Emmanuel Lorin

Atoms and molecules in intense laser fields: gauge invariance of theory and models

Gauge invariance was discovered in the development of classical electromagnetism and was required when the latter was formulated in terms of the scalar and vector potentials. It is now considered to be a fundamental principle of nature, stating that different forms of these potentials yield the same physical description: they describe the same electromagnetic field as long as they are related to each other by gauge transformations. Gauge invariance can also be included into the quantum description of matter interacting with an electromagnetic field by assuming that the wavefunction transforms under a given local unitary transformation. The result of this procedure is a quantum theory describing the coupling of electrons, nuclei and photons. Therefore, it is a very important concept: it is used in almost every field of physics and it has been generalized to describe electroweak and strong interactions in the standard model of particles. A review of quantum mechanical gauge invariance and general unitary transformations is presented for atoms and molecules in interaction with intense short laser pulses, spanning the perturbative to highly nonlinear non-perturbative interaction regimes. Various unitary transformations for a single spinless particle time-dependent Schrodinger equation (TDSE) are shown to correspond to different time-dependent Hamiltonians and wavefunctions. Accuracy of approximation methods involved in solutions of TDSEs such as perturbation theory and popular numerical methods depend on gauge or representation choices which can be more convenient due to faster convergence criteria. We focus on three main representations: length and velocity gauges, in addition to the acceleration form which is not a gauge, to describe perturbative and non-perturbative radiative interactions. Numerical schemes for solving TDSEs in different representations are also discussed. A final brief discussion of these issues for the relativistic time-dependent Dirac equation for future super-intense laser field problems is presented.

Saturation of the nonlinear refractive index in atomic gases

Motivated by the ongoing controversy on the origin of the nonlinear index saturation and subsequent intensity clamping in femtosecond filaments, we study the atomic nonlinear polarization induced by a high-intensity and ultrashort laser pulse in hydrogen by numerically solving the time dependent Schrodinger equation. Special emphasis is given to the efficient modeling of the nonlinear polarization at central laser frequency corresponding to 800 nm wavelength. Here, the recently proposed model of the Higher-Order Kerr Effect (HOKE) and two versions of the Standard model for femtosecond filamentation, including either a multi-photon or tunnel ionization rate, are compared. We find that around the clamping intensity the instantaneous HOKE model does not reproduce the temporal structure of the nonlinear response obtained from the quantum mechanical results. In contrast, the non-instantaneous charge contributions included in the Standard models ensure a reasonable quantitative agreement. Therefore, the physical origin for the observed saturation of the overall electron response is confirmed to mainly result from contributions of free or nearly free electrons.

Numerical Solution of the Time-Dependent Dirac Equation in Coordinate Space without Fermion-Doubling

Abstract The validation and parallel implementation of a numerical method for the solution of the time-dependent Dirac equation is presented. This numerical method is based on a split operator scheme where the space–time dependence is computed in coordinate space using the method of characteristics. Thus, most of the steps in the splitting are calculated exactly, making for a very efficient and unconditionally stable method. We show that it is free from spurious solutions related to the fermion-doubling problem and that it can be parallelized very efficiently. We consider a few simple physical systems such as the time evolution of Gaussian wave packets and the Klein paradox. The numerical results obtained are compared to analytical formulas for the validation of the method.

Attosecond pulse generation from aligned molecules?dynamics and propagation in H2+

The dynamics and propagation effects in attosecond (asec) pulse generation from high-order harmonic generation (HHG) of aligned one- dimensional (1D) H + molecules are investigated from numerical solutions of fully coupled Maxwell and time-dependent Schrodinger equations (Maxwell- TDSEs), in the highly nonlinear nonperturbative regime of laser-molecule interaction. Density, laser-phase and propagation length effects are studied on the total electric field and nonlinear polarization from the Maxwell-TDSE for intense few cycle (800 nm) laser pulses interacting with a 1D H + gas. We show how single and double asec pulses can be generated and propagated as a function of the phase of individual harmonics created by ultrashort intense laser pulses in aligned H + molecules. We find furthermore extension of maximum HHG plateaux with increasing gas pressure.

