Comparative study of photoionization of atomic hydrogen by solving the one- and three-dimensional time-dependent Schrödinger equations

We develop a numerical scheme for solving the one-dimensional (1D) time-dependent Schrödinger equation (TDSE), and use it to study the strong-field photoionization of the atomic hydrogen. The photoelectron energy spectra obtained for pulses ranging from XUV to near infrared are compared in detail to the spectra calculated with our well-developed code for accurately solving the three-dimensional (3D) TDSE. For XUV pulses, our discussions cover intensities at which the ionization is in the perturbative and nonperturbative regimes. For pulses of 400 nm or longer wavelengths, we distinguish the multiphoton and tunneling regimes. Similarities and discrepancies between the 1D and 3D calculations in each regime are discussed. The observed discrepancies mainly originate from the differences in the transition matrix elements and the energy level structures created in the 1D and 3D calculations.     

非均匀光纤中暗孤子传输特性研究

由于变系数非线性Schrödinger方程的增益、色散和非线性项都是变化的, 根据方程这一特点可以研究光脉冲在非均匀光纤中的传输特性. 本文利用Hirota方法, 得到非线性Schrödinger方程的解析暗孤子解. 然后根据暗孤子解对暗孤子的传输特性进行讨论, 并且分析各个物理参量对暗孤子传输的影响. 经研究发现, 通过调节光纤的损耗、色散和非线性效应都能有效的控制暗孤子的传输, 从而提高非均匀光纤中的光脉冲传输质量. 此外, 本文还得到了所求解方程的解析双暗孤子解, 最后对两个暗孤子相互作用进行了探讨. 本文得到的结论有利于研究非均匀光纤中的孤子控制技术.     

A Vector Fokas-Lenells System from the Coupled Nonlinear Schrödinger Equations

With the aid of the spectral gradient method of Fuchssteiner, the compatible pair of Hamiltonian operators for the coupled NLS hierarchy is rediscovered. This result enables us to construct a hierarchy, which contains a vector generalization of Fokas-Lenells system. The vector Fokas-Lenells system is shown to be bi-Hamiltonian and to possess a Lax pair.     

Photoelectron momentum distributions of Ne and Xe dimers in counter-rotating circularly polarized laser fields

The strong-field ionization of dimers is investigated theoretically in counter-rotating circularly polarized laser fields. By numerically solving the two-dimensional (2D) time-dependent Schrödinger equation (TDSE) with the single-electron approximation (SEA) frame, we present the photoelectron momentum distributions (PMDs) and photoelectron angular distribution (PADs) of aligned Ne and Xe dimers. It is found that the PMDs and PADs strongly depend on the time delays by counter-rotating circularly polarized laser pulses. The results can be explained by the ultrafast photoionization model and the evolution of electron wave packets for Ne and Xe dimers. Besides, We make a comparison of PMDs between Ne atom and Ne dimer.     

求解带有锥形交叉区域薛定谔方程的改进界面跃迁格式

改进一种欧拉框架下的界面跃迁格式.原格式是为求解带有多个能级,不同能级之间存在锥形交叉区域的薛定谔方程提出的.石墨烯中的电子输运与上述过程相似.Kammerer等人提出带有跳跃算子的数值格式能够更好地保持石墨烯中电子输运的能量守恒.引入这种跳跃算子可以改进原有的界面跃迁格式.改进格式的数值结果取得了预期的效果.     

Evaluating the robustness of rogue waves under perturbations

Rogue waves, and their periodic counterparts, have been shown to exist in a number of integrable models. However, relatively little is known about the existence of these objects in models where an exact formula is unattainable. In this work, we develop a novel numerical perspective towards identifying such states as localized solutions in space-time. Importantly, we illustrate this methodology for different perturbations of nonlinear Schrödinger models. In particular, in addition to benchmarking known solutions (and confirming their numerical propagation under controllable error) this enables the continuation of such solutions over parametric variations to non-integrable models. As a result, we can answer in the positive the question about the parametric robustness of Peregrine-like waveforms and even of generalizations thereof on a cnoidal wave background.     

