Generalized Pseudospectral Method for Solving the Time-Dependent Schrodinger Equation Involving the Coulomb Potential

We present an accurate and efficient generalized pseudospectral method for solving the time-dependent Schrodinger equation for atomic systems interacting with intense laser fields. In this method, the time propagation of the wave function is calculated using the well-known second-order split-operator method implemented by the numerically exact, fast transform between the grid and spectral representations. In the grid representation, the radial coordinate is discretized using the Coulomb wave discrete variable representation （CWDVR）, and the angular dependence of the wave function is expanded in the Gauss-Legendre-Fourier grid. In the spectral representation, the wave function is expanded in terms of the eigenfunctions of the field-free zero-order Hamiltonian. Calculations on the high order harmonic generation and ionization dynamics of hydrogen atom in strong laser pulses are presented to demonstrate the accuracy and efficiency of the present method. This new algorithm will be found more computationally attractive than the close-coupled wave packet method using CWDVR and/or methods based on evenly spaced grids.     

THz wave emission from argon in two-color laser field

Terahertz(THz) wave emission from argon atom in a two-color laser pulses is studied numerically by solving the one-dimensional(1D) time-dependent Schr ¨odinger equation. The THz spectra we obtained include both discontinuous and continuum ones. By using the special basis functions that we previously proposed, our analysis points out that the discontinuous and continuum parts are contributed by bound–bound and continuum–continuum transition of atomic energy levels. Although the atomic wave function is strongly dressed during the interaction with laser fields, our identification for the discontinuous part of the THz wave shows that the transition between highly excited bound states can still be well described by the field-free basis function in the tunneling ionization regime.     

数值求解三维含时Schrodinger方程及其应用

发展了一套高精度、高效率的伪谱方法,以非微扰的方式求解真实原子三维含时Schrodinger方程.该方法选用二阶劈裂算符作为时间演化算子,分别选择能谱表象和坐标表象作为含时波函数演化的两个表象.在坐标表象下波函数的径向部分使用库仑波函数离散变量表象来离散;角向波函数展开在两维的Gauss-Legendre-Fourier格点上.以H原子的光激发和光电离过程为例,进行了数值计算并和解析解进行了比对.结果表明二者符合很好.该方法很好地处理了库仑奇点问题.还计算了强激光辐照H原子的多光子电离过程,并和其他的数值方案进行了比较.结果表明,在计算收敛的前提下本方法计算效率更高.

Abstract：

We present an accurate and effective pseudospectral method for solving the three-dimensional time-dependent Schrodinger equation involving the Coulomb potential. In this method, the Hamiltonian is evaluated by exploiting the two representations of the wave function. One is a grid representation, in which the angular dependence of the wave function is expanded in a two-dimensional Gauss-Legendre-Fourier grid in the coordinate space of polar and azimuthal angles. The radial coordinate is discretized using a discrete variable representation constructed from the Coulomb wave function (CWDVR) . The other is a spectral representation, in which the wave function is expanded in a set of square integrable functions chosen as the eigenfunctions of a zero-order Hamiltonian. The time of propagation of the wave function is calculated using the well-known second-order split-operator method implemented through the transform between the grid and spectral representations. Calculations on the photo-absorption strength of hydrogen atom are presented to demonstrate the accuracy of present method in low energy limit by the time-dependent wave-packet propagation method. As another example, the present method is applied to multiphoton ionization of H atom. For a wide range of field parameters, ionization rates calculated using the present method are in excellent agreement with those from other accurate numerical calculations. The new algorithm will be found more efficient than the close coupled wave packet method using CWDVR and/or methods based on evenly spaced grids.     

Binding energies of impurity states in strained wurtzite GaN/Al_xGa_(1-x)N heterojunctions with finitely thick potential barriers

Ground state binding energies of donor impurities in a strained wurtzite GaN/AlxGal_xN heterojunction with a po- tential barrier of finite thickness are investigated using a variational approach combined with a numerical computation. The built-in electric field due to the spontaneous and piezoelectric polarization, the strain modification due to the lattice mismatch near the interfaces, and the effects of ternary mixed crystals are all taken into account. It is found that the binding energies by using numerical wave functions are obviously greater than those by using variational wave functions when impurities are located in the channel near the interface of a heterojunction. Nevertheless, the binding energies using the former functions are obviously less than using the later functions when impurities are located in the channel far from an interface. The difference between our numerical method and the previous variational method is huge, showing that the former should be adopted in further work for the relevant problems. The binding energies each as a function of hydrostatic pressure are also calculated. But the change is unobvious in comparison with that obtained by the variational method.     

