One-Dimensional Chain of n-Level Atoms and Discrete Nonlinear Schrödinger Equation

The Hamiltonian of one-dimensional chain of n-level atoms is represented in terms of Boson operators by using the Dyson-Maleev transformation and it is shown that the finite-ladder effect disappears when n tends toward infinity.In this way,it is found that the Heisenberg equation of motion of this system is exactly described in the coherent state representation by the dark discrete nonlinear Schrödinger(DNLS)equation.It is also briefly shown that the DNLS equation has some general soliton solutions.This indicates that this simple system has richness of nonlinear waves.

An Analytical Study for Controlling Chaos in Duffing Oscillator with Weak Periodic Perturbation

We obtain a general unstable periodic solution near the homoclinic orbit of the Duffing oscillator with weak periodic perturbation by using the direct perturbation technique.Theoretical analysis reveals that the stable periodic orbits are embedded in the Melnikov chaotic attractor.The corresponding numerical results show that fitting control parameters into the stability conditions can control chaos in the system,and the phase difference between the two sinusoidal forces added to the Duffing equation plays an important role in controlling chaos.

Analytic Study of a Bose-Einstein Condensate in Waveguide with an Obstacle Potential

We investigate the exact solutions of one-dimensional (1D) time-independent Gross-Pitaevskii equation(GPE),which governs a Bose-Einstein condensate (BEC) in the magnetic waveguide with a square-Sech potential.Boththe bound state and transmission state are found and the corresponding spatial configurations and transport propertiesof BEC are analyzed.It is shown that the well-known absolute transmission of the linear system can occur in theconsidered nonlinear system.

Laser Controlling Wavepacket Trains of a Paul Trapped Ion

We have studied the quantum and classical motions of a single Paul trapped ion interacting with a timeperiodic laser field. By using the test-function method, we construct n exact solutions of quantum dynamics that describe the generalized squeezed coherent states with the expectation orbits being the corresponding classical ones. The spacetime evolutions of the exact probability densities show some wavepacket trains. It is demonstrated analytically that by adjusting the laser intensity and frequency, we can control the center motions of the wavepacket trains. We also discuss the other physical properties such as the expectation value of energy, the widths and heights of the wavepackets, and the resonance loss of stability.

Exact Vortex Clusters of Two-Dimensional Quantum Fluid with Harmonic Confinement

A family of exact analytical solutions of vortices in quantum fluid governed by a two-dimensional time-dependent Schr?dinger equation is presented,which describes different kinds of vortex structures.The dynamics of different vortex clusters,such as the single vortex,vortex pair,vortex dipole and vortex trimer in a two-dimensional quantum fluid are analytically studied based on these exact solutions.The time evolutions of the wave of such vortices are demonstrated,and the orbits of motion of singular points in the vortices are also explored.The interactions of vortices in many-vortex clusters are discussed.A repulsive interaction between vortices with the same topological charge,and inter-annihilation and inter-creation of vortices with opposite topological charge,are shown.

Variational-Integral Perturbation Corrections for Hydrogen Atoms in Magnetic Fields

A variational-integral perturbation method(VIPM)has been established by combining the variational perturbation with the integral perturbation.We present the results obtained by the application of VIPM to the ground and excited states associated with the Zeeman effect on a hydrogen atom.We construct correction wave functions and calculate energy corrections for the ground state and some high excited states.These values are compared with results of Smith and Jason et al.;our results are more accurate than those from Smith and Jason,especially in the 4f excited state where we find a very interesting result that when magnetic fields increase,the energy level trap appears.

Bound States of the Perturbed Schrödinger Systems

A bound state of the Schrödinger system is disturbed by an additional potential.It is shown that the system is still in a bound state if and only if its energy correction is restricted to the corresponding value.This infers some previous series solutions in quantum mechanics to be unbound states for the order i>1 of perturbation.The corrected bound states of the system are given and their physical meaning is discussed simply.Application of the method to Stark effect reveals its advantage for researches of quantum chaos.