Radial position-momentum uncertainties for the infinite circular well and Fisher entropy

We show how the product of the radial position and momentum uncertainties can be obtained analytically for the infinite circular well potential. Some interesting features are found. First, the uncertainty Δr increases with the radius R and the quantum number n, the n-th root of the Bessel function. The variation of the Δr is almost independent of the quantum number n for $n>4$ and it will arrive to a constant for a large n, say $n>4$. Second, we find that the relative dispersion $\mathrm{\Delta }r/〈r〉$ is independent of the radius R. Moreover, the relative dispersion increases with the quantum number n but decreases with the azimuthal quantum number m. Third, the momentum uncertainty Δp decreases with the radius R and increases with the quantum numbers $m>1$ and n. Fourth, the product $\mathrm{\Delta }r\mathrm{\Delta }{p}_r$ of the position-momentum uncertainty relations is independent of the radius R and increases with the quantum numbers m and n. Finally, we present the analytical expression for the Fisher entropy. Notice that the Fisher entropy decreases with the radius R and it increases with the quantum numbers $m>0$ and n. Also, we find that the Cramer–Rao uncertainty relation is satisfied and it increases with the quantum numbers $m>0$ and n, too.     

环形振子势的精确解

将环形振子势的Schrödinger方程在球坐标系中进行变量分离.然后求解了角向方程和径向方程,给出了精确的能谱方程.获得了用普遍的缔合Legendre多项式表示的归一化的角向波函数和用合流超几何函数表示的归一化的径向波函数.     

Time-evolution-proof Scattering Data for the Focusing and Defocusing Zakharov-Shabat Systems

In this article we give sufficient conditions on the scattering data of a defocusing or focusing Zakharov-Shabat system in order that its potential is square integrable. For a dense subset of integrable as well as square integrable potentials, we show that the scattering data actually satisfy these sufficient conditions.     

Rogue waves: New forms enabled by GPU computing

A method to determine parameters governing periodic Riemann theta function rogue-wave solutions to the nonlinear Schrödinger equation is presented. A map of parameter values leading to candidate solutions is developed. In addition to candidate solutions, an overview of qualitative aspects of the solution space can be gained from this map. Based on these findings, several new extreme wave solutions are presented. Although the computations required to determine the map are quite demanding, it is shown that these computations can be efficiently accelerated with a parallel computing architecture. A general purpose computing on a graphics processor unit (GPGPU) implementation yielded a 400× acceleration over a single threaded high level implementation. This acceleration enabled exploration and examination of the solution space, which otherwise would not have been possible. In addition, the solution methodology presented here can be extended to explore other classes of solutions.     

Nonrelativistic Shannon information entropy for Kratzer potential

The Shannon information entropy is investigated within the nonrelativistic framework. The Kratzer potential is considered as the interaction and the problem is solved in a quasi-exact analytical manner to discuss the ground and first excited states. Some interesting features of the information entropy densities as well as the probability densities are demonstrated.The Bialynicki–Birula–Mycielski inequality is also tested and found to hold for these cases.     

Exact solutions of the Gerdjikov-Ivanov equation using Darboux transformations

We study the Gerdjikov-Ivanov (GI) equation and present a standard Darboux transformation for it. The solution is given in terms of quasideterminants. Further, the parabolic, soliton and breather solutions of the GI equation are given as explicit examples.     

氦原子1s2p态的有限元近似能量

用有限元方法近似计算了1s2p态氦原子单态和三重态的能量,所得结果的相对误差:三重态为10^-6,单态为10^-4.这一结果比Schertzer^[1]对基态氦原子的相应结果稍好.有限元法导致的大型广义矩阵特征值问题,对于基态是对称的,而对于1s2p态是非对称的,给求解带来了难度.由波函数的图形说明,在有界区域上求Schrödinger方程的近似解是合理的.     

