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arXiv:1211.5438 · Physica Scripta

A parabolic model for dimple potentials

Melike Cibik Aydin, Haydar Uncu, Coşkun Deniz

Published Aug 20, 2013• 3 citations
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Abstract

We study the truncated parabolic function and demonstrate that it is a representation of the Dirac δ function. We also show that the truncated parabolic function, used as a potential in the Schrödinger equation, has the same bound state spectrum, tunneling and reflection amplitudes as the Dirac δ potential, as the width of the parabola approximates to zero. Dirac δ potential is used to model dimple potentials which are utilized to increase the phase-space density of a Bose–Einstein condensate in a harmonic trap. We show that a harmonic trap with a δ function at the origin is a limiting case of the harmonic trap with a symmetric truncated parabolic potential around the origin. Hence, the truncated parabolic is a better candidate for modeling the dimple potentials.

DOI
10.1088/0031-8949/88/03/035006

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We study the truncated parabolic function and demonstrate that it is a representation of the Dirac δ function. We also show that the truncated parabolic function, used as a potential in the Schrödinger equation, has the same bound state spectrum, tunneling and reflection amplitudes as the Dirac δ potential, as the width of the parabola approximates to zero. Dirac δ potential is used to model dimple potentials which are utilized to increase the phase-space density of a Bose–Einstein condensate in a harmonic trap. We show that a harmonic trap with a δ function at the origin is a limiting case of the harmonic trap with a symmetric truncated parabolic potential around the origin. Hence, the truncated parabolic is a better candidate for modeling the dimple potentials.
Bu tezde; Bose-Einstein yoğuşukluğunu elde etmekte manyetik tuzaklamaya ek olarak kullanılan gamze potansiyel için teorik modeller ele alınmıştır. Tezin ilk aşamasında gamze potansiyelleri betimlemek için literatürde kullanılmış Dirac delta modeli tanıtılmıştır. Ancak, Dirac delta modelinde gamze potansiyelin genişliği ve derinliği ayrı birer parametre olarak kullanılamamaktadır. Bu yüzden, bu tezde gamze potansiyellerin, Dirac delta fonksiyonunu özel bir limit olarak içeren, kesilmiş parabolik fonksiyon ile modellenebileceği gösterilmiştir. Bu modelden yararlanarak, gamze potansiyel eklenmiş harmonik tuzak içindeki Bose-Einstein yoğuşuk maddesinin; kritik sıcaklık, yoğuşukluk oranı gibi termodinamik nicelikleri hesaplanmıştır. Ek kısmında ise kesilmiş parabolik fonksiyonun Dirac delta fonksiyonu için bir temsil oluşturduğu ispatlanmıştır. In this thesis; theoretical models of the dimple potential, which are used for obtaining Bose-Einstein condensate in addition to harmonic trap, are considered. In the first part of the thesis, the Dirac delta models which are used for describing the dimple potentials in the literature are introduced. However, one can not incorporate both the dimple depth and width together as different parameters into Dirac d models. Therefore, in this thesis, it is tried to show that the dimple potentials can be modeled by the truncated parabolic function that includes Dirac delta function as a special limit. Using this model, the thermodynamical quantities like critical temperature, condensate fraction of a Bose-Einstein condensate are calculated. In the appendix, it is proven that the truncated parabolic function is a representation for the Dirac delta function.
We study Bose-Einstein condensation in a harmonic trap with a dimple potential. We specifically consider the case of a tight and deep dimple potential which is modeled by a Dirac $\ensuremath{\delta}$ function. This allows for simpler, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schr\"odinger equation is used instead of the Gross-Pitaevski equation. Calculating the atomic density, chemical potential, critical temperature, and condensate fraction, the role of the relative depth of the dimple potential with respect to the harmonic trap in large condensate formation at enhanced temperatures is clearly revealed.
We study Bose—Einstein condensation in a linear trap with a dimple potential where we model dimple potentials by Dirac δ function. Attractive and repulsive dimple potentials are taken into account. This model allows simple, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schrödinger equation is used instead of the Gross—Pitaevski equation. We calculate the atomic density, the chemical potential, the critical temperature and the condensate fraction. The role of the relative depth of the dimple potential with respect to the linear trap in large condensate formation at enhanced temperatures is clearly revealed. Moreover, we also present a semi-classical method for calculating various quantities such as entropy analytically. Moreover, we compare the results of this paper with the results of a previous paper in which the harmonic trap with a dimple potential in 1D is investigated.
We investigate Bose-Einstein condensation of noninteracting gases in a harmonic trap with an off-center dimple potential. We specifically consider the case of a tight and deep dimple potential which is modelled by a point interaction. This point interaction is represented by a Dirac delta function. The atomic density, chemical potential, critical temperature and condensate fraction, the role of the relative depth and the position of the dimple potential are analyzed by performing numerical calculations.
We investigate Bose-Einstein condensation of noninteracting gases in a harmonic trap with an offcenter dimple potential. We specifically consider the case of a tight and deep dimple potential, which is modeled by a point interaction. This point interaction is represented by a Dirac delta function. The atomic density, chemical potential, critical temperature and condensate fraction, and the role of the relative depth and the position of the dimple potential are analyzed by performing numerical calculations.
This paper presents several cases in which the effects of the addition of a delta function potential on bound states can be computed exactly. In the case of the one dimensional Schrödinger equation, a Green’s function technique is used to compute an exact implicit expression that gives the effect of the one−dimensional delta function potential on the bound states eigenvalues. It is also shown that the weak coupling limit of the implicit expression agrees with the perturbative treatment of the delta function. It is pointed out that this technique cannot be applied to obtain an exact treatment of the delta function potential δ3(r) in the three−dimensional Schrödinger equation. Instead, an alternate definition of the delta function in three dimensions as a limit of a sequence of square well potentials is used. Unlike the one dimensional case, the exact effect on the eigenvalue does not agree in the limit of weak coupling with the perturbative results. This indicates that the three−dimensional delta function potential should not be used to approximate strongly coupled short range potentials for which perturbation theory is inadequate.