Influence of laser fields on the vibrational population of molecules and its wave-packet dynamical investigation

The time-dependent wave packet method is used to investigate the influence of laser-fields on the vibrational population of molecules.For a two-state system in laser fields,the populations on different vibrational levels of the upper and lower electronic states are given by wavefunctions obtained by solving the Schro¨dinger equation with the splitoperator method.The calculation shows that the field parameters,such as intensity,wavelength,duration,and delay time etc.can have different influences on the vibrational population.By varying the laser parameters appropriately one can control the evolution of wave packet and so the vibrational population in each state,which will benefit the light manipulation of atomic and molecular processes.     

Controlling paths of high-order harmonic generation by orthogonal two-color fields

Controlling paths of high-order harmonic generation from H^2+ is theoretically investigated by numerically solving the time-dependent Schrodinger equation based on the Born-Oppenheimer approximation in orthogonal two-color fields.Our simulations show that the change of harmonic emission paths is dependent on time-dependent distribution of electrons.Compared with one-dimensional linearly polarized long wavelength laser,multiple returns are suppressed and short paths are dominant in the process of harmonic emission by two-dimensional orthogonal ω/2ω laser fields.Furthermore,not only are multiple returns weaken,but also the harmonic emission varies from twice to once in an optical cycle by orthogonalω/1.5ωlaser fields.Combining the time-frequency distributions and the time-dependent electron wave packets probability density,the mechanism of controlling paths is further explained.As a result,a 68-as isolated attosecond pulse is obtained by superposing a proper range of the harmonics.     

Interference of dissociating wave packets in the I2 molecule driven by femtosecond laser pulses

The interference between two dissociating wave packets of the I2 molecule driven by femtosecond laser pulses is theoretically studied by using the time-dependent quantum wave packet method.Both the internuclear distanceand velocity-dependent density functions are calculated and discussed.It is demonstrated that the interference pattern is determined by the phase difference and the delay time between two pump pulses.With two identical pulses with a delay time of 305 fs and a FWHM of 20 fs,more interference fringes can be observed,while with two pump pulses with a delay time of 80 fs and a FWHM of 20 fs,only a few interference fringes can be observed.     

Matter-wave solutions of Bose—Einstein condensates with three-body interaction in linear magnetic and time-dependent laser fields

We construct, through a further extension of the tanh-function method, the matter-wave solutions of Bose-Einstein condensates (BECs) with a three-body interaction. The BECs are trapped in a potential comprising the linear magnetic and the time-dependent laser fields. The exact solutions obtained include soliton solutions, such as kink and antikink as well as bright, dark, multisolitonic modulated waves. We realize that the motion and the shape of the solitary wave can be manipulated by controlling the strengths of the fields.     

Control of photoassociation reaction F+H→HF with ultrashort laser pulse

The laser-induced vibrational state-selectivity of product HF in photoassociation reaction H+F$\rightarrow$HF is theoretically investigated by using the time-dependent quantum wave packet method. The population transfer process from the continuum state down to the bound vibrational states can be controlled by the driving laser. The effects of laser pulse parameters and the initial momentum of the two collision atoms on the vibrational population of the product HF are discussed in detail. Photodissociation accompanied with the photoassociation process is also described.     

Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method

In this paper,an approximate function for the Galerkin method is composed using the combination of the exponential B-spline functions.Regularized long wave equation(RLW)is integrated fully by using an exponential B-spline Galerkin method in space together with Crank–Nicolson method in time.Three numerical examples related to propagation of single solitary wave,interaction of two solitary waves and wave generation are employed to illustrate the accuracy and the efficiency of the method.Obtained results are compared with some early studies.     

Generalized Pseudospectral Method for Solving the Time-Dependent Schrödinger Equation Involving the Coulomb Potential

We present an accurate and efficient generalized pseudospectral method for solving the time-dependent Schrödinger equation for atomic systems interacting with intense laser fields. In this method, the time propagation of the wave function is calculated using the well-known second-order split-operator method implemented by the numerically exact, fast transform between the grid and spectral representations. In the grid representation, the radial coordinate is discretized using the Coulomb wave discrete variable representation (CWDVR), and the angular dependence of the wave function is expanded in the Gauss-Legendre-Fourier grid. In the spectral representation, the wave function is expanded in terms of the eigenfunctions of the field-free zero-order Hamiltonian. Calculations on the high order harmonic generation and ionization dynamics of hydrogen atom in strong laser pulses are presented to demonstrate the accuracy and efficiency of the present method. This new algorithm will be found more computationally attractive than the close-coupled wave packet method using CWDVR and/or methods based on evenly spaced grids.     

