Singular Points, Closed Orbits, Stable Manifolds and Unstable Manifolds of Second Order Autonomous Birkhoff Systems

Aim To study singular points, closed orbits, stable manifolds and unstable manifolds of a second order autonomous Birkhoff system. Methods Qualitative methods of ordinary differential equation were used. Results and Conclusion The criteria for singular points, closed orbits and hyperbolic equilibrium points of a second order autonomous Birkhoff system are given. Moreover the stability of equilibria, stable manifolds and unstable manifolds are obtained.

On the detection of closed orbits in time-dependent systems by using double-pulse lasers

In the photodetachment of atoms or negative ions by a double-pulse laser, the first pulse of the double-pulse laser generates waves and the delayed second pulse may detect them. The phenomenon of the excitation and detection of waves by a double-pulse laser can be used to identify the closed orbits in the system. We demonstrate this phenomenon with a negative hydrogen ion(H^-) by analyzing the total population excited by a double-pulse laser in a timedependent field for different physical parameters. By analyzing the total excited population using a double-pulse laser, we can uncover all the closed orbits existing in the system. We demonstrate that this can be realized by scanning the first pulse position and the time delay between the two pulses.

The Closed Orbits of a Class of Cubic Vector Fields in R^(3)

In this paper,we investigate the isolated closed orbits of two types of cubic vector fields in R^3 by using the idea of central projection transformation,which sets up a bridge connecting the vector field X(x)in R^3 with the planar vector fields.We have proved that the cubic vector field in R^3 can have two isolated closed orbits or one closed orbit on the invariant cone.As an application of this result,we have shown that a class of 3-dimensional cubic system has at least 10 isolated closed orbits located on 5 invariant cones,and another type of 3-dimensional cubic system has at least 26 isolated closed orbits located on 13 invariant cones or 26 invariant cones.

The closed-orbit and the photoabsorption spectra of the Rydberg hydrogen atom between two parallel metallic surfaces

Using the closed orbit theory, we study the classical motion and calculatethe photoabsorption spectra of Rydberg hydrogen atom between two parallel metallic surfaces. The results show that the metallic surfaces have a significant effect on the photoabsorption process. When the distances between the hydrogen atom and the two metallic surfaces are close to a critical value dc, the number of the closed orbits is the greatest. When the distance larger or smaller than dc, the number of the closed orbits decreases and the absorption spectra are shown to exhibit a damping oscillation. This work is an interesting new application of closed-orbit theory and is of potential experimental interest.

Some Properties of Homogeneous Vector Fields

For planar analytic homogcneous vector fields, the existence of periodic orbits and the noncxistence of limit sets arc verilied. It is concluded that spacial analytic homlogencous vector tleld of order in has no limit sets for any m>1. Similar results arc extended to highel-dimensional polynomial homogeneous vector fields under certain conditions.

Semiclassical Analysis of Quarter Stadium Billiards

An expansion method for stationary states is applied to obtain the eigenfunctions and the eigenenergies of the quarter stadium billiard, and its nearest energy-level spacing distribution is obtained. The histogram is consistent with the standard Wigner distribution, which indicates that the stadium billiard system is chaotic. Particular attention is paid to pursuing the quantum manifestations of such classical chaos. The correspondences between the Fourier transformation of quantum spectra and classical orbits are investigated by using the closed-orbit theory. The analytical and numerical results are in agreement with the required resolution, which corroborates that the semiclassical method provides a physically meaningful image to understand such chaotic systems.

Photodetachment of negative ion in a gradient electric field near a metal surface

Based on closed-orbit theory, the photodetachment of H- in a gradient electric field near a metal surface is studied. It is demonstrated that the gradient electric field has a significant influence on the photodetachment of negative ions near a metal surface. With the increase of the gradient of the electric field, the oscillation in the photodetachment cross section becomes strengthened. Besides, in contrast to the photodetachment of H- near a metal surface in a uniform electric field, the oscillating amplitude and the oscillating region in the cross section of a gradient electric field also become enlarged. Therefore, we can use the gradient electric field to control the photodetachment of negative ions near a metal surface. We hope that our results will be useful for understanding the photodetachment of negative ions in the vicinity of surfaces, cavities, and ion traps.

Photodetachment of H^- Near a Dielectric Surface

Using the closed orbit theory, the photodetachment cross section of H- near a dielectric surface has been derived and calculated. The results show that the dielectric surface has great influence on the photodetachment process of negative ion near the ionization threshold. Above the ionization threshold, the photodetachment cross section starts to oscillate. With the increase of the energy, the oscillating amplitude decreases and the oscillating frequency increases. The oscillation in the photodetachment cross section of H- in the presence of a dielectric surface is either larger or smaller than the photodetachment of H- without the surface. As the photon energy is larger than the critical value Epc, the oscillatory structure disappeared and the cross section approaches to the case of the photodetachment of H- without any external fields. For a given detached-electron energy, the photodetachment cross section becomes decreased with the increase of the ion-surface distance. Besides, the dielectric constant has great influence on the photodetachment of H-. With the increase of the dielectric constant, the oscillation in the cross section becomes increased. As the dielectric constant increases to infinity, the cross section is the same as the photodetachment of H- near a metal surface. This study provides a new understanding on the photodetachment process of H- in the presence of a dielectric surface.

