Time-dependent Dirac Equation Research Articles

The time-dependent Dirac equation for atoms exposed to short intense electromagnetic pulses

The time dependent Dirac equation for hydrogen-like systems exposed to intense laser pulses, is solved numerically and compared to the corresponding prediction from the non-relativistic Schrödinger equation. It is found that virtual electron-positron pairs are instrumental for a correct description of effects beyond the dipole approximation. Relativistic effects in the ionization dynamics of highly charged systems are studied

Creation of multiple electron-positron pairs in arbitrary fields

We examine the spontaneous breakdown of the matter vacuum triggered by an external force of arbitrary strength and spatial and temporal variations. We derive a nonperturbative framework that permits the computation of the complete time evolution of various multiple electron-positron pair probabilities. These time-dependent probabilities can be computed from a generating function as well as from solutions to a set of ratelike equations with coupling constants determined by the single-particle solutions to the time-dependent Dirac equation. This approach might be of relevance to the planned experiments to observe for the first time the laser-induced breakdown process of the vacuum.

Solution of the Dirac equation for hydrogenlike systems exposed to intense electromagnetic pulses

The time-dependent Dirac equation is solved numerically and compared to the corresponding prediction of the nonrelativistic Schr\"odinger equation for hydrogenlike systems exposed to intense laser pulses. It is found that for a correct description of effects beyond the dipole approximation, virtual electron-positron pairs can be very important. Relativistic effects in the ionization dynamics of highly charged systems are studied.

Calculation of electron-positron production in supercritical uranium-uranium collisions near the Coulomb barrier

The time-dependent Dirac equation was solved for zero-impact-parameter bare U-U collisions in the monopole approximation using a mapped Fourier grid matrix representation. A total of 2048 states including bound, as well as positive- and negative-energy states for an $N=1024$ spatial grid were propagated to generate occupation amplitudes as a function of internuclear separation. From these amplitudes spectra were calculated for total inclusive positron and electron production, and also the correlated spectra for $({e}^{+},{e}^{\ensuremath{-}})$ pair production. These were analyzed as a function of nuclear sticking time in order to establish signatures of spontaneous pair creation, i.e., QED vacuum decay. Subcritical Fr-Fr and highly supercritical Db-Db collisions both at the Coulomb barrier were also studied and contrasted with the U-U results.

New calculations for heavy-ion collisions with super-critical fields

An adaptation of the mapped Fourier grid method for the Dirac equation is applied to the problem of positron production in quasi-molecular collisions between fully stripped heavy ions with super-critical Coulomb fields. The resonance parameters for the 1Sσ vacancy which dives into the filled negative-energy continuum are calculated on the basis of the two similar analytic continuation methods of smooth exterior scaling and complex absorbing potential with Padé extrapolation. Previous results in monopole approximation for the U-Cf system as a function of internuclear separation are extended to a coupled-channel approximation to investigate resonance broadening. Calculations for dynamical positron production on the basis of the time-dependent Dirac equation for U-U collisions with artificially delayed nuclear trajectories are also presented, and explanations for the observed positron spectrum are provided.

Complete solutions for a class of time-dependent Dirac equations

Using a unitary transformation, the Dirac equation with a time-dependent linear potential is transformed into a Schrödinger equation for a particle moving in an effective external magnetic field, which can be solved by the Lewis–Riesenfeld invariant theory. The wavefunction is written in a complete form. It is also shown that in the non-relativistic limit, matrix elements of the velocity operator go to the classical velocity when there is an arbitrary external potential and a magnetic field.

Exact solutions of time-dependent Dirac equations and the quantum–classical correspondence

Exact solutions to the Dirac equations with a time-dependent mass and a static magnetic field or a time-dependent linear potential are given. Matrix elements of the coordinate, momentum and velocity operator are calculated. In the large quantum number limit, these matrix elements give the classical solution.

