The Quantum Motion Of Few Trapped Ions Interacting With Laser

Ion traps have been widely used in researches of science and technology. Espe-cially, in recent years, as a powerful method, ion traps get many extensive applica-tions in the quantum logic operation, quantum calculation, quantum information,and the preparation of quantum states, which leads to more and more attentionsfocused on the dynamics of trapped ions. However, the motions of trapped ions sensi-tively depend on the system parameters and initial conditions. Very small variationsof them can lead to large deviations of the trajectories of the ions and even force theions into loss of the stability, which makes the motions of the ions uncontrollable.Quantum states of the Paul trapped ions are governed by the SchrSdinger equa-tion, which is not exactly soluble for a many-ion system. For applicable purposeslots of approximation methods and perturbation theories were developed, in thisthesis, the dynamics of a trapped ion and the corresponding quantum-mechanical.treatment are investigated more extensively, and We report the n×n′coherence-like state solutions of two Coulomb-c0rrelated trapped ions are reported. The maincontents is presented as follows:In the first chapter, the basic principle of the paul trap is briefly introduced,and the history and progress on the researches of few trapped ions is presented.In the second chapter, We study quantum motions around a classical hetero-clinic point of a single trapped ion interacting with a strong laser, standing wave.We construct a set of exact coherent states of the quantum system.When the classical reference frame is fixed at the heteroclinic point, the classicalgeneral solution and quantum expectation orbit show that the motion of microscopicsubsystem is regular and may be unstable. The instability of the classical solutionleads to the collapse of quantum wavepackets for sufficiently great time. We calculatethe expectation energy and find that the adjacent energy level spacing will tend tozero but the whole energy may tend to infinity as time increasing. However, thenonphysical infinity of the expectation values of the position, momentum and energycan be suppressed by adjusting the system parameters and initial conditions. Thecorresponding control condition is established. In the case where the classical reference frame initially nears and adiabaticallytends to the heteroclinic point along a heteroclinic orbit, it is shown that the quan-turn expectation position and momentum agree with the classically chaotic solutions.They are bounded if and only if some relationships between the initial conditionsand system parameters are satisfied. These relationships lead Melnikov functionto simple zero, indicating the existence of quantum chaos. By using the quantumchaotic wavefunction, we illustrate the collapse property of quantum wavepacketsand the expectation energy, and find the level crossing and quantum resonance intime evolution analytically and numerically. The exponentially rapid growths of thequantum expectation energy and the Heisenberg uncertainty are exhibited that im-ply the instability of the quantum system. However, the asymptotic behavior of thequantum chaotic system is resided in the regular motion of its asymptotic system.Therefore the above-mentioned control condition for its asymptotic system is a goodtheoretical scheme for suppressing the instability of chaotic motion.In the third chapter, We report the n×n′coherence-like state solutions forn, n′=1, 2,..., and of two Coulomb-correlated ions confined in a one-dimensionalPaul trap with a time-dependent harmonic potential. Any one of the n′exact solu-tions of the center-of-mass motion describes a generalized coherent state. For a smalldriving strength the n approximate solutions of relative motion are constructed,which describe the harmonic oscillations of the two ions around the classical equi-librium position.In the fourth chapter, some conclusive remarks on the work presented in thisthesis and the outlook on the prospect for the dynamics of trapped ions are given.