Regular And Chaotic Motions Of A Single Paul-trapped Ion Interacting With Laser Pulses

Ion trap has been widely used since it has combined with laser-cooling technology. Not only it is applied as a powerful tool to prove the fundamental theory of quantum mechanics, for example, preparing non-classical state and observing quantum jump, but also it has potential applications in new technology fields such as the quantum logic operation, quantum computation and quantum information. However, the motions of trapped ions sensitively depend on the system parameters and initial conditions. Very small variations of the latter can lead to large deviation of the trajectories of the ions, even to the uncontrollable chaotic motion. Because the classical instability and chaotic feature have correspondences in quantum motion of trapped ions, it is necessary and importance to find the new properties of classical motion.This paper is organized as the following four parts. In the first chapter, the basic principle of the Paul trap is briefly introduced, and the history and progress on the researches of a single trapped ion is presented.In the second chapter, we consider a single Paul trapped ion under the pseudopotential approximation, which interacts with a flashing ratchet potential formed by two laser standing waves. We apply the method of integral-equation to construct the exact solution of the classical equation, and use the numerical technique to plot the orbits on phase space and to calculate the current defined by the average velocity. By combining the analytical results with numerical ones, we investigate the regular and chaotic motions of the system. Compared to the case of the flashing lattice potential, two important ratchet effects are found here: At first, the flashing ratchet potential changes the chaotic region on the parameter space such that we can make the regular motion of ion the chaotic one or on the contrary, by adjusting the parameters of the second laser. Secondly, by analyzing the evolutions of the average current versus the laser parameters, we find that during many pulse periods the presence of the flashing ratchet potential can lead the trapped ion to transport along a single direction and in the mean sense. When the potential strength is increased to the chaotic region, the current decreases its amplitude evidently and varies its direction, indicating the onset of chaotic motion.In the third chapter, we investigate the regular and chaotic motions of a single Paul-trapped ion under, the pseudopotential approximation, which interacts with a double-δ-Kicked periodic potential. We apply the method of integral-equation to construct the exact solution of the classical equation and use the numerical technique to plot the orbits on phase space. Combining the analytical results with the numerical ones, we found two interesting conclusions: At first, there are regular motions of the doubleδ-Kicked system for the resonance case of the corresponding singleδ-Kicked system. Secondly, when the time interval between the doubleδpulses becomes shorter, the regular motion of the system is transformed to the chaotic one, and the speed of classical diffusion of the average energy is related to the degree of chaos. It is shown that the resonance may lead to loss of the stability and the instability can be controlled by adjusting the laser wave vector.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 ions are given.