Explicit schemes for time propagating many-body wavefunctions

Accurate theoretical data on many time-dependent processes in atomic and molecular physics and in chemistry require the direct numerical solution of the time-dependent Schr\"odinger equation, thereby motivating the development of very efficient time propagators. These usually involve the solution of very large systems of first order differential equations that are characterized by a high degree of stiffness. We analyze and compare the performance of the explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have exactly the same stability function, therefore sharing the same stability properties that turn out to be optimum. Their respective accuracy however differs significantly and depends on the physical situation involved. In order to test this accuracy, we use a predictor-corrector scheme in which the predictor is either Fatunla's or Arnoldi's algorithm and the corrector, a fully implicit four-stage Radau IIA method of order 7. We consider two physical processes. The first one is the ionization of an atomic system by a short and intense electromagnetic pulse; the atomic systems include a one-dimensional Gaussian model potential as well as atomic hydrogen and helium, both in full dimensionality. The second process is the decoherence of two-electron quantum states when a time independent perturbation is applied to a planar two-electron quantum dot where both electrons are confined in an anharmonic potential. Even though the Hamiltonian of this system is time independent the corresponding differential equation shows a striking stiffness. For the one-dimensional Gaussian potential we discuss in detail the possibility of monitoring the time step for both explicit algorithms. In the other physical situations that are much more demanding in term of computations, we show that the accuracy of both algorithms depends strongly on the degree of stiffness of the problem.Comment: 24 pages, 14 Figure

Static field limit of excitation probabilities in laser-atom interactions

We consider the interaction of atomic hydrogen, in its ground state, with an electromagnetic pulse whose duration is fixed in terms of the number of optical cycles. We study the probability of excitation of the atom in the static field limit i.e. for field frequencies going to zero. Despite the fact that the well-known Born–Fock adiabatic theorem is valid only for a system whose energy spectrum is discrete, we show that it is still possible to use this theorem to derive, in the low frequency limit, an analytical formula which gives the probability of transition to any excited state of the atom as a function of the field intensity, the carrier envelope phase and the number of optical cycles within the pulse. The results for the probability of excitation to lowlying excited states, obtained with this formula, agree with those we get by solving the timedependent Schrödinger equation. The domain of validity is discussed in detail

Multiresolution schemes for time-scaled propagation of wave packets

We present a detailed analysis of the time scaled coordinate approach and its implementation for solving the time-dependent Schr\"odinger equation describing the interaction of atoms or molecules with radiation pulses. We investigate and discuss the performance of multi-resolution schemes for the treatment of the squeezing around the origin of the bound part of the scaled wave packet. When the wave packet is expressed in terms of B-splines, we consider two different types of breakpoint sequences: an exponential sequence with a constant density and an initially uniform sequence with a density of points around the origin that increases with time. These two multi-resolution schemes are tested in the case of a one-dimensional gaussian potential and for atomic hydrogen. In the latter case, we also use Sturmian functions to describe the scaled wave packet and discuss a multi-resolution scheme which consists in working in a sturmian basis characterized by a set of non-linear parameters. Regarding the continuum part of the scaled wave packet, we show explicitly that, for large times, the group velocity of each ionized wave packet goes to zero while its dispersion is suppressed thereby explaining why, eventually, the scaled wave packet associated to the ejected electrons becomes stationary. Finally, we show that only the lowest scaled bound states can be removed from the total scaled wave packet once the interaction with the pulse has ceased

Ionisation of H₂O by a strong ultrashort XUV pulse:a model within the single active electron approximation

We present and discuss a new computationally inexpensive method to study, within the single active electron approximation, the interaction of a complex system with an intense ultrashort laser pulse. As a first application, we consider the one photon single ionisation of the highest occupied molecular orbital of the water molecule by a laser pulse. The ionisation yield is calculated for different orientations of the molecule with respect to the field polarization axis and for different carrier envelope phases of the pulse, and compared against predictions of another single active electron approach.Comment: 24 pages, 13 figure

