Relativistic calculation of the electron-momentum shift in tunneling ionization

Abstract

We describe a procedure for the solution of the time-dependent Dirac equation. The procedure is based on the relativistic generalization of the matrix iteration method. We use this procedure to study electron-momentum distribution along the laser-beam propagation direction for the process of the tunneling ionization of a hydrogen atom. We found, in agreement with the experimental observations [C. Phys. Rev. Lett. 106, 193002 (2011)], that relativistic effects lead to appreciable deviation of the distribution from the strict left-right symmetry present in the nonrelativistic case. The expectation value of the momentum along the laser-beam propagation direction grows linearly with intensity and follows closely the behavior of the expectation value of the kinetic energy divided by the speed of light. These features agree with the experimental results [C.Keldysh's pioneering work [1] and its subsequent develop ments [2,3] laid out a basis of our current understanding of the main features of the strong field ionization of atoms and molecules. According to this theory ionization phenomena can be separated into two broad classes: the multiphoton and tunneling ionization. Whether an ionization process can be characterized as a multiphoton or a tunneling one depends on the value of the Keldysh parameter y = Wy/2I/E (where w and E are the frequency and strength of the laser field, respectively, and I is the ionization potential of the target in atomic units). Values of y <£ 1 correspond to the tunneling regime, which interests us in the present work.Original Keldysh theory relied on a completely nonrela tivistic approach. A relativistic modification of this theory was proposed in [4]. It is clear that such relativistic modification is indispensable if our goal is the proper description of the ionization phenomena occurring for the field intensities of the order of several units of 1 x 1022 W /cm 2 which are currently available [5], or if we are interested in ionization driven by high-frequency electromagnetic fields. However, even for the infrared (IR) laser fields of much lower intensity, relativistic effects can manifest themselves. Even though the momentum of a photon is small, for the processes with participation of a large number of photons the total momentum delivered to the atom can be non-negligible. Smeenk et al. [6] report results of the study of the ionization process driven by the laser pulses with wavelengths of 800 and 1400 nm under experimental conditions where a large number of photons (30-50) is absorbed. The experiment demonstrated that the average electron-momentum component along the laser prop agation direction grows linearly with the laser field intensity and is nearly equal to the average photoelectron energy divided by the speed of light. Another example is provided by the recent study [7] of momentum sharing between the photoelectron and the parental ion in the processes of single-and multiphoton ionization. It was found [7] that momentum sharing is very different for the two processes, with intens...

Supercontinuum in ionization by relativistically intense and short laser pulses: Ionization without interference and its time analysis

“…Recently, due to the experimental results reported in Ref. [62] concerning the effects related to the radiation pressure exposed on the atomic or ionic systems by intense laser pulses, we observe an increased interest in theoretical investigations of ionization with the relativistic effects accounted for [47,[63][64][65][66][67]. We shall discuss below that the signatures related to the radiation pressure are present in the developed above theoretical approach [cf.…”

“…Recently, we observe a significant progress in solving numerically the relativistic Dirac equation (see, e.g., Refs. [43][44][45][46][47]). Even so, for very large intensities of laser pulses available today (of the order of 10 20 W/cm 2 and larger) and for high energies of photoelectrons (i.e., a few keV and higher) such solutions are not achievable.…”

“…(26) is not shifted by the vector (q 0 − p 0 )n, independently of the gauge choice. This term, among others, is responsible for the so-called radiation pressure [61] which has recently been discussed in the literature (see, e.g., [47,[62][63][64][65][66][67]). …”

Nondipole effects in atomic dynamic interference

“…Recently, several theoretical studies discussed the nondipole asymmetrical angular distributions [17,[35][36][37][38][39]. A related topic is the photon-momentum transfer and partition in the atomic and molecular ionization [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56].…”

Strong Field Theories beyond Dipole Approximations in Nonrelativistic Regimes

“…Thanks to the detection technics with high resolutions developed recently, Smeenk et al [9] and Ludwig et al [10] reported that photoelectrons acquire the momentum shift along the direction of laser propagation (termed as longitudinal momentum shift in the rest of the letter) during tunneling ionization, and this photon momentum transfer is also reported in the time-dependent Dirac equation simulation [11], time-dependent Schrödinger equation (TDSE) simulations [12,13], classical trajectory Monte Carlo calculations [14]. Some characters of the longitudinal momentum shift [15][16][17][18][19][20] were explored by calculations of strong field approximation (SFA).…”