Time-dependent Schroedinger Equation Research Articles

Molecules in intense laser fields: Beyond the dipole approximation

The time-dependent Schroedinger equation is solved for a Born-Oppenheimer (static nuclei) three-dimensional H{sub 2}{sup +} in super intense laser fields (I=4x10{sup 18}, 10{sup 19}, and 4x10{sup 19} W/cm{sup 2}) at wavelength {lambda}{sub L}=45 nm and 25 nm to assess the influence of nondipolar (magnetic) effects on high order harmonic generation spectra in molecules. It is found that even harmonics appear due to the magnetic field component direction perpendicular to the electric field polarization with intensities about two orders of magnitude less than the odd harmonics emitted along the electric field polarization. The even harmonics exhibit plateaus with cutoffs which exceed in intensity the odd harmonic plateaus and maximum energies predicted by semiclassical electron recollision models. Although the spectra are weak, the wavelength of the recollision electron in the maximum energy regions correspond to subatomic dimensions and the corresponding emitted photons have subnanometer wavelengths.

Dynamics of a nanowire superlattice in an ac electric field

With a one-band envelope function theory, we investigate the dynamics of a finite nanowire superlattice driven by an ac electric field by solving numerically the time-dependent Schroedinger equation. We find that for an ac electric field resonant with two energy levels located in two different minibands, the coherent dynamics in nanowire superlattices is much more complex as compared to the standard two-level description. Depending on the energy levels involved in the transitions, the coherent oscillations exhibit different patterns. A signature of barrier-well inversion phenomenon in nanowire superlattices is also obtained.

Complex integration steps in decomposition of quantum exponential evolution operators

We generalize previous high-order exponential split operator methods for solving time-dependent Schroedinger equations [A.D. Bandrauk, H. Shen, Chem. Phys. Lett. 176 (1991) 428] by introducing complex integration steps ( a + i b) with real positive part a. We show that this new procedure avoids real negative steps which occur generally in high-order split operator methods. New highly accurate splitting schemes are thus derived and the efficiency of these is demonstrated in the calculation of the eigenstates of the one-electron molecular ion H 2 + .

Stationary and dynamical aspects of two-magnon states in disordered ferromagnetic chains

We study the nature of collective two-spin excitations in disordered $S=1∕2$ ferromagnetic chains. Using a direct diagonalization scheme, we characterize the two-magnon eigenstates by computing their spacial extent, two-point correlation and the average distance between the excited spins within the allowed energy band. We found that, due to the effective excitation interaction imposed by the exclusion rule, the low-energy two-magnon states display strong spin-spin correlations as compared to the more localized high-energy states. We further solve the time-dependent Schroedinger equation to follow the time evolution of an initially localized two-magnon state. We show that the effective one-magnon wave packet develops power-law tails with distinct exponents for the left and right tails, with the distribution function of the spin-spin distance decaying as $P(d)\ensuremath{\propto}1∕d$. We show that the average distance between the two excited spins evolves in time diffusively, while the wave-packet dispersion evolves superdiffusively.

Fourier’s Law from Schrödinger Dynamics

We consider a class of one-dimensional chains of weakly coupled many level systems. We present a theory which predicts energy diffusion within these chains for almost all initial states, if some concrete conditions on their Hamiltonians are met. By numerically solving the time dependent Schrödinger equation, we verify this prediction. Close to equilibrium we analyze this behavior in terms of heat conduction and compute the respective coefficient directly from the theory.

Intensity dependence of theH2+ionization rates in Ti:sapphire laser fields above the Coulomb-explosion threshold

Ionization rates of the hydrogen molecular ion H{sub 2}{sup +} under linearly polarized pulse of intense laser fields are simulated by direct solution of the fixed-nuclei time-dependent Schroedinger equation for the Ti:sapphire laser lines {lambda}=790 and 800 nm at high intensities starting from just above the Coulomb explosion threshold (i.e., 6.0x10{sup 13}, 1.0x10{sup 14}, 3.2x10{sup 14}, and 1.4x10{sup 15} W cm{sup -2}). Results obtained in this research exhibit a high degree of complexity for the R-dependent enhanced ionization rates for the H{sub 2}{sup +} system in these intense laser fields. The R-dependent ionization peaks move towards small internuclear distances and their structure becomes simpler and smoother with the increase in the intensity of the laser pulse, i.e., with the decrease in the Keldysh parameter. Results obtained in this research are comparable to and even more reliable than the results of other theoretical calculations reported recently and successfully simulate the experimental ionization data.

