Landau Damping Of Collective Mode In Quasi-one-dimensional And Quasi-two-dimensional Repulsive Bose Einstein Condensate

Bose-Einstein condensate (Bose-Einstein condensate, BEC) is a cutting-edgedresearch topic in physics concerned by people in recent years. It not only provides amacro system to study condensed matter and quantum many-body theory, but alsohas a wide application prospects in the field of atom laser, precision measurement,quantum communication and quantum information processing. The research ofelementary excitations is one of the basic topics in statistical physics and condensedmatter physics, and collective excitations are the most basic excitation modes in BEC.The interaction between particles not only leads to the generation of collectiveexcitations, but also the decay of their amplitudes (which is called as damping). Thecollective excitation amplitude of boson gases in trapped potential decay with timemainly through the mechanics of Landau damping, that is, a quasi-particle absorbsthe collective mode and then generate a new quasi-particle. The precision calculationof Landau damping may be used to explain many experimental phenomena, test anddevelop the quantum many-body theory for condensed matter.Although there exists a lot of studies on the Landau damping of collectiveexcitations in trapped BEC made by many famous laboratories and theoretical groupsin the world, the experimental phenomenon of damping are not yet well understoodtheoretically up to now due to the complexity of research issues. In addition, thestudy of quasi-one-dimensional and quasi-two-dimensional system is of greatimportance, because the system is relatively simple and easy to research, also thereare little approximation in the research, the calculation results are reliable and can beused to detect theory easily. In this paper, we improved the theoretical calculation method and made athorough study on the Landau damping of collective excitations in low-dimensionalsystems of trapped BEC. The research results are as follows:We have studied the Landau damping of collective excitations inquasi-one-dimensional and quasi-two-dimensional repulsive interactions87Rb atomicgas in trapped potential. By using Hartree-Fock-Bogoliubov(HFB) approximationtheory, we solved the eigen functions and eigen values of Bogiliubov-deGennes(BdG) equations satisfied by collective excitations particles andquasi-particles. On the basis of Bogiliubov energy spectrum, we calculated thethree-mode coupling matrix element. By using Bogiliubov transformation andFourier transformation, we obtained the formula of collective-mode energycorrection in Landau mechanism. Different from previous methods, we consideredthe actual relaxation of elementary excitations in the application of the energycorrection formula, deduce the Landau damping rate formula and calculated theLandau damping rate by using iterative approach. We had a detailed discussion onthe dependence among the Landau damping, temperature, trapping frequency,particle number, and condensates length. At the condition of the same atom, sametemperature, same collective mode, same atom density. our theoretic results are inagreement with the experiment one.