Nonlinear coherent modes of trapped Bose-Einstein condensates

Abstract

Nonlinear coherent modes are the collective states of trapped Bose atoms, corresponding to different energy levels. These modes can be created starting from the ground state condensate that can be excited by means of a resonant alternating field. A thorough theory for the resonant excitation of the coherent modes is presented. The necessary and sufficient conditions for the feasibility of this process are found. Temporal behaviour of fractional populations and of relative phases exhibits dynamic critical phenomena on a critical line of the parametric manifold. The origin of these critical phenomena is elucidated by analyzing the structure of the phase space. An atomic cloud, containing the coherent modes, possesses several interesting features, such as interference patterns, interference current, spin squeezing, and massive entanglement. The developed theory suggests a generalization of resonant effects in optics to nonlinear systems of Bose-condensed atoms. 03.75. Fi

Two-stage continuation algorithms for Bloch waves of Bose–Einstein condensates in optical lattices

“…Any nontrivial solution of (1.5) which minimizes the energy functional (1.6) is called a nonlinear eigenstate [5] or nonlinear coherent mode [6] associated with the nonlinear eigenvalue μ. Bloch waves are nonlinear eigenstates of the form (1.7) where φ k (x) is a complex periodic function of period 2π and k the Bloch wavenumber. Substituting (1.7) into (1.5), we obtain the following equation for each Bloch wave state φ k : (1.9) where ν Ω denotes the volume of Ω.…”

Nonequilibrium Bose systems and nonground-state Bose-Einstein condensates

“…which could be considered as a normalization condition, instead of Eq. (15). For two field variables, we have to fix not less and not more than two normalization conditions.…”

“…By employing the representative ensemble, as is done in the present paper, we introduce two Lagrange multipliers, µ 0 and µ 1 , for two normalization conditions (14) and (15). These multipliers are not obliged to be equal, but are to be such that to render the whole theory completely selfconsistent.…”

Small-Amplitude Nonlinear Modes under the Combined Effect of the Parabolic Potential, Nonlocality and PT Symmetry

“…Presence of the external potential or nonlocality enriches the variety of possible nonlinear modes allowing to observe the so-called higher (alias multi-pole or excited) nonlinear modes [6,[23][24][25][26][27][28] apart from the best studied fundamental (alias ground state) nonlinear modes [5,[29][30][31][32]. Nonlocal nonlinearities have been considered in combination with the double-well [33,34] and the periodic potentials [35][36][37][38].…”

“…In our present study, we make a step towards filling this gap and investigate nonlocal nonlinear modes in a complex parabolic P T -symmetric potential of the form V(x) = x 2 − 2iαx, where α is a real coefficient (for a recent review of the available results on nonlinear P T -symmetric systems see [39]). For α = 0 the potential is reduced to the real parabolic potential whose modes are fairly well studied in the local case [23][24][25][26][27][28]40], but have attracted much less attention in the nonlocal context.…”