Radiative reaction in ultra-intense laser–atom interaction

The influence of the radiation of an initially bound electron in an ultra-intense laser field on its dynamics and the consequent corrections to its spectral response are investigated within the relativistic classical Lorentz–Dirac approach. We find appreciable damping of the electron motion in the direction of polarization of the laser field, and a consequent reduction of the electron oscillation frequency. This yields a further boost in the electron velocity in the direction of propagation of the laser pulse, which can also be understood as due to the reabsorption of emitted radiation mainly with momentum in the direction of electron motion. The increase in the redshift of the electron motion is visible in the spectral response detected perpendicular to the laser propagation direction. There has been considerable interest in predictions of the behaviour of atoms in strong electromagnetic fields [1], given the availability of very intense laser pulses in a number of research laboratories [2]. Theoretical studies of free electrons in super-intense electromagnetic fields, employing both classical and quantum descriptions, go back to the advent of laser sources in the 1960s [3]. But it is only recently that the interaction of atoms with super-intense laser fields has been investigated experimentally using femtosecond laser sources. Electrons are accelerated to relativistic energies in the regime starting at 1018 W cm−2 for near-infrared radiation. The electrons oscillate with relativistic energies in such a laser pulse and efficient high-order harmonic generation has been observed [4]. The emitted radiation is relatively intense and the question arises as to how the presence of this field affects the motion of the electrons. In fact, the influence of radiative reaction within a rigorous treatment is a serious, unsolved problem in both classical and quantum electrodynamics [5, 6]. However, a consistent extension of ordinary Maxwell dynamics can be developed: Dirac showed, in his classical relativistic theory of the electron [7], that the emission of radiation leads to a mass renormalization and to forces which generalize the non-relativistic Abraham–Lorentz model. This Lorentz–Dirac model has been investigated in great detail [8] and, though it shows somewhat puzzling properties when small distances or time-scales are involved, has been accepted as describing radiative effects well. A breakdown of this classical model is expected to happen when intrinsically quantum and, in particular, proper QED effects, such as pair creation, are important. § Present address: Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 25, 6020 Innsbruck, Austria. E-mail address: [email protected] ‖ E-mail address: [email protected] 0953-4075/98/030075+09$19.50 c © 1998 IOP Publishing Ltd L75 L76 Letter to the Editor In this letter we investigate the dynamics of free and initially bound electrons at such high laser intensities that the amount of emitted radiation due to the enormous acceleration by the laser field is sufficiently strong to influence significantly the dynamics of the electron. This ‘back action’ of the radiation results, as expected, in the damping of electron motion; however, this is only the case in the polarization direction of the linearly polarized laser field. This decrease in the electron velocity leads to a reduction of the electron oscillation frequency in the polarization direction. In the propagation direction, the relativistically moving electron generally follows the laser pulse, but is slightly slower, i.e. slips behind and oscillates with a strongly redshifted frequency. As a consequence of this decrease of the electron oscillation frequency due to damping in the polarization direction, the electron has to fall behind the laser pulse more slowly, i.e. has to move faster in the direction of the laser pulse. Therefore damping in the polarization direction implies a stronger drift of the electron in the propagation direction of the laser pulse and vice versa. The damping in the polarization direction follows from the fact that the radiated field is essentially phase shifted by π with respect to the applied field. The counterintuitive extra boost can be understood as due to the reabsorption of emitted radiation, because the latter is predominantly emitted in the direction of the ultra-relativistic electron motion along the propagation direction of the laser pulse. Also we present results suggesting an increase in the redshift in the Monte Carlo averaged radiated spectrum for detection perpendicular to the direction of propagation of the laser pulse. The approach employed here is based on a classical Monte Carlo simulation of the relativistic laser–atom interaction [9], but now with additional modifications due to radiative reaction [5, 7, 11]. Quantum mechanical calculations appear to be impossible for present numerical facilities for highly relativistic intensities and relatively low frequencies of interest here [10]. However, classical dynamics in this regime should produce sensible results because ionization occurs at about the order of the atomic unit in time and thus many orders of magnitude faster than typical relaxation rates among quantum mechanical bound states and laser-induced superpositions of those. Free-electron dynamics is sufficiently well described by classical theory. In the weakly relativistic regime, the quantum ionization behaviour in more than one-dimensional models including the magnetic field component (Rathe et al in [10]) of the laser field has already confirmed the classical ionization behaviour (Keitel and Knight in [9]). The role of the nucleus for the laser intensities of interest here is merely to determine the initial condition for the electron position and momentum, as within one laser cycle the electron is driven many atomic units away from the nucleus. Thus, we will initially consider the dynamics of a free electron and for the evaluation of the spectral response we average over a microcanonical ensemble of initial electron positions and momenta which simulate the hydrogenic ground state [9]. We emphasize that the physical situation discussed here is quite different from processes investigated recently by Hartemann and Kerman [12]. They consider free electrons accelerated to highly relativistic velocities which interact with a strong laser field very similar to recent beam experiments at SLAC [13]. In contrast to the laser–atom interaction studied here, in that case radiative forces become large compared to the laser field forces due to the Lorentz boost. Before proceeding to a discussion of the radiation spectra, we recall that the spatial part of the covariant Lorentz–Dirac equation of motion for the 3-vectors of position r and momentum p including radiative reaction of a classical electron in a linearly polarized laser field is given in atomic units by [11]