Modelling laser-atom interactions in the strong field regime

We consider the ionisation of atomic hydrogen by a strong infrared field. We extend and study in more depth an existing semi-analytical model. Starting from the time-dependent Schroedinger equation in momentum space and in the velocity gauge we substitute the kernel of the non-local Coulomb potential by a sum of N separable potentials, each of them supporting one hydrogen bound state. This leads to a set of N coupled one-dimensional linear Volterra integral equations to solve. We analyze the gauge problem for the model, the different ways of generating the separable potentials and establish a clear link with the strong field approximation which turns out to be a limiting case of the present model. We calculate electron energy spectra as well as the time evolution of electron wave packets in momentum space. We compare and discuss the results obtained with the model and with the strong field approximation and examine in this context, the role of excited states.Comment: 11 pages, 5 figure

Strong field approximation within a Faddeev-like formalism for laser-matter interactions

We consider the interaction of atomic hydrogen with an intense laser field within the strong-field approximation. By using a Faddeev-like formalism, we introduce a new perturbative series in the binding potential of the atom. As a first test of this new approach, we calculate the electron energy spectrum in the very simple case of a photon energy higher than the ionisation potential. We show that by contrast to the standard perturbative series in the binding potential obtained within the strong field approximation, the first terms of the new series converge rapidly towards the results we get by solving the corresponding time-dependent Schroedinger equation.Comment: 7 pages, 1 figur

Ionization and excitation of the excited hydrogen atom in strong circularly polarized laser fields

In the recent work of Herath et al. [T. Herath, L. Yan, S. K. Lee, and W. Li, Phys. Rev. Lett. 109, 043004 (2012)PRLTAO0031-900710.1103/PhysRevLett.109.043004] the first experimental observation of a dependence of strong-field ionization rate on the sign of the magnetic quantum number m [of the initial bound state (n,l,m)] was reported. The experiment with nearly circularly polarized light could not distinguish which sign of m favors faster ionization. We perform ab initio calculations for the hydrogen atom initially in one of the four bound substates with the principal quantum number n=2, and irradiated by a short circularly polarized laser pulse of 800nm. In the intensity range of 1012-1013W/cm2 excited bound states play a very important role, but also up to some 1015W/cm2 they cannot be neglected in a full description of the laser-atom interaction. We explore the region that with increasing intensity switches from multiphoton to over-the-barrier ionization and we find, unlike in tunneling-type theories, that the ratio of ionization rates for electrons initially counter-rotating and corotating (with respect to the laser field) may be higher or lower than 1

Explicit schemes for time propagating many-body wave functions

Accurate theoretical data on many time-dependent processes in atomic and molecular physics and in chemistry require the direct numerical ab initio solution of the time-dependent Schrödinger equation, thereby motivating the development of very efficient time propagators. These usually involve the solution of very large systems of first-order differential equations that are characterized by a high degree of stiffness. In this contribution, we analyze and compare the performance of the explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have exactly the same stability function, therefore sharing the same stability properties that turn out to be optimum. Their respective accuracy, however, differs significantly and depends on the physical situation involved. In order to test this accuracy, we use a predictor-corrector scheme in which the predictor is either Fatunla's or Arnoldi's algorithm and the corrector, a fully implicit four-stage Radau IIA method of order 7. In this contribution, we consider two physical processes. The first one is the ionization of an atomic system by a short and intense electromagnetic pulse; the atomic systems include a one-dimensional Gaussian model potential as well as atomic hydrogen and helium, both in full dimensionality. The second process is the decoherence of two-electron quantum states when a time-independent perturbation is applied to a planar two-electron quantum dot where both electrons are confined in an anharmonic potential. Even though the Hamiltonian of this system is time independent the corresponding differential equation shows a striking stiffness which makes the time integration extremely difficult. In the case of the one-dimensional Gaussian potential we discuss in detail the possibility of monitoring the time step for both explicit algorithms. In the other physical situations that are much more demanding in term of computations, we show that the accuracy of both algorithms depends strongly on the degree of stiffness of the problem