product of the solutions of the transverse and the longitudinal components of the GPE. The coupling between these two equations is the functional of both the transverse and the longitudinal profiles. It is shown that the CPM can be used to obtain a new class of three‐dimensional solitary waves solutions of the GPE, which governs the dynamics of Bose‐Einstein condensates. By imposing an external controlling potential, the desired time‐dependent shape of the localized BECs is obtained. The stability of the exact solutions was checked with direct simulations of the time ‐dependent, three‐dimensional GPE. Our simulations show that the localized condensates are stable with respect to perturbed initial conditions.
We present two families analytical solutions to the extended Gross-Pitaevskii equation (GPE), which describe the dynamics of bright and dark solitons in Bose-Einstein condensations(BECs) with varying coefficients of two and three-body interactions in a parabolic and complex potential. The properties of the dynamical evolutions of the bright and dark matter-wave solitons are investigated. It is interesting that the matter-wave can be controlled by applying the external magnetic field and a background.
The possibility of the decomposition of the three-dimensional (3D) Gross–Pitaevskii equation (GPE) into apair of coupled Schrödinger-type equations, is investigated. It is shown that, under suitable mathematical conditions, it is possible to construct the exact controlled solutions of the 3D GPE from the solutions of a linear 2D Schrödinger equation coupled with a 1D nonlinear Schrödinger equation (the transverse and longitudinal components of the GPE, respectively). The coupling between these two equations is the functional of the transverse and the longitudinal profiles. The applied method of nonlinear decomposition, called the controlling potential method (CPM), yields the full 3D solution in the form of the product of the solutions of the transverse and longitudinal components of the GPE. It is shown that the CPM constitutes a variational principle and sets up a condition on the controlling potential well. Its physical interpretation is given in terms of the minimization of the (energy) effects introduced by the control. The method is applied to the case of a parabolic external potential to construct analytically an exact BEC state in the form of a bright soliton, for which the quantitative comparison between the external and controlling potentials is presented.
The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE.