A detailed exposition of highly efficient and accurate method for the propagation of the time-dependent Schrödinger equation [50] is presented. The method is readily generalized to solve an arbitrary set of ODE's. The propagation is based on a global approach, in which large time-intervals are treated as a whole, replacing the local considerations of the common propagators. The new method is suitable for various classes of problems, including problems with a time-dependent Hamiltonian, nonlinear problems, non-Hermitian problems and problems with an inhomogeneous source term. In this paper, a thorough presentation of the basic principles of the propagator is given. We give also a special emphasis on the details of the numerical implementation of the method. For the first time, we present the application for a non-Hermitian problem by a numerical example of a one-dimensional atom under the influence of an intense laser field. The efficiency of the method is demonstrated by a comparison with the common Runge-Kutta approach.

Markovian dynamics of open quantum systems are described by the L-GKS equation, known also as the Lindblad equation. The equation is expressed by means of left and right matrix multiplications. This formulation hampers numerical implementations. Representing the dynamics by a matrix-vector notation overcomes this problem. We review three approaches to obtain such a representation. The methods are demonstrated for a driven two-level system subject to spontaneous emission.

Coherent control of harmonic generation was studied theoretically. A specific harmonic order was targeted. An optimal control theory was employed to find the driving field where restrictions were imposed on the frequency band. Additional restrictions were added to suppress undesired outcomes, such as ionization and dissociation. The method was formulated in the frequency domain. An update procedure for the field based on relaxation was employed. The method was tested on several examples demonstrating the generation of high frequencies from a driving field with a restricted frequency band.

A propagation method for the time dependent Schrödinger equation was studied leading to a general scheme of solving ode type equations. Standard space discretization of time-dependent pde's usually results in system of ode's of the formwhere G is a operator ( matrix ) and u is a time-dependent solution vector. Highly accurate methods, based on polynomial approximation of a modified exponential evolution operator, had been developed already for this type of problems where G is a linear, time independent matrix and s is a constant vector. In this paper we will describe a new algorithm for the more general case where s is a time-dependent r.h.s vector. An iterative version of the new algorithm can be applied to the general case where G depends on t or u. Numerical results for Schrödinger equation with time-dependent potential and to non-linear Schrödinger equation will be presented.

The ω sampling is the same as the DCT sampling (Cf. Eq. (C5)). The functional has to be expressed by the terms of the discrete optimization space:where J d is a discretized variant of J. Theǭ(ω) dependence in J is reformulated by discrete means;J energy is replaced by