Discrete solitons in zigzag waveguide arrays with different types of linear mixing between nearest-neighbor and next-nearest-neighbor couplings

We study discrete solitons in zigzag discrete waveguide arrays with different types of linear mixing between nearest-neighbor and next-nearest-neighbor couplings. The waveguide array is constructed from two layers of one-dimensional (1D) waveguide arrays arranged in zigzag form. If we alternately label the number of waveguides between the two layers, the cross-layer couplings (which couple one waveguide in one layer with two adjacent waveguides in the other layer) construct the nearest-neighbor couplings, while the couplings that couple this waveguide with the two nearest-neighbor waveguides in the same layer, i.e., self-layer couplings, contribute the next-nearest-neighbor couplings. Two families of discrete solitons are found when these couplings feature different types of linear mixing. As the total power is increased, a phase transition of the second kind occurs for discrete solitons in one type of setting, which is formed when the nearest-neighbor coupling and next-nearest-neighbor coupling feature positive and negative linear mixing, respectively. The mobilities and collisions of these two families of solitons are discussed systematically throughout the paper, revealing that the width of the soliton plays an important role in its motion. Moreover, the phase transition strongly influences the motions and collisions of the solitons.     

On Darboux transformations for the derivative nonlinear Schrödinger equation

We consider Darboux transformations for the derivative nonlinear Schrödinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, the parabolic and soliton solutions of the derivative nonlinear Schrödinger equation are given as explicit examples.     

A new trial equation method for finding exact chirped soliton solutions of the quintic derivative nonlinear Schrödinger equation with variable coefficients

In this work, we propose an efficient generalization of the trial equation method introduced recently by Liu [Appl. Math. Comput. 217 (2011) 5866] to construct exact chirped traveling wave solutions of complex differential equations with variable coefficients. The effectiveness of the proposed method has been tested by applying it successfully to the quintic derivative nonlinear Schrödinger equation with variable coefficients. As a result, a class of chirped soliton-like solutions including bright and kink solitons is derived for the first time. Compared with previous work of Liu in which unchirped solutions were given, we obtain exact chirped solutions which have nontrivial phase that varies as a function of the wave intensity. These localized structures characteristically exist due to a balance among the group-velocity dispersion, self-steepening and competing cubic-quintic nonlinearity. Parametric conditions for the existence of envelope solutions with nonlinear chirp are also presented. It is shown that the chirping can be effectively controlled through the variable parameters of group-velocity dispersion and self-steepening.     

Electromagnetic Radiation of Accelerated Charged Particle in the Framework of a Semiclassical Approach

The problem of the electromagnetic radiation produced by charge distributions in the framework of a semiclassical approach proposed in the work by Bagrov, Gitman, Shishmarev, and Farias Jr. [J. Synchrotron Rad. 27 , 902–911 (2020)] is addressed here. In this approach, currents, generating the radiation are considered classically, while the quantum nature of the radiation is kept exactly. Quantum states of the electromagnetic field are solutions of Schrödinger's equation, and relevant quantities to the problem are evaluated with the aid of transition probabilities. This construction allows us to introduce the quantum transition time in physical quantities and assess its role in radiation problems by classical currents. Radiated electromagnetic energies are studied in detail and a definition for the rate at which radiation is emitted from sources is presented. In calculating the total energy and rate radiated by a pointlike charged particle accelerated by a constant and uniform electric field, it is discovered that these results are compatible with results obtained by other authors in the framework of the classical radiation theory under an appropriate limit. Numerical and asymptotic analyses of the results are also performed.     

Quantum dynamics in a time-dependent cylindrical trap

Solutions to the Schrödinger equation are examined for a particle inside a cylindrical trap of a circular time-dependent cross-section. Analytical expressions for energy and momentum expectation values are derived with respect to the exact solutions; and the adiabatic and sudden change of the boundary are discussed. The density profile as a function of time in a given observation point, resembles the diffraction-in-time pattern observed in a suddenly released particle but with an enhanced fringe visibility. Numerical computations are presented for both contracting and expanding boxes.