Calculations for electron transitions on a three-dimensional lattice in relativistic heavy-ion collisions

In order to calculate the cross section for electron-positron pair production with capture of the electron into the K shell of the target in relativistic scattering of heavy nuclei, we apply the semiclassical method and solve the time-dependent Dirac equation in time-reversal symmetry, starting with a 1s{sub 1/2} state at the target. The solution is carried out numerically on a three-dimensional lattice in coordinate space. Cross sections for excitation, ionization, electron transfer, and pair production with capture are obtained for collisions of U{sup 92+}({gamma}=1.5), Au{sup 79+}({gamma}=2), and U{sup 92+}({gamma}=10 000) on U{sup 91+}(1s{sub 1/2}) and U{sup 92+}, where {gamma} is the Lorentz factor, and compared with experimental data and other theoretical results.

Electric-field-induced relativistic Larmor-frequency reduction

Using the numerical solution to the time-dependent Dirac equation we show that the effect of relativity on the usual Larmor period for an electron in a magnetic field can be enhanced drastically if a suitably scaled and aligned static electric field is added to the interaction. This electric field does not change the electron's speed but leads to an elliptical spin precession due to relativity. This spin precession is accompanied by a position-dependent spin distribution.

Dynamics of ionization mechanisms in relativistic collisions involving heavy and highly-charged ions

The dynamics of mechanisms associated with the ionization of inner-shell electrons in relativistic collisions involving heavy and highly-charged ions is investigated within a nonperturbative approach formulated explicitly in the time domain. The theoretical treatment is based on the exact numerical solution of the time dependent Dirac equation for two Coulomb centers on a lattice in momentum space. We present results for ionization in encounters between 100 MeV/u Au79+ projectile ions impinging on a hydrogen-like uranium target. By directly visualizing the collision dynamics we identify a new ionization mechanism in which electrons are emitted from the internuclear region preferentially in the transverse direction with respect to the projectile trajectory. A striking characteristic of this ionization mechanism is that the velocity of the electron is higher than the projectile velocity.

Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential

We consider the Dirac operator with a long-range potential V(x). Scalar, pseudo-scalar and vector components of V(x) may have arbitrary power-like decay at infinity. We introduce wave operators with time-independent modifiers. These modifiers are pseudo-differential operators whose symbols are, roughly speaking, constructed in terms of approximate eigenfunctions of the stationary problem. We derive and solve eikonal and transport equations for the corresponding phase and amplitude functions. From an analytical point of view, our proof of the existence and completeness of the wave operators relies on the limiting absorption principle and radiation estimates established in the paper. This allows us to fit the long-range scattering theory for the Dirac operator into the framework of smooth perturbations. Finally, we find the asymptotics for large times t of solutions u(x, t) of the time-dependent Dirac equation.

Critique of the Wigner tunneling speed and a proposed alternative

In the context of superluminal propagation of wave packets through potential barriers, the tunneling speed is usually characterized by the Wigner velocity. We propose an alternative speed that takes into account the interference between the incoming and the reflected waves and leads to a better estimation of arrival time for a wave packet entering the tunneling region. This arrival time is derived by an extrapolation from inside the barrier. The analytical theory is based on the stationary phase approximation whose validity is justified by a comparison with the numerical solution of the time-dependent Dirac equation.

Dynamical signature in spatial spindistributions of relativistic electrons

The time-dependent Dirac equation has been solved for anelectron in external static electric,static magnetic and time-dependent electromagnetic fields. We giveseveral examples of howthe dynamics of the spin variables can couple to the relativisticorbital motion and investigate thetime evolution of the spatial spin distribution as a function ofthe position for a relativisticquantum state.

Effects of relativity on the time-resolved tunneling of electron wave packets

We solve numerically the time-dependent Dirac equation for a quantum wave packet tunneling through a potential barrier. We analyze the spatial probability distribution of the transmitted wave packet in the context of the possibility of effectively superluminal peak and front velocities of the electron during tunneling. Both the Dirac and Schr\"odinger theories predict superluminal tunneling speeds. However, in contrast to the Dirac theory the Schr\"odinger equation allows a possible violation of causality. Based on an analysis of the tunneling process in full temporal and spatial resolution, we introduce an instantaneous tunneling speed that can be computed inside the potential barrier.