Spontaneous Emission of a Polarized Atom in a Medium Between Two Parallel Mirrors

Using the photon closed orbit theory, the spontaneous emission rate of a polarized atom in a medium between two parallel mirrors is derived and calculated. It is found that the spontaneous emission rate of a polarized atom between the mirrors is related to the atomic position and the polarization direction. The results show that in the vicinity of the mirror, the variation of the spontaneous emission rate depends crucially on the atomic polarization direction. With the increase of the polarization angle, the oscillation in the spontaneous emission rate becomes decreased. For the polarization direction parallel to the mirror plane, the oscillation is the greatest; while for the perpendicular polarization direction, the oscillation is nearly vanished. The agreement between our result and the quantum electrodynamics result suggests the correctness of our calculation. This study further verifies that the atomic spontaneous emission process can be effectively controlled by changing the polarization orientation of the atom.

Photodetachment of H^- in a Static Electric Field near a Surface

The photodetachment of H^- in a static electric field near a surface is investigated based on the closed orbit theory. It is found the distance between the ion and the surface modulates the cross section of photodetachment. For an elastic surface perpendicular to electric field, the detachment spectrum displays a staircase structure, in contrast with the smooth oscillation when only the electric field exists.

Semiclassical Calculation of Conductance Transmission through an Open Equilateral Triangular Billiards

We study the transport property passing through a weakly open equilateral triangular billiards by using the semiclassicM method. We extend the Green function and the transport matrix theory to include the multiple scattering effect at the boundary and the diffractions of the pair of the lead apertures. For analysing the structure of semiclassical pseudo path kinks, the geometric and the special dynamical symmetries of the system are simultaneously taken into account. The conductance is calculated by Landauer formula as a function of the electron＇s Fermi wave number. Its Fourier transformation, the quantum path-length spectrum, is qualitatively in accordance with the results of the classical trajectories, which indicates that such approach provides an obvious improvement of the semiclassical description.

Semiclassical Calculation of Recurrence Spectra of Rydberg Hydrogen Atom Near a Metal Surface

Using closed orbit theory, we give a clear physical picture description of the Rydberg hydrogen atom near a metal surface and calculate the Fourier transformed recurrence spectra of this system at different scaled energies below ionization threshold. The results show that with the increase of the scaled energy, the number of the closed orbit increases greatly. Some of the orbits are created by the bifurcation of the perpendicular orbit. This case is quite similar to the Rydberg atom in an electric field. When the scaled energy increases furthermore, chaotic orbits appear. This study provides a different perspective on the dynamical behavior of the Rydberg atom near a metal surface.

Photodetachment of H^- near a hard wall with arbitrary laser polarization direction

The photodetachment of H-near a hard wall is investigated with linear polarized laser light travelling in arbitrary direction θL with respect to the z axis. An analytical formula for the total cross section is derived using semi-classical closed orbit theory, which consists of two terms, i.e., the smooth background term and the oscillatory term with an extra factor 2(θL). This factor controls oscillations in the total photodetachment cross section. The amplitude of oscillation is maximum at θL = 0 when the laser polarization direction is perpendicular to the wall and it approaches zero at θL = π /2when the laser polarization direction is parallel to the wall. It is also observed that the total cross section depends on the source–wall distance and it reduces to a free space case when the wall is at infinite distance from the source.

Control of the photodetachment of H^-near a metallic sphere surface by an elastic interface

According to the closed-orbit theory, we study the influence of elastic interface on the photodetachment of H- near a metallic sphere surface. First, we give a clear physical description of the detached electron movement between the elastic interface and the metallic sphere surface. Then we put forward an analytical formula for calculating the photodetachment cross section of this system. Our study suggests that the photodetachment cross section of H is changed with the distance between the elastic interface and H^-. Compared with the photodetachment cross section of H^- near a metallic sphere surface without the elastic interface, the cross section of our system oscillates and its oscillation is strengthened with the decrease of the distance from the elastic interface to H^-. In additon, our calcuation results suggest that the influence of the elastic interface becomes much more significant when it is located in the lower half space rather than in the upper half space. Therefore, we can control the photodetachment of H^- near a metallic sphere surface by changing the position of the elastic interface. We hope that our work is conducive to the understanding of the photodetachment process of negative ions near interfaces, cavities and ion traps.