A split-step numerical method for the time-dependent Dirac equation in 3-D axisymmetric geometry

Abstract A numerical method is developed to solve the time-dependent Dirac equation in cylindrical coordinates for 3-D axisymmetric systems. The time evolution is treated by a splitting scheme in coordinate space using alternate direction iteration, while the wave function is discretized spatially on a uniform grid. The longitudinal coordinate evolution is performed exactly by the method of characteristics while the radial coordinates evolution uses Poissons integral solution, which allows to implement the radial symmetry of the wave function. The latter is evaluated on a time staggered mesh by using Hermite polynomial interpolation and by performing the integration analytically. The cylindrical coordinate singularity problem at r = 0 is circumvented by this method as the integral is well-defined at the origin. The resulting scheme is reminiscent of non-standard finite differences. In the last step of the splitting, the remaining equation has a solution in terms of a time-ordered exponential, which is approximated to a higher order than the time evolution scheme. We study the time evolution of Gaussian wave packets, and we evaluate the eigenstates of hydrogen-like systems by using a spectral method. We compare the numerical results to analytical solutions to validate the method. In addition, we present three-dimensional simulations of relativistic laser–matter interactions, using the Dirac equation.

Resonantly enhanced pair production in a simple diatomic model

A new mechanism for the production of electron-positron pairs from the interaction of a laser field and a fully ionized diatomic molecule in the tunneling regime is presented. When the laser field is turned off, the Dirac operator has resonances in both the positive and the negative energy continua while bound states are in the mass gap. When this system is immersed in a strong laser field, the resonances move in the complex energy plane: the negative energy resonances are pushed to higher energies while the bound states are Stark shifted [F. Fillion-Gourdeau et al., J. Phys. A 45, 215304 (2012)]. It is argued here that there is a pair production enhancement at the crossing of resonances by looking at a simple one-dimensional model: the nuclei are modeled simply by Dirac delta potential wells while the laser field is assumed to be static and of finite spatial extent. The average rate for the number of electron-positron pairs produced is evaluated and the results are compared to the one and zero nucleus cases. It is shown that positrons are produced by the resonantly enhanced pair production mechanism, which is analogous to the resonantly enhanced ionization of molecular physics. This phenomenon could be used to increase the number of pairs produced at low field strength, allowing the study of the Dirac vacuum.

Numerical solution of the time-independent Dirac equation for diatomic molecules: B-splines without spurious states

Two numerical methods are used to evaluate the relativistic spectrum of the two-centre Coulomb problem (for the H + and Th 179+ diatomic molecules) in the fixed nuclei approximation by solving the single particle time-independent Dirac equation. The first one is based on a min-max principle and uses a two-spinor formulation as a starting point. The second one is the Rayleigh-Ritz variational method combined with kinematically balanced basis functions. Both methods use a B-spline basis function expansion. We show that accurate results can be obtained with both methods and that no spurious states appear in the discretization process.

Convection Systems with Stiff Source Terms

The aim of this paper is the numerical treatment of some convection systems with stiff relaxation source-terms. We will first define the notion of stiffness for such systems and will select some prototypical and physical problems. We will introduce a new numerical method in order to solve accurately this type of systems. Numerical comparisons will be performed on the evoked problems.

On the numerical approximation of one-dimensional nonconservative hyperbolic systems

Abstract Attempts to define weak solutions to nonconservative hyperbolic systems have lead to the development of several approaches, most notably the path-based theory of Dal Maso, LeFloch, and Murat (DLM) and the vanishing viscosity solutions described by Bianchini and Bressan. While these theories enable us to define weak solutions to nonconservative hyperbolic systems, difficulties arise when numerically approximating these systems. Specifically, in the neighborhood of a discontinuity, the numerical solutions tend to not converge to the theoretically specified weak solution of the system. This convergence error is easily seen in the numerical approximation of Riemann problems, in which the error appears and propagates at the formation of discontinuity waves. In this paper we investigate several methods to numerically approximate nonconservative hyperbolic systems, we discuss why these convergence errors arise, and by using recent results established by Alouges and Merlet we give an approximate description of what weak solutions these numerical solutions converge to. We then propose several strategies for the design of numerical schemes which reduce these convergence errors.

Frozen Gaussian approximation based domain decomposition methods for the linear Schrödinger equation beyond the semi-classical regime

The paper is devoted to develop efficient domain decomposition methods for the linear Schrodinger equation beyond the semiclassical regime, which does not carry a small enough rescaled Planck constant for asymptotic methods (e.g. geometric optics) to produce a good accuracy, but which is too computationally expensive if direct methods (e.g. finite difference) are applied. This belongs to the category of computing middle-frequency wave propagation, where neither asymptotic nor direct methods can be directly used with both efficiency and accuracy. Motivated by recent works of the authors on absorbing boundary conditions (Antoine et al. (2014) 13 and Yang and Zhang (2014) 43), we introduce Semiclassical Schwarz Waveform Relaxation methods (SSWR), which are seamless integrations of semiclassical approximation to Schwarz Waveform Relaxation methods. Two versions are proposed respectively based on Herman-Kluk propagation and geometric optics, and we prove the convergence and provide numerical evidence of efficiency and accuracy of these methods.