Symplectic-algorithm Computation Of One- Dimensional Model Atoms In Intense Laser Fields

Recently, with the development of laser technology, we can get the laser pulse with the maximum intensity about 1020w/cm2 in the experiment, which has made the research of the laser technology become extremely and foreland basic subject in the domain of physics, chemistry, material, medicine and bioscience etc. Any research based on the laser theory and application sums up the research on the interaction between the laser and the matter. The photon emission under the laser, i.e., high-order harmonics generation, is an important area in the researching of the interaction between the intensity laser and matter. The high-order harmonic not only provides a new source of XUV and the soft X-ray radiations, but also shows a new possible way for extremely short pulses.In the traditional quantum mechanics, the interaction with the laser and matter is regarded as the perturbation of the coulomb field and is dealt with perturbation theory. When the electron locates at the ground state of the hydrogen atom, its coulomb electronic field is 5.14×109 V /cm, corresponding to the light intensity3.51×101 6 W /cm2, which cannot reach up in the previous laser machine, so the previous result from the perturbation theory is accord to the one from the experiment. But when intensity of the laser field reaches so much as overruns the coulomb electronic field, some experimental phenomena cannot be explained by the perturbation theory. That is to say, the traditional perturbation theory in the quantum mechanics will be invalidated for the interaction with the laser and atom. In the past decade, various nonperturbative methods have been developed and adopted, i.e., Floquet theory method, R-matrix method, etc, but these methods cannot be applied to the super-short and super-intense laser field. The methods solving numerically time-dependent Schr(o|¨)dinger equation (TDSE) have been applied to describe the interaction between the super-short and super-intense laser and atom, that is, time-dependent Schr(o|¨)dinger equation is directly discretized using finite difference, discrete variable representation, finite elements, B-splines or a basis set expansion approach. In the basis set expansion approach, the basis function set may be chosen as the Eigen state of free-field, Volkov state and so on.Time-dependent Schr(o|¨)dinger equation implies all the physical messages about the atom, the laser field and the interaction with the laser and atom, so solving numerically Time-dependent Schr(o|¨)dinger equation is one of the popular and the reasonable methods for the research about the interaction with the laser and atom in the present theoretical research. In the interaction with the laser and atom, the electronic influence in the direction along the polarization of the laser field is greater than by other directions, so the one-dimensional model atom is chosen in the research about the interaction between the laser and atom and makes the problem simpler.There is symplectic structure in the Hamilton system. Hamilton canonical equation maintains symplectic structure after symplectic transformation. The solutions of the Hamilton canonical equation can be generated from symplectic group. Based on this, in the early 1980's, Ruth and Feng Kang presented the symplectic algorithm of the Hamilton system, which maintains the symplectic structure of the system. After that, many people have investigated on the symplectic structure. Feng Kang and his research group presented the methods generating the symplectic schemes, i.e., generating function methods and power series expansion methods and investigated on the algorithm of the conservation quantity and the preserving-volume, contacted structure and contacted algorithm, etc. Sun Geng and J. M. Sanz-Serna al. studied the symplectic Runge-Kutta method, and Yashida presented the symmetrical power methods that construct the symplectic schemes of the separable Hamilton system, and etc. Now symplectic algorithm has applied to many fields, such as the astronomy, aerography, geognosy, plasmas, molecular dynamics and quantum system, and has shown the obvious advantage in the long time and multi-step computation than the traditional nonsymplectic algorithm.In the quantum mechanics textbook, it is point out that, quantum system is an infinite-dimensional Hamilton system; time-evolution of the solution of time-dependent Schr(o|¨)dinger equation is unitary. This is equivalent to the properties of the norm conversation of the wave function and symplectic product conservation. So the time dependent Schr(o|¨)dinger equation is discretized into a finite dimensional canonical equation, whose norm of the wave function is a conservation quantity, and it should be solved by symplectic scheme, which keeps the norm conservation, which is defined as the preserving-structure algorithm for time dependent Schr(o|¨)dinger equation.In this paper, the time-dependent Schr(o|¨)dinger equation of the atom is solved in the intense laser field by the numerical basis set expansion method. Firstly the method expands the time-evolution wave function based on a maturity basis function, and then we can get the coefficient of the expansion by symplectic algorithm. Two parts will be presented about how to choose the basis function here: 1).Computing the discrete Eigen-function by the improved symplectic shooting method; 2).Computing the continuum Eigen-function by diagonalization of matrix, and then discrete it by the pseudo-spectrum expansion method. The linear combination of the tow part can be the maturity basis function. The advantage of this basis function is that, it not only can describe the discrete states of the atom, but also can describe the continuum states of the atom commendably.The method above is used to solve the wave function, high-order harmonics and the ionization rate of the model atom in the intense laser field. We have investigated the problem of time-evolution 1-D model atom under the soft score Coulomb potential in the intense laser field.