Photodetachment dynamics of H^- ion in a harmonic potential plus a time-dependent oscillating electric field

The photodetachment dynamics of H^- ion in a harmonic potential plus an oscillating electric field is studied using the time-dependent closed orbit theory. An analytical formula for calculating the photodetachment cross section of this system is put forward. It is found that the photodetachment cross section of this system is nearly unaffected for the weak oscillating electric field strength, but oscillates complicatedly when the oscillating electric field strength turns strong. In addition, the frequency of the harmonic potential and the oscillating electric field （the frequency of the harmonic potential and the frequency of the oscillating electric field are the same in the paper, unless otherwise stated.） can also affect the photodetachment dynamics of this system. With the increase of the frequency in the harmonic potential and the oscillating electric field, the number of the closed orbits for the detached electrons increased, which makes the oscillatory structure in the photodetachment cross section much more complex. Our study presents an intuitive understanding of the photodetachment dynamics driven by a harmonic potential plus an oscillating electric field from a space and time dependent viewpoint. This study is very useful in guiding the future experimental research for the photodetachment dynamics in the electric field both changing with space and time.

Semiclassical calculation of ionisation rate for Rydberg helium atoms in an electric field

The ionisation of Rydberg helium atoms in an electric field above the classical ionisation threshold has been examined using the semiclassical method, with particular emphasis on discussing the influence of the core scattering on the escape dynamics of electrons. The results show that the Rydberg helium atoms ionise by emitting a train of electron pulses. Unlike the case of the ionisation of Rydberg hydrogen atom in parallel electric and magnetic fields, where the pulses of the electron are caused by the external magnetic field, the pulse trains for Rydberg helium atoms are created through core scattering. Each peak in the ionisation rate corresponds to the contribution of one core-scattered combination trajectory. This fact further illustrates that the ionic core scattering leads to the chaotic property of the Rydberg helium atom in external fields. Our studies provide a simple explanation for the escape dynamics in the ionisation of nonhydrogenic atoms in external fields.

Autocorrelation Function of Hydrogen Atoms in Magnetic Fields

The autocorrelation function is an important quantity that can reflect the dynamical properties of the Rydberg wave packet and can be measured in experiments. Applying time-dependent perturbation theory and rotating wave approximation, we derive the autocorrelation function of the double-pulse laser describing the evolution of a Rydberg wave packet of hydrogen atoms in magnetic fields. The resulting expression is written as a sum of the modified Caussian terms. Each Caussian term comes from a parent semiclassical closed orbit. It provides a direct explanation and experimentally controllable measurement scheme, which allows us therefore to recognize the closed orbit and to determine its returning time in high precision.

Absorption and Recurrence Spectra of Li Rydberg Atom in Perpendicular Electric and Magnetic Fields

We develop the semi-closed orbit theory from two degrees of freedom to three non-separable degrees of freedom and put forward a new model potential for the Li Rydberg atom, which reduces the study of the system to an effective one-particle problem. Using this model potential and the closed orbit theory for three degrees of freedom, we caiculate the recurrence spectra of Li Rydberg atom in perpendicular electric and magnetic fields. The closed orbits in the corresponding classicai system have also been obtained. The Fourier transformed spectra of Li atom have ailowed direct comparison between the resonance peaks and the scaied action values of closed orbits, whereas the nonhydrogenic resonance can be explained in terms of the new orbits created by the core scattering. Our result is in good agreement with the quantum spectra, which suggests that our calculation is correct.

Absorption and Recurrence Spectra of Nonhydrogenic Rydberg Atom Near a Metal Surface

Multielectron to theoretical treatments atoms near a metal surface are essentially more complicated than hydrogen atom with regard By using the semicalssical dosed orbit theory generalized to the multielecton atoms, we study the dynamical properties of the Rydberg lithium atom near a metal surface. The photoabsorption spectra and recurrence spectra of this system have also been calculated. Considering the effect of the ionic core potential of the Rydberg lithium atom, the number of the closed orbits increases, which leads to more peaks in the recurrence spectra than the case of hydrogen atom near a metal surface. This result shows that the core-scattered effects play an important role in nonhydrogenic atoms. This study is a new application of the closed-orbit theory and is of potential experimental interest.

The influence of interface modifier on photodetachment of negative ions in an electric field near a metal surface

Based on closed-orbit theory, the influence of an interface modifier on the photodetachment of H^- in an electric field near a metal surface is studied. It is demonstrated that the interface strengthens the oscillations in the photodetachment cross section. However, when the electric field environments are different, the strengthening oscillations are caused by different sources. When the electric field direction is upward, the interface enhances the oscillations by shortening the period and the action of the closed orbit. When the electric field direction is downward, the interface strengthens the oscillations either by extending the coherent energy range or by increasing the total number of the closed orbits. We hope that our results will be conducive to the understanding of the photodetachment process of negative ions near interfaces, cavities and ion traps.