| The developments of laser theory and technique give rise to influence on the traditional atomic and molecular physics in deeply. The central issue interested is gradually turned to dynamical problem rather than the structure of the energy level. Particularly, with the help of the short laser pulse and intense laser in experiment made preparing and probing highly excited Rydberg stats in possible, and promoted, therefore, relative research and explorations in following basic areas: the interactions between laser and particles of matter, the interference evolution of wave-function in spatial and temporal domains, correlations of many body theory, the spectroscopy for intense field, regular-chaos transitions and correspondence between classical mechanics and quantum mechanic, etc.As a sensitive probative tool of interaction of the fields and particles Rydberg atoms have widely used to many fields due to possess the characteristics of long-live time, wide scattering cross section and sensitive to applied external fields. For example, atom-chip experiments that realize control and storage of the Rydberg stats have opened the way to investigations at the frontier of solid state physics and atomic physics. Aside from their relative ease of use and potential as atom interferometers, they provide an ideal environment in which atomic ensemble can be integrated with, and coupled to, devices on a microcircuit. The dynamic measurement in external static fields may simulate strong effects in celestial physics subject to ultra conditions. The Rydberg wave packet excited by a short laser pulse has accomplished squeezed state in different dimensions and reached a minimum-uncertainty thus is deemed best way for discussing classical limit of the quantum mechanics. Because of the spatial dimensions of highly excited Rydberg atoms or molecules are often about 10-9m even 10-6 m therefore are theoretical model for studying the transform process in micro-cavity and escape behaviors of particles from a potential, which is important in design of micro-electronics devices. For theoretical works the system of Rydberg atom in strong fields is typical example demonstrating classical-quantum correspondence and exhibiting the transition from regular motion to chaos motion. This system became one of a few available low dimensional systems for understanding quantum chaos. Since 70th years last century the development of physics has to confront with the challenges from complicated atomic and molecular system having higher dimensional degree of freedom and strong correlation between particles called nonlinear or non-integrable system. Variety of the classical method got resurgence as the full quantum treatment fails in exactly solving this problem. For example, in exploring the photo-absorption problem in strong fields the perturbation theory is no longer available, and does not separate variables, moreover, the spatial dimensional scope of the considered physics processes here is from 10-15m( characteristic dimension of atomic nucleus) to 10-6m (a typical measurement of a Rydberg atom when its principal number equals 100),while the temporal range expands from 10-16(the characteristic time of a quantum transition) to 10-9( typical live-time of Ryberg atom). It is impossible to calculate the evolutions of the state functions. Closed-orbit theory is available semi-classical approach to spectrally analyze the non-integrable system with a finite resolution, because it reveals that multiple-periodic phenomena hidden in quantum transition. The calculation method can be referred to what called in space region-splitting consistent and iterative approach: In atomic nucleus region calculate wave function by quantum mechanics, such as Coulomb scattering and Core-scattering. The impacts of the external fields can be ignored comparison with the Coulomb interaction of the core. In region far away from the nucleus the electron motion is along with classical trajectories, the wave functions can be built according to quasi-classical approximation. The inner and outer solutions are connected in proper position in the Coulomb region around the core. To ensure accuracy of the result, the asymptotic expand is introduced for inner solution and carried out the stationary phase approximation. Since the theory was proposed in 1987, it has got improved and developed. The studying system had altered from simple atom (hydrogen) or negative ion ( H- ) to multiple-electron atom (such as Lithium, Helium) even molecules H2 and NO. The studying problem had greatly developed from the calculations of the photo-absorption cross section or average density of the oscillator strength to the analyses of recurrence spectra, the investigations of dynamical properties of the atom near a metal surface, transform process in micro-cavity and spontaneous emission and decay of Rydberg atom in medium environment. The research method has extended from original the electron closed orbit to resent the photon closed orbit, and complimented quantum defect theory, model potential, scaled variables recurrence spectroscopy, quantum spectral function, the uniform approximation and harmonic inversion. etc. The closed-orbit theory is based on the Gutzwiller's periodic orbit theory. There are obvious advantages