Dirac theory of ring-shaped electron distributions in atoms

The time-dependent Dirac equation is solved numerically on a space-time grid for an atom in a strong static magnetic field and a laser field. The resonantly induced relativistic motion of the atomic electron leads to a ringlike spatial probability density similar to the features that have been recently predicted [Wagner, Su, and Grobe, Phys. Rev. Lett. 84, 3282 (2000)] based on a phase-space method. We further demonstrate that spin-orbit coupling for a fast-moving electron in such an atom becomes significant and the time dependence of the spin can dephase even if initially aligned parallel to the direction of the static magnetic field.

Ion-ion collision dynamics in two-dimensional Cartesian space

Ion-ion collision dynamics are studied by direct solution of the time-dependent Schroedinger and Dirac equations on a two-dimensional Cartesian lattice. The time-dependent Dirac equation is solved in both its two- and four-component forms using difference offsetting on the lattice to mitigate the fermion doubling pathology. For Pd{sup 46+} on Pd{sup 45+}, which energetically corresponds to U{sup 92+} on U{sup 91+} in a three-dimensional space, total inelastic probabilities are calculated at a fixed ion velocity and several impact parameters. In this intermediate energy range, the total inelastic probabilities from the Schroedinger and two- and four-component Dirac equations are in reasonable agreement, although the detailed collision dynamics reveal interesting differences. (c) 2000 The American Physical Society.

Free electron-positron pair production in collisions of Au (10.8 GeV/u) with Cu , Ag and Au

Total and differential cross-sections for the production of free electron-positron pairs are calculated for the reactions Au79+ (10.8 GeV/u) on Cu29+, Ag47+ and Au79+. The methods used are the semiclassical approach and the solution of the time-dependent Dirac equation by a coupled channel method with free wavepakets describing the created fermions. The obtained total cross-sections are in good agreement with the experimental data. The differential cross-sections give informations about the correlation between the electron and positron.

Relativistic theory of atom-laser interactions at very high laser intensities

The high-frequency approximation of Kristic and Mittleman is considered in detail as a basis for the relativistic theory of atom-laser interactions. The properties of the 3D potentials are discussed. Within a one-dimensional model similar to that employed by Kylstra, Ermolaev, and Joachain in ab initio calculations on the time-dependent Dirac equation, the electron mass-shift due to dressing by a superstrong laser field is investigated. In the full domain of the laser parameters, the frequency ω and the peak field strength ℰ0, the 1D bound states exhibit remarkable features. The numerical calculations show the existence of a very wide intermediate range of the field strengths where, in the zeroth order of the high-frequency approximation, the binding is stabilized by the field.

Exact Dirac-equation calculation of ionization induced by ultrarelativistic heavy ions

The time-dependent Dirac equation can be solved exactly for ionization induced by ultrarelativistic heavy-ion collisions. Ionization calculations are carried out in such a framework for a number of representative ion-ion pairs. For each ion-ion pair, the computed cross section consists of two terms: a constant energy independent term and a term whose coefficient is $\mathrm{ln}\ensuremath{\gamma}.$ Scaled values of both terms are found to decrease with increasing Z of the nucleus that is ionized.

Classical versus quantum dynamics for a driven relativistic oscillator

We compare the time evolution of the quantum-mechanical spatial probability density obtained by solving the time-dependent Dirac equation with its classical counterpart obtained from the relativistic Liouville equation for the phase-space density in a regime in which the dynamics is essentially relativistic. For a resonantly driven one-dimensional harmonic oscillator, the simplest nontrivial model system to perform this comparison, we find that, despite the nonlinearity induced by relativity, the classical ensemble description matches the quantum evolution remarkably well.