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arXiv:1508.04341 · Journal of Physics A Mathematical and Theoretical

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

R. Cipolatti, Leonardo Villegas-Lelovsky, Ming-Chiang Chung, C. Trallero‐Giner

Published Feb 22, 2016• 5 citations
arXivHTMLPDF

Abstract

Abstract Employing a general variational method and perturbation theory, we derived explicit solutions for the description of one-dimensional two species Bose–Einstein condensates confined by a harmonic trap potential in an optical lattice. We consider the system of two coupled Gross–Pitaevskii equations (GPE) and derive explicit expressions for the chemical potentials and wavefunctions in terms of the atom–atom interaction parameters and laser intensity. We have compared our results with the numerical solutions of the GPE and performed a quantitative analysis for the both considered methods. We underline the importance of the obtained explicit solutions to characterize the density profile or degree of miscibility of the two components.

DOI
10.1088/1751-8113/49/14/145201

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Abstract Employing a general variational method and perturbation theory, we derived explicit solutions for the description of one-dimensional two species Bose–Einstein condensates confined by a harmonic trap potential in an optical lattice. We consider the system of two coupled Gross–Pitaevskii equations (GPE) and derive explicit expressions for the chemical potentials and wavefunctions in terms of the atom–atom interaction parameters and laser intensity. We have compared our results with the numerical solutions of the GPE and performed a quantitative analysis for the both considered methods. We underline the importance of the obtained explicit solutions to characterize the density profile or degree of miscibility of the two components.
The ground state solutions of two-component Bose-Einstein condensates (BECs) in Bessel optical lattices (BLs) are studied by means of the balance condition between BLs strength and inter-atomic interactions. We consider a quasi-two-dimensional (2D) BECs system，strongly confined in longitudinal direction and weakly trapped in the radial direction in the transverse plane，which obeys 2D Gross-Pitaevskii equation (GPE) derived from its 3D counterpart. Analytically we obtained the atom number distribution，atom number and energy of ground state and give the parameter ranges. Compared to single-component BEC，two-component BECs exhibit a rich variety of ground state structures. These structures depend upon various parameters，in particular the inter-atomic interactions and the BLs strength. Neglecting the inter-component atomic interaction，the corresponding results of single-component situation are given.
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Based on the Gross-Pitaevskii theory, for a Bose-condensed gas confined in a harmonic trap and in a one-dimensional (1D) optical lattice, the relationship of the Gaussian width of the sub-condenses and the intensity of 1D-optical lattices potential is studied by using the G-P energy functional and the variational method. For the ideal non-interacting model, the Gaussian width of the sub-condenses is analyzed. Concerning interacting Bose-condensed atoms, the Gaussian approximate model is advanced and the analytic expression of the Gaussian width with the intensity of optical lattices potential is obtained. And then, those analytical results are compared with the straightforward numerical calculation. It indicates that the Gaussian approximate model is closer to the results of the numerical solution, and both are reaching unanimity with the intensity of ID-optical lattices potential increasing.
We identify all possible classes of solutions for two-component Bose-Einstein condensates (BECs) within the Thomas-Fermi (TF) approximation and check these results against numerical simulations of the coupled Gross-Pitaevskii equations (GPEs). We find that they can be divided into two general categories. The first class contains solutions with a region of overlap between the components. The other class consists of non-overlapping wavefunctions and also contains solutions that do not possess the symmetry of the trap. The chemical potential and average energy can be found for both classes within the TF approximation by solving a set of coupled algebraic equations representing the normalization conditions for each component. A ground state minimizing the energy (within both classes of states) is found for a given set of parameters characterizing the scattering length and confining potential. In the TF approximation, the ground state always shares the symmetry of the trap. However, a full numerical solution of the coupled GPEs, incorporating the kinetic energy of the BEC atoms, can sometimes select a broken-symmetry state as the ground state of the system. We also investigate effects of finite-range interactions on the structure of the ground state.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.
The properties of traps for exciton polaritons in a semiconductor microcavity with an embedded quantum well have been considered. The behavior of the two-component Bose-Einstein condensate of photons and excitons described by the coupled system of Gross-Pitaevskii equations has been investigated. The analytical solutions for weak-confinement traps have been found in the Thomas-Fermi approximation. In the case of strong confinement, the behavior of the condensate has been investigated and constraints on the possible values of the chemical potential of the system have been obtained. The wavefunctions and generally different spatial profiles of the coupled photon and exciton condensates have been found.