in both above theories, they are suitable to Rydberg bounded stats not only, but also suitable to ionization stats above ionization threshold no matter whether the system is chaotic or not. Thus they provide a useful tool for discussing the classical-quantum correspondence.Another effective method for studying non-integrable system possessing regular-chaotic mixed characters is Poincarésurface of section in classical phase space. The coordinates and the momentum in the classical Hamilton canonical equation are continuous variables respect to time t, the motion of particles in phase space results in flow. The classical trajectories in phase space often have very involved global features which are to apparent contradiction to the locally smooth flow. Poincaréproposed to deal with this problem by intercepting the flow at discrete times, which arises to a series of map, every crossed point indicates a map. Because the dimension of the section is one less than that of the flow, for many purposes the map is easier to analyze and understand. There are an infinite number of ways to do this. Poincarésurface of section is a general tool to analyze the classical properties of dynamic system.In this thesis, the dynamic properties of Rydberg Lithium atoms in strong external electric field have been studied by above theoretical methods. The Poincarésurfaces of section and recurrence spectra are calculated for several selected scaled energies. By comparing with numerical results of Rydberg hydrogen atom shown the non-Coulomb core-scatted effects. The hydrogen in applied electric field is an Integrable, and does not appear chaos. Only in applied magnetic field and when the scaled energy is high enough to arise chaos. The case of lithium atom is entirely different; the core-scattering brings instinctively chaos. This is illustrated by the evolution of the Poincarésurface of section respect to external fields and scaled energies. Chaos belongs to pure classical mechanics in concept. In order to understand the quantum manifestation of the classical chaos, we perform first time the analysis on correspondence between classically Poincarésurface of section and quasi-quantum recurrence spectra. It is helpful to gain an insight into the implication of quantum chaos from new evidence. This is author's first motivation.The thesis is divided into five chapters: The first chapter is introduction, which chiefly introduces the background of the closed-orbit theory and its development in application areas and treatment methods. The second chapter is devoted to classically analyze the dynamics. We calculate the evolutions of Poincarésurface of section respect to scaled energy which demonstrate that the electron motion is gradually changed from regular-chaotic mixed state into full chaos. In order to study the core-scattering effects, as comparison we calculate the Poincarésurfaces of section of hydrogen atom in electric field and in magnetic field, respectively. It is clearly that the nucleus core-scattering arises to chaos, and only applied external magnetic field is strong enough to result in chaotic motion. To eliminate the Coulomb singularity in Hamiltonian we adopt semi-parabolic coordinates and introduce scaled variables to conveniently analyze the classical-quantum correspondence as same the traditional treatment method in closed-orbit theory. In the third chapter we present the theoretical computation of the semi-classical (in other words, quasi-quantum) recurrence spectra for applied external parallel electric and magnetic fields. First the primary closed orbits at given scaled energies are specified, which also contain great number combined orbits. Computing the Jacobian and Marslov index for every orbit and summing to obtain the strength spectrum, its Fourier transformation is recurrence spectrum. Since the chaotic properties are related with the behaviors in long range. Summation here requires take more contributions from the longer closed orbits into account. Contrast to the case of hydrogen atoms the recurrence spectra of the Lithium atoms is more complicated, which also is referred to core-scattering. In the next chapter the classical-quantum correspondence is investigated in detail. We analyze the bifurcation of the closed trajectories. For convenience of comparison we have given the recurrence spectra at selected scaled energies as the same in second chapter. The features of spectra sensitive depend on the initial conditions indicate the system is chaotic. When the full chaos is reached corresponding recurrence considerably reduced, which demonstrates that the classical chaos rises to localize the quantum wave function in partly. Other property need to emphasize is the role of the periodic orbits that are fixed points in Poincarésections and stand for invariant of the chaotic system. In last chapters we give some important conclusions. Finally, we complement some explanations. In order to emphasize our major goal we have not further researched Poincarésection, the stability of the maps, the Lyapunov index, statistical distribution of the energy levels, bifurcation effects and auto-ionization phenomena are not concerned. These problems are required continuous explore in future.
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