Local Absorbing Boundary Conditions For The Time-dependent Schr(?)dinger Equation Of Laser-atom Interaction

Schr(o|¨)dinger equation uncovers the basic regulation of the motive particle in the microcosmic world, which equation generally is expressed bywhere H(o|¨) is the Hamiltonian operator. In the interaction of laser and atom, the time-dependent Schr(o|¨)dinger equation includes all the contents of physics referring to the atom, laser field and the interaction of laser and atom, so numerical solution of the time-dependent of Schr(o|¨)dinger equation of laser and atom is an effective method. Recent years, with the development of laser technique, the intensity of laser rapidly exceeds the intensity of Coulomb field which bounds the electrons. Many novel phenomena are observed by people. High order harmonic emission is an important nonlinear phenomenon in the process of the interaction of laser and atom. The general characteristic of high order harmonic spectrum is first the rapid decreasing from the basic frequency, then a plateau of the spectrum and at the area there are no great differences of harmonic intensities, and last a cutoff and a sharp decreasing. The high order harmonics with low energy are mainly generated by the interaction of the bound states. For the plateau area, the high order harmonics are mainly generated by the interaction of the bound states. The interaction of the continuum states takes little effect to the generation of high order harmonics. According to the classical three step model theory, when a laser field irradiates an atomic system, the electron is ionized by tunneling through the barrier formed by the Coulomb potential and laser , then it is accelerated and may arrive at the state of continuous states of ionization because of the laser pulse, when the quasi-electron would get the extra energy with the oscillation of the laser field and its motion could be described by the classical theory, and with the laser field inversing its direction the electron may recombines with the parent ion and emits a harmonic photon.For one thing, in the whole process of the generation of high-order harmonics,the electronic wave packets which take the part in the generation of harmonics mostly concentrate in the finite space and are not far away from the nuclear, and the outgoing wave packets have little effects to it.For another, the interaction between laser and atom is an"open system"and its calculating space is infinite. The larger the problem space we are researching, the higher the ability of the storage required of the computer. We are often limited by the storage and calculation speed of the computer. So the space of the interaction of laser and atom we researching should be finite. In order to equalize the limit space and the limitless space, we should deal with the boundary of the limit space with a special method, so that the wave packet propagating to the outside border remains outside-propagating at the border and has low reflections and the wave packet in the inner space would not produce the aberrance. So to solve of the time-dependent Schr(o|¨)dinger accurately, not only should we select the smaller calculating step, but also should we use the rational boundary condition to simulate the information of the motion and emission of wave packet of electron in laser field and to avoid the unphysical reflection of the wave packet of electron at the border.There are several works done by us as following:First, we deduce the time-dependent Schr(o|¨)dinger equation of the interaction of laser and atom in the length gauge through the Hamiltonian function of the system. The direction of the electron forced by the laser following the direction of the electronic field of laser is considered and the rationality of one-dimension model atom used in the computation is introduced. We numerically solve the time-independent Schr(o|¨)dinger equation with the splitting-operator fast Fourier transform algorithm to obtain the ground state and the excited states of the atom, and detailed process of the deduction and substantiation of the method is showed in our paper. At the same time we take the zero boundaries as the example to introduce the Crank-Nicholson difference method of solving the time-dependent Schr(o|¨)dinger equation.Second, we set forth the theory of the local absorbing boundary condition. On the boundaries, the wave functions are approximated by the wave packet by the superposition of plane waves. With the one-way dispersion relation and rational function approximation, the local boundary condition of laser and atom is established.We present the theory of exterior complex scaling absorbing boundary condition. The transformation of the complex coordinates on the boundary area is used, and by the characteristic of the large amplitude of laser field, the wave packet of electron on the boundary decreases quickly on the boundary area, so the aim of reducing the boundary reflection is acquired. In addition, the theory of masking function absorbing boundary condition is also presented briefly. And there is a conclusion of these boundary conditions in our paper.Third, the expression of the linearly polarized laser field in the dipole approximation and the one-dimension atomic model are introduced, and the derivation of the power spectrum of the high order harmonics in the length and the acceleration gauges is showed. We numerically simulate the evolution of wave packet of electron with the local absorbing boundary condition in the finite calculating space. High-order harmonic spectrum from a hydrogen atom exposed to a linearly polarized laser pulse is simulated with the local absorbing boundary condition. The results of computation demonstrate local absorbing boundary condition could avoid spurious harmonics effectively in the small calculating l interval. High-order harmonic spectra obtained with the other absorbers (cos1/8 masking function, exterior complex scaling, and zero boundary) as boundary condition in the limited computational region are also simulated. It indicates that with high order harmonic spectrum is calculated with the different boundary conditions, the different computational intervals to avoid unphysical harmonics should be selected. The comparative analysis shows that local absorbing boundary condition provides a better way to solve the problem of boundary reflection of the laser and atom in the finite calculating space.