Bose–Einstein condensates in optical lattices: Mathematical analysis and analytical approximate formulae

Abstract

We show that the GPE with cubic nonlinearity, as a model to describe the one dimensional Bose-Einstein condensates loaded into a harmonically confined optical lattice, presents a set of ground states which is orbitally stable for any value of the self-interaction (attractive and repulsive) parameter and laser intensity. We also derive a new formalism which gives explicit expressions for the minimum energy E min and the associated chemical potential µ 0 . Based on these formulas, we generalize the variational method to obtain approximate solutions, at any order of approximation, for E min , µ 0 and the ground state.

One-dimensional cubic-quintic Gross-Pitaevskii equation for Bose-Einstein condensates in a trap potential

“…Explicit formulae (10), (11) and the approximate relationships ( 14), (15) are among the main results of our work. Nevertheless, more explicit expressions for the order parameter ϕ λ , the energy and the chemical potential as a function of the atom-atom interaction term would be desirable.…”

Two-species Bose–Einstein condensate in an optical lattice: analytical approximate formulae

“…In Ref. 25 it is shown that the perturbation theory for one component reproduces quite well the chemical potential if |Λ 1 | 2. In the present case, the inter-species interaction plays the role as an effective non linear term given by Λ 1 |ψ 1 | 2 + Λ 12 |ψ 2 | 2 .…”

“…Notice that if λ 1 = λ 2 , the respective roots are different even in the case V 0 = 0, N 1 = N 2 = N and m 1 = m 2 . Indeed, by subtracting the first equation from the second one in (25), we obtain:…”

“…With arguments of convexity, we can show that the (real) solution of minimal energy is unique provided that λ 1 , λ 2 and λ 3 are positive. Moreover, since the system (6) has the properties of conservation of energy and mass (i.e., the number of particles), we can prove the orbital stability 25,28 of ground states.…”

Collective modes of trapped spinor Bose–Einstein condensates

“…are reported in [35,40]. In the absence of an optical lattice, namely, for = V 0 0 , in [34][35][36] it was shown that the range of validity of the GPE perturbation approach, equation (7), is limited by a universal criterion given in terms of the dimensionless self-interaction parameter L < | | 3 g J . In the presence of the optical lattice, ¹ V 0 0 , the range of validity of equation ( 14) depends additionally on two parameters, V 0 and d l 0 , yielding a twodimensional parameter map.…”