用伪谱方法计算强激光场中一维原子的阈上电离谱

引入非线性空间变换,用伪谱方法求解了一维原子在强激光场中的薛定谔方程,再利用B样条函数和傅立叶级数的线性组合构造原子未微扰的本征函数,计算了一维原子在强激光场中的阈上电离谱,其结果与分裂算符法得到的结果符合得很好.     

Laser-assisted positron-impact ionization of atomic hydrogen

We study the ionization of atomic hydrogen by a fast positron in the presence of an external linearly polarized laser field. We concentrate on the limit of a small momentum transfer and describe the fast positron's continuum states by Volkov wave functions. The ejected electron is described by a Coulomb-Volkov wave function. We are limited to small laser intensities such that the dressed state of the target is treatable within the time-dependent perturbation theory, even though the laser intensity is still quite high by laboratory standards. Numerical results for the triply differential cross sections and their dependencies on laser-field parameters are discussed and compared with the results of laser-assisted ionization by electron impact.     

Influence of quantum transitions in the continuum on ionization of atoms in strong fields

The tight-binding method is used to analyze the ionization of a hydrogenlike atom by an intense monochromatic laser field. The orthogonal and normalized basis in which the solution of the time-dependent Schrödinger equation is expanded contains unperturbed wave functions of the discrete spectrum and generalized Coulomb wave functions of the continuum. In the solution of the coupled equations we make use of the fact that the bound-free and free-free transitions are efficient in different regions of complex time. Simplified equations are constructed and investigated. Results of calculations for ionization of a hydrogen atom from its ground state and of the energy distribution of the electrons in strong and superstrong linearly polarized fields are presented. It is shown that in this case the ground state decays completely, and free-free transitions play a defining role in the dynamics of the process. It is established that the total probability of population of the upper Rydberg states abutting the continuum does not exceed 0.05. The range of applicability of the approach is discussed. A comparison with numerical results obtained by other authors is given.     

Extracting the phase distribution of the electron wave packet ionized by an elliptically polarized laser pulse

We use an interferometic scheme to extract the phase distribution of the electron wave packet from above-threshold ionization in elliptically polarized laser fields. In this scheme, an electron wave packet released from a circularly polarized laser pulse acts as a reference wave and interferes with the electron wave packet ionized by a time-delayed counter-rotating elliptically polarized laser field. The generated vortex-shaped interference pattern in the photoelectron momentum distribution enables us to extract the phase distribution of the electron wave packet in the elliptically polarized laser pulse with high precision. By artificially screening the ionic potential at different ranges when solving the time-dependent Schördinger equation, we find that the angle-dependent phase distribution of the electron wave packet in the elliptically polarized laser field shows an obvious angular shift as compared to the strong-field approximation, whose value is the same as the attoclock shift. We also show that the amplitude of the angle-dependent phase distribution is sensitive to the ellipticity of the laser pulse, providing an alternative way to precisely calibrate the laser ellipticity in the attoclock measurement.     

Photoionization of NaK molecule with a double-well potential in femtosecond pump--probe pulse laser fields

The method of time-dependent quantum wave packet dynamics is used to calculate the femtosecond pump--probe photoelectron spectra and study the wave packet dynamic processes of the double-minimum potential state 6¹∑⁺ of NaK in intense laser fields. The evolutions of the wave packet and the photoelectron energy spectra with time and internuclear distance are described in detail. The wave packet dynamic information of the 6¹∑⁺ state can be extracted from the photoelectron energy spectra.     

The effect of field modulation on the vibrational population of the photoassociated NaK and its dynamics

This paper presents calculation results for the photoassociation of a NaK molecule with a two-color modulated laser and gives a detailed analysis about them.For the two-step photoassociation process in intense fields,the effect of two-color modulated laser parameters,such as relative phase,envelope period,and laser intensity,on the population of the molecular electronic state can be obtained by solving the time-dependent Schrodinger equation through the quantum wave packet method.The numerical simulation shows not only that the influence of laser parameters on the vibrational distribution presents some regularity,but also that a higher population in the ground electronic state can be realized through adjusting these laser parameters.