الفهرس 
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Abstract
Experimentally, harmonically trapped Bose Einstein condensation in optical lat- tices are widely used as a simulator to investigate many phenomena which are diffi- cult to study in their natural appearance. Among other, Spielman et al have been used trapped BEC in a combined 3D harmonic-1D optical potential in investigating Superfluid-Mott insulator quantum phase transition from the behavior of the conden- sate fraction as a function of the lattice depth. The experimental usage required two conditions which are loading the condensate must be adiabatic and the BEC phase transition must be 2nd order.
In this thesis, we consider the effect of rotation on applicability of using BEC in one dimension optical lattice to investigate SF-MI transition by using the behavior of the condensate fraction as a function of the lattice depth and rotation rate. More- over, the rotating trapped BEC in a combined potential can be extended to study binning vortecies in multilayer superconductivity type II and quantum Hall effect. Theoretically, the condition of the adiabatic loading can be investigated by studying the behavior of the entropy under rotation, while the order of phase transition can be investigated by using the behavior of the heat capacity under rotation.
Our system is formed by adiabatically loading three dimensional BEC into 1D optical lattice and subjected to rotate with angular velocity Ω about the z axis. This system is known as a rotating condensate in a quasi two dimensional trap. We trace the experimentally accessible parameters for which the temperature dependence on the thermodynamical parameters becomes perceivable. A semiclassical approxima- tion is employed to calculate the condensate fraction, heat capacity and entropy of the system. This approximation can include the correction due to the finite size effect.(finite size effect of the system is required by the experimental usage). The calculated results show that the rotating condensates in a quasi two dimensional can be used partially to investigate the SF-MI transition. The condensate fraction is still vanishes by increasing the lattice depth as verified in Spielman’s experiment through the phase transition from SF-MI. But the adiabatic loading is lost at some critical ro- tating velocity in spite of the phase transition is still 2nd order. The latter is achieved from the behavior of the heat capacity under rotation. This concluded remark is obtained from our numerical results for the entropy-temperature and heat capacity behavior.