Asymmetry interactions of electrons and positrons in strong pulsed laser fields

The interaction of two electrons and two positrons in the field of two strong pulsed laser waves outside the framework of the dipole approximation (with an accuracy of relativistic corrections ) has been studied theoretically. The possibility of significant asymmetry in the interactions of particles and antiparticles in the external laser fields is shown. This asymmetry of interactions essentially depends on the magnitude and sign of the phase shifts of the waves relative to each other, and also on the intensities of the waves. The ranges of phase shifts and field intensities in which the interaction of electrons and positrons have qualitatively different character from anomalous repulsion to attraction are determined.


Introduction
There are many works devoted to research of interaction of electrons in the presence of an electromagnetic field (see, the works [1][2][3][4][5][6][7][8][9]). The possibility of electron attraction in the presence of a plane electromagnetic wave was firstly assumed by Oleinik [4]. However, the theoretical proof of the attraction possibility was given by Kazantsev and Sokolov for interaction of classical relativistic electrons in the field of a plane wave [5]. We also note the paper [6]. It is very important to point out, that attraction of classical electrons in the field of a plane monochromatic electromagnetic wave is possible only for particles with relativistic energies. In the authors works (see, review [3], articles [7][8][9]) the possibility of attraction of nonrelativistic electrons (identically charged ions) in the pulsed laser field was shown. Thus, in the review [3] the following processes were discussed: interaction of electrons (light ions) in the pulsed field of a single laser wave; interaction of nonrelativistic electrons in the pulsed field of two counterpropagating laser waves moving perpendicularly to the initial direction of electrons motion; the interaction of nonrelativistic light ions moving almost parallel to each other in the propagation direction of the pulsed field of two counter-propagating laser waves moving in parallel direction to ions; interaction of two nonrelativistic heavy nuclei (uranium 235), moving towards each other perpendicularly to the propagation direction of two counter-propagating laser waves. The effective force of interaction of two hydrogen atoms (after their ionization) in the pulsed field of two counter-propagating laser waves was considered in [7]. Influence of pulsed field of two co-propagating laser waves on the effective force of interaction of two electrons and two identically charged heavy nuclei was studied in [8]. The main attention is focused on the study of the influence of phase shifts of the pulse peak of the second wave relatively to the first on the effective force of particles interaction. The phase shift allows to increase duration of electron's confinement at a certain averaged effective distance by 1,5 time in comparison with the case of one and two counter-propagating pulsed laser waves. Interaction of two classical nonrelativistic electrons in the strong pulsed laser field of two light mutually perpendicular waves, when the maxima laser pulses coincide, was studied in [9]. It is shown that the effective force of electron interaction becoming the attraction force or anomalous repulsion force after approach of electrons to the minimum distance. In this paper, in contrast to the previous papers, the effective interaction of two electrons and two positrons in the field of mutually perpendicular strong laser waves is studied with allowance for their phase shifts of the pulses. The ranges of intensities and phase shifts of waves are determined for which the effective interaction of two particles (electrons) and two antiparticles (positrons) is essentially asymmetric.

Equations of particles interaction in pulsed field of two laser waves
We study interaction of two nonrelativistic electrons (positrons) moving towards each other along the axis x in a field of two linearly polarized pulsed electromagnetic waves. Waves propagate perpendicularly to each other. The first wave propagates along the axis z , the second wave propagates along the axis x (see figure 1). We assume the strength of electric and magnetic field in following form: (5) ( ) (6) where ij ϕ are the phases of corresponding wave ( 1, 2 i = ) and corresponding particle ( 1, 2 j = ); 0i E and 0i H are the strength of electric and magnetic field in the pulse peak, respectively; i δτ are the phase shifts of pulse peaks of the first and second waves; