Coherent And Partially Coherent Matter Waves And Their Evolutions

The rapid progress of techniques for laser cooling and trapping of atoms, and the realization of Bose-Einstein condensate in laboratories, attract many in-terests in the study of the coherent and partially coherent matter waves. The existing theories deal with the Bose-Einstein condensate (BEC) at the absolute zero temperature or at finite temperatures in the method of two-fluid theory by solving the GP (Gross-Pitaeviskii) equation. The matter wave field is separated into the condensate part and the non-condensate part, and the dynamics inside the matter wave system is studied. When the evolution and propagation prop-erties of the matter wave are studied, the focus of the theory is how the matter wave propagates as a whole instead of the inner dynamics. This thesis is focused on the evolution of the coherent and partially coherent matter wave, as a result we study the matter wave field as a whole.In optics, the tensor method is very concise and effective in the treatment of 3D propagation problems. It is applicable not only in the study of the propaga-tions in the individual x, y, z directions, but also very useful in the treatment of the situations when the propagation in each coordinate direction is coupled. The asymmetric experiment setups make the cold atom matter wave have anisotropic and inhomogeneous properties. The regular scalar ABCD formula can not de-scribe this astigmatic matter wave, neither deal with its evolution problems. Based on the similarities between the matter wave and the light beams, starting from the Heisenberg equation, this thesis extends the tensor method to the mat-ter wave field. The tensor ABCD law for the general astigmatic matter wave is derived analytically. After performing the calculation, it is found that, during the evolution, the atom number density profile and the constant phase front of the general astigmatic matter wave rotate continuously.Cold atom systems and non-ideal BECs can be regarded as partially coherent matter wave, we derived the tensor ABCD law for partially coherent matter wave. This ABCD law is applicable to analysis the evolution problem of the cold atom matter wave system which has partial coherence and widely exist in experiments. The tensor ABCD law describes the evolution of the first order correlation function of the partially coherent matter wave. Using this ABCD law, we calculate the coherence length of the partially coherent matter wave in the gravitational field. It is found the coherence increases during the evolution of the matter wave.One of the major differences between the bosonic atom and the photon is that the atom has mass and can have collision interaction with each other, but the photon doesn't. This thesis studied the nonlinear evolution of the matter wave with repulsive interaction among atoms. The effective complex curvature method in nonlinear optics is extended to the matter wave field. The effective matter wave probability amplitude is calculated in this method. A comparison between the interacting matter wave and the ideal matter wave (matter wave without interaction among atoms) in the matter wave focus example is given. In this thesis, we also carry out the numerical calculation to obtain the phase distribution of the interacting matter wave. It is found that for a Gaussian shaped matter wave, the central phase lags behind its surrounding points initially, but after the evolution, it is advanced. The phase of the interacting matter wave changes with the spatial coordinate more than the ideal matter wave.Our results extended the theoretical methods in optics to the field of matter wave, and built the correspondence between the matter wave and the light. The theoretical methods developed in this thesis are very simple and effective in the treatment of the evolution of the coherent and partially coherent matter waves, and can be very fruitful in the further study of the various properties of matter wave and their related applications.