Nonlinear Quantum Effect Of The Coherent Matter Wave

The nonlinear effect in quantum mechanics plays a significant role in many fields, such as many-body interaction in solid physics. After the realization of Bose Einstein condensates (BECs) in laboratory, it is widely applied to investigate many nonlinear quantum effects, such as matter wave soliton, nonlinear four wave mixing, et.al.. The BECs is here used to study another nonlinear quantum effect, namely, nonlinear tunneling.It is possible for a quantum particle transmitting through a potential barrier higher than its kinetic energy, which is a distinguished feature from the classical counterpart. This phenomenon is called quantum tunneling, which is widely stud-ied theoretically and experimentally since the quantum mechanics is established. At the same time, based on the quantum tunneling effect, many quantum devices, such as tunneling diode, scanning tunneling microscope, have been developed. In quantum mechanics, time is a parameter describing the evolution of the system rather than an operator. It is therefore impossible to define the mean value of the time. In the study of the quantum tunneling effect, the tunneling time has been proposed with many kinds of indirect definition, such as phase time, dwell time. All these tunneling times, however, are not accepted unanimously, which result in debating for a long time. Except for the debate about time definition, the time itself is one significant physical quantity to be treated theoretically and experimentally. Therefore it has theoretical and practical significance to study the quantum tunneling time.To begin with, the dissertation reviews study history and the current status of the tunneling time and lists all kinds of definition of tunneling time. In chapter II, a general equation describing a two-level atomic BECs transmitting through laser beam is constructed. Considering a BECs in a one dimensional atomic waveguide, the 3D equation can be reduced to 1D vector nonlinear Schrodinger equation. In the second half of chapter II, ignoring the nonlinear interaction, we obtain the plane wave solution and the wave packet solution for the situation of a two-level atom tunneling a barrier, by solving the 1D vector nonlinear Schrodinger equation. Then, the tunneling time of the ground-state atom traversing a barrier is discussed.Chapter III mainly works on the reliability of adiabatic approximation in the tunneling problem. First of all, we compare the exact solution with the adiabatic approximation solution in the vector model, and find the tunneling time in these two cases is utterly different. After analyzing, the reason for this is revealed that the ground state wave function and the excited state wave function of the two-level atom interfere with each other in the optical field. Afterwards, we expand the exact solution according to the small quantity method, and preserve the first and second order, in which case the exact solution matches well with the approximate solution.Chapter IV studies how the two-body collision between atoms affects on the tunneling time. With the nonlinear interaction, it is impossible to obtain the analytic solution. Therefore, we simulate the tunneling process of the BEC wave packet via solving the nonlinear Schrodinger equation numerically. Firstly, we consider that a BEC wave packet traverses a barrier. The simulation result tells that the nonlinear interaction has a great influence on the tunneling time. The nonlinear interaction not only increases the center velocity of the BEC wave packet, but also modifies the height and shape of the barrier, which would impact on the tunneling time. After that, the case for the BEC wave packet traverses a barrier with arbitrary shape is studied. In contrast to the rectangular barrier, the tunneling time will not be saturated as the width of the barrier increases.Chapter V studies the various phenomena when the ultracold atom wave packet traverses a red-detuning Gaussian beam theoretically and experimentally. As presented in many works, an ultracold wave packet will show focusing, trans-verse cooling, longitudinal advancement and other phenomena. And the focusing phenomenon indicates that a red-detuning Gaussian beam can act as an excellent atom-optical lens.