Asymmetries in strong-field photoionization by few-cycle laser pulses: Kinetic-energy spectra and semiclassical explanation of the asymmetries of fast and slow electrons

Using numerical solutions of a time-dependent Schroedinger equation for a hydrogen atom in a linearly polarized few-cycle laser field, we calculate the left-right photoelectron kinetic-energy spectra measured by two opposing detectors placed along the laser polarization vector, with laser focus in the center. The fastest electrons show huge asymmetries strongly dependent on the laser carrier-envelope (CE) phase which confirms the recent theoretical results [D. B. Milosevic et al., Opt. Express 11, 1418 (2003)], obtained from a modified strong field approximation model which includes rescattering by the Coulomb potential. This asymmetry can also be explained by a simple semiclassical model in which the electron after tunneling through a potential barrier returns to the proton and is elastically backscattered in the presence of the laser field thus acquiring energy close 10U{sub p} where U{sub p} is the electron ponderomotive energy in the laser field. We also present a semiclassical interpretation of counterintuitive left-right asymmetries of slow electrons discussed in our previous work [Phys. Rev. A. 70, 013815 (2004)]. Our analysis shows that the Coulomb attraction from the proton must be included in the standard tunneling model in order to account for the CE phase dependent angular asymmetry seen in our previous numericalmore » calculations.« less

Low-order above-threshold ionization in intense few-cycle laser pulses

The low-energy end of the spectrum of photoelectrons detached from hydrogenic ions exposed to an intense low-frequency few-cycle pulse is calculated within the strong-field approximation (SFA). The effect on the detached photoelectron of the Coulomb field of the nucleus is taken into account quasiclassically. The results are compared with those of an ab initio solution of the time-dependent Schroedinger equation, for the case of an He{sup +} ion irradiated by a 400-nm pulse of 1x10{sup 16} W cm{sup -2} peak intensity. Many of the features of the ab initio spectra can be understood within the SFA.

Exact norm-conserving stochastic time-dependent Hartree–Fock

We derive an exact single-body decomposition of the time-dependent Schroedinger equation for N pairwise-interacting fermions. Each fermion obeys a stochastic time-dependent norm-preserving wave equation. As a first test of the method we calculate the low energy spectrum of Helium. An extension of the method to bosons is outlined.

Nonlinear photon processes in molecules at high intensities-route to XUV—attosecond pulse generation

Numerical solutions of the time-dependent Schroedinger equation, TDSE, for a non-Born Oppenheimer H2+ molecule exposed to the combination of a sub-femtosecond (attosecond) XUV pulse and a short (∼5fs) intense (I≥1013W/cm2) 800nm pulse allow for studying the nonlinear nonperturbative response of electrons and nuclei in molecules exposed to extreme laser conditions. New nonlinear phenomena, such as high order harmonic generation (HOHG), are considerably enhanced. This is a consequence of nonlinear photon processes and offers new routes for ‘attosecond’ pulse generation. Coupling the molecular TDSE to the photon Maxwell equations leads to the synthesis of even shorter attosecond and wavelength pulses, as new ultrashort sources for time-resolved XUV-Raman spectroscopy.

A quantum-classical treatment of non-adiabatic transitions

Abstract We have carried out molecular dynamics simulations of non-adiabatic processes with the help of a newly formulated potentially exact quantum-classical approach derived from a method proposed earlier [J. Chem. Phys. 118 (2003) 5302]. In this method, time-dependent Schroedinger equation is solved by representing Ψ on a moving Gauss–Hermite DVR grid, the motion of grid-centre being handled classically, but self consistently with the quantum evolution of the wavefunction. Electronic transitions are allowed anywhere in the configuration space among any number of coupled states. We have tested the method on three model problems proposed by J.C. Tully [J. Chem. Phys. 93 (1990) 1061]. These models are relevant to a wide range of gas-phase and condensed-phase phenomena occurring even at low energies. Excellent agreement of computed transition probabilities with corresponding quantum mechanical (DVR/FFT) results even in the deep quantum regime and its cost-efficiency (computational) are encouraging.

Electromagnetic dissociation ofB8and the astrophysicalSfactor forBe7(p,γ)8B

We present S factor data obtained from the Coulomb dissociation of 83 MeV/nucleon 8B, and analyze 7Be longitudinal momentum distributions measured at 44 and 81 MeV/nucleon using a potential model, first-order perturbation theory, and dynamical solution of the time-dependent Schroedinger equation. Comparing our results with independent continuum-discretized coupled channels calculations, we study the reaction model and beam energy dependence of the E2 contribution to the dissociation cross section. By fitting radiative capture and Coulomb breakup data taken below relative energies of 400 keV with potential models constrained by 7Li + n and 7Be + p elastic scattering data, we examine the mutual consistency of recent S17 measurements and obtain a recommended value for S17(0) of 18.6 +/- 0.4 (experimental) +/- 1.1 (extrapolation) eV b (1 sigma). This result is in good agreement with recent experimental determinations of the asymptotic normalization coefficient of the valence proton wave function in 8B.

Quantum shutter transient solutions and the delay time for theδpotential

The analytical solution to the time-dependent Schroedinger equation for tunneling using cutoff plane-wave initial conditions is in general given by the sum of two types of terms that exhibit a transient behavior. The time evolution of the probability density for the {delta} potential is compared with the free case to investigate in this case the role of these transient terms for the delay time. We find, by a dynamical calculation, that the delay time arises from the interference between these transient terms and we show that at very long times it goes into the phase delay time, given by the energy derivative of the phase of the transmission amplitude.

Correlation effects in two-photon single and double ionization of helium

We study the effects of two-photon absorption in the region of the double continuum of helium, with extreme ultraviolet (xuv) laser fields. The nonrelativistic time-dependent Schroedinger equation is solved for the two active electron systems, interacting with strong laser pulses. The dynamic calculations include electronic correlations. The final double continuum states are calculated by treating the electronic term 1/r{sub 12} within the zero- and first-order perturbation theory. We calculate the total single and double ionization probabilities and cross sections for photon energies of 45 and 57 eV, as well as the electron energy distributions in the double continuum. The effects of electronic correlations are discussed in the context of ultrashort pulses.

Born−Oppenheimer Time-Dependent Systems: Perturbative vs Nonperturbative Diabatization

In this article a time-dependent molecular system is considered. The theoretical treatment is characterized by the fact that here, for the first time, the adiabatic framework is assumed to contain singular nonadiabatic coupling terms or, in other words, conical intersections. The aim is to derive under these conditions, from first principles and in a rigorous manner, the time-dependent nuclear Schroedinger equations to be solved. We arrived at two different versions: (1) the “ordinary” version, which is essentially similar to the one encountered within the time-independent scheme and demands the fulfillment of the spatial Curl condition (Baer, M. Chem. Phys. Lett. 1975, 35, 112), and (2) a novel version, which introduces the time−space configuration and consequently demands the fulfillment of a “four”-component Curl condition, reminiscent of the one in special relativity (R. Baer, M. Baer, D. K. Hoffman, and D. J. Kouri, to be published). Both versions lead to the same number of Schroedinger equations, b...

Theory of phase characterization of harmonics through interference with the fundamental

We present calculations, based on the solution of the time-dependent Schroedinger equation for the hydrogen atom in a discrete basis demonstrating the possibility of extracting the phase profile of a harmonic through its interference with the fundamental. We examine two aspects of the interference: the generated harmonic radiation and the photoelectron energy spectra (if the harmonic is above the ionization threshold). We have focused on the third and the eleventh harmonics (of a 1.5 eV fundamental), where our results demonstrate a reconstruction of the phase profile with deviations of a few percent, on the basis of the modulation of the radiation. In the case of the 11th harmonic, we have also examined the PES modulation, and the phase profile calculated can be of the same quality, although more sensitive.

High-order variational perturbation theory for the free energy.

In this paper we introduce a generalization to the algebraic Bender-Wu recursion relation for the eigenvalues and the eigenfunctions of the anharmonic oscillator. We extend this well known formalism to the time-dependent quantum statistical Schrödinger equation, thus obtaining the imaginary-time evolution amplitude by solving a recursive set of ordinary differential equations. This approach enables us to evaluate global and local quantum statistical quantities of the anharmonic oscillator to much higher orders than by evaluating Feynman diagrams. We probe our perturbative results by deriving a perturbative expression for the free energy, which is then subject to variational perturbation theory as developed by Kleinert, yielding convergent results for the free energy for all values of the coupling strength.

Schr$ouml$dinger equation for joint bidirectional motion in time

The conventional, time-dependent Schroedinger equation describes only unidirectional time evolution of the state of a physical system, i.e., forward or, less commonly, backward. This paper proposes a generalized quantum dynamics for the description of joint, and interactive, forward and backward time evolution within a physical system. [...] Three applications are studied: (1) a formal theory of collisions in terms of perturbation theory; (2) a relativistically invariant quantum field theory for a system that kinematically comprises the direct sum of two quantized real scalar fields, such that one field evolves forward and the other backward in time, and such that there is dynamical coupling between the subfields; (3) an argument that in the latter field theory, the dynamics predicts that in a range of values of the coupling constants, the expectation value of the vacuum energy of the universe is forced to be zero to high accuracy. [...]

Collective Excitations of a One-Dimensional Fermi Gas Under Harmonic Confinement

We study the spectrum of collective excitations of a spin-polarized Fermi gas confined in a one-dimensional harmonic trap at zero temperature. In the collisionless regime we evaluate exactly the dynamic structure factor, while in the collisional regime we solve analytically the linearized equations of hydrodynamics in the Thomas-Fermi approximation. We also verify the validity of the Thomas-Fermi theory by solving numerically a time-dependent nonlinear Schroedinger equation with a fifth-order interaction term. We find that in both the collisionless and the collisional regime the excitation frequencies of the Fermi gas are multiples of the trap frequency, analogously to the case of the one-dimensional homogeneous Fermi fluid where the velocities of zero and first sound coincide. Due to boson-fermion dynamical mapping our results for the spectrum apply as well to a one-dimensional Bose gas with hard-core point-like interactions (“Tonks gas”).

Harmonic generation in ring-shaped molecules

We study numerically the interaction between an intense circularly polarized laser field and an electron moving in a potential which has a discrete cylindrical symmetry with respect to the laser pulse propagation direction. This setup serves as a simple model, e.g., for benzene and other aromatic compounds. From general symmetry considerations, within a Floquet approach, selection rules for the harmonic generation [O. Alon Phys. Rev. Lett. 80 3743 (1998)] have been derived recently. Instead, the results we present in this paper have been obtained solving the time-dependent Schroedinger equation ab initio for realistic pulse shapes. We find a rich structure which is not always dominated by the laser harmonics.