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IOSR Journal of Engineering

Theoretical Analysis of Wave Function under the Influence of Harmonic Oscillator Potential in BEC Experiments

Noori.H.N. Al-Hashimi Al-Hashimi

Published Mar 1, 2014• 1 citation
Publisher

Abstract

This paper will focus on theoretical study of BEC in one dimension with weakly attractive interactions in particular in a symmetric harmonic oscillator potential.In this case, the ground state may not be symmetric state, which is in contrast to a BEC with repulsive interaction.The behaviors of the wave function under the influences of different external trapping potentials usually used in experiments that lead to produced Bose-Einstein condensation in ultra cold gases is analysis.Several values of harmonic trapping potentials are used.These analyses give us the overall view of the region of confinement that the external trapping potentials have employed and the influence of some terms in Gross-Pitaevskii equation.

DOI
10.9790/3021-04312832

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This paper will focus on theoretical study of BEC in one dimension with weakly attractive interactions in particular in a symmetric harmonic oscillator potential.In this case, the ground state may not be symmetric state, which is in contrast to a BEC with repulsive interaction.The behaviors of the wave function under the influences of different external trapping potentials usually used in experiments that lead to produced Bose-Einstein condensation in ultra cold gases is analysis.Several values of harmonic trapping potentials are used.These analyses give us the overall view of the region of confinement that the external trapping potentials have employed and the influence of some terms in Gross-Pitaevskii equation.
This paper will focus on theoretical treatment of external trapping potentials which are usually used in experimental that lead to produced Bose-Einstein condensation BEC in ultra cold gases.Several types of trapping potentials such as one dimensional BEC in a harmonic oscillator potential (HOP), one dimensional BEC in a double well potential DWP, and one dimensional BEC in a harmonic plus optical lattice potential HOLP are analyze.These analyses give us the overall view of the region of confinement that the external trapping potentials have employed.
The dynamics of weakly interacting three-dimensional Bose-Einstein condensates (BECs), trapped in external axially symmetric plus anharmonic distortion potential are studied. Within a variational approach and time-dependent Gross-Pitaevskii equation, the coupled condensate width equations are derived. By modulating anharmonic distortion of the trapping potential, nonlinear features are studied numerically and illustrated analytically, such as mode coupling of oscillation modes, and resonances. Furthermore, the stability of attractive interaction BEC in both repulsive and attractive anharmonic distortion is examined. We demonstrate that a small repulsive and attractive anharmonic distortion is effective in reducing (extending) the condensate stability region since it decreases (increases) the critical number of atoms in the trapping potential.
In this paper, we discuss theoretically the behavior of ideal Bose-Einstein gases in a 3D box and in a harmonic trap. We make use of the mean filed theory based on the time dependent Gross-Pitaevskii equation which describes the main features of Bose-Einstein condensation. We investigate and discuss the density profiles and the ground state properties of a trapped condensate confined in harmonic trap. We found out that in an attractive interaction, the gas tends to increase its central density in order to lower the interaction energy.
We have studied the ground state energy of one-dimensional macroscopic quantum oscillator by adding gravity term as a time-independent external potential. The macroscopic quantum oscillator itself is based on discussion about Gross-Pitaevskii equation (GPE) describing Bose-Einstein condensate in anisotropic parabolic trap. That gravitational potential displaces the symmetry axis of Bose-Einstein condensation (BEC) with respect to symmetry axis of magnetic field that creates trapping potential of the Bose-Einstein condensate (BECs) in research on atomic lasers. Since the Gross-Pitaevskii equation can be treated as macroscopic quantum oscillator, we use time-independent perturbation theory as in ordinary quantum mechanics to find the ground state energy correction by assuming that other terms are very small. We also formulate the appropriate coefficients of expansion in the solution of one-dimensional GPE by considering the characteristics of harmonic oscillator eigenfunctions and find that the existence of the gravity can only be included if the higher coefficients are considered. Keywords: Gravitational potential, Gross-Pitaevskii, Quantum Oscillator.
Abstract By exact numerical solutions of the Gross–Pitaevskii (GP) equation in 3D, we assess the validity of 1D and 2D approximations in the study of Bose–Einstein condensates confined in harmonic trap potentials. Typically, these approximations are performed when one or more of the harmonic frequencies are much greater than the remaining ones, using arguments based on the adiabatic evolution of the initial approximated state. Deviations from the 3D solution are evaluated as a function of both the effective interaction strength and the ratio between the trap frequencies that define the reduced dimension where the condensate is confined. The observables analyzed are both of stationary and dynamical character, namely, the chemical potential, the wave function profiles, and the time evolution of the approximated 1D and 2D stationary states, considered as initial states in the 3D GP equation. Our study, besides setting quantitative limits on approximations previously developed, should be useful in actual experimental studies where quasi-1D and quasi-2D conditions are assumed. From a qualitative perspective, 1D and 2D approximations certainly become valid when the anisotropy is large, but in addition the interaction strength needs to be above a certain threshold.