Exact Quantum Motions Of A Single Trapped Ion Interacting With Laser In Lamb-Dicke Regime

As one of the most representative device to study the modern atomic and molecular, ion trap has been widely used in researches of science and technology. Especially after the combination with laser-cooling technology, ion trap has a wider application. Not only it is applied as a powerful tool to prove the fundamental theory of quantum mechanics, more extensive applications of ion trap are stud-ied in some new technology fields, such as the quantum logic operation, quantum computation, quantum information and the preparation of quantum states, which leads to more and more attentions focused on the dynamics of trapped ion. How-ever, the motion of trapped ion sensitively depends on the external environment parameters and the initial conditions of itself, very small variations of them can lead to large deviations of the trajectory of the ion, even to the chaotic motion, which makes the motion of the ion uncontrollable. Therefore, more extensively investigation about the dynamic of trapped ion and the corresponding quantum-mechanical treatment appear particularly important. This paper is organized as the four parts and the main contents is presented as follows.The first chapter introduces the basic principle of the Paul trap, and the history and progress on the researches of few trapped ions.In the second chapter, we investigate the secular motion of a single Paul-trapped ion in the Lamb-Dicke regime, which interacts with a sequence of standing laser pulses. We get an limited forms of exact solution of the classical mechanics, and an exact quantum mechanics solution of the system and the time-varied energy spectrum that is discontinuous by using the ansatz method. Based on the wave-packet trains and energy expectation value described by the exact solution, we find that:a) The motion of the ion will not present chaos in the Lamb-Dicke regime. b) The center, height and width of the wave-packet trains depend on the strength and wave vector of laser pulses, the deformation and spread of the wave-packet trains can be controlled by adjusting the strength of laser. c) Energy expectation values of the ion occur jump at the instantaneous switching on the laser pulses. For the time intervals switching off the laser pulses some narrow bands of the energy are generated. d) When the strength and the wave vector of laser pulses reach critical value, the system changes its stability.In the third chapter, we investigate the secular motion of a single Paul-trapped ion in the Lamb-Dicke regime, which interacts with the laser standing wave. We get the infinite series forms of exact solution of the classical mechanics, and an exact quantum mechanics solution of the system and the time-varied energy spectrum that is continuous the same by using the ansatz method. and the exact quantum solution of the system by using the ansatz method. Using the approximate solution of Mathieu equation, we analyze the impact of laser standing wave on wave-packet trains. a) The height and width of the wave-packet trains periodically oscillate in small range,the effect of laser is very small, the spread of the wave-packet trains almost has no deformation. b) The laser field has effect on the center of the wave-packet trains, larger intensity and smaller frequency of laser lead to smaller amplitude and larger frequency of wave-packet trains, on the contrary, smaller intensity and larger frequency of laser cause larger amplitude and smaller frequency of wave-packet trains.In the fourth chapter, some conclusive remarks on the work presented in this thesis and the outlook on the prospect for the dynamics of trapped ion is given.In this article, the author mainly concentrated in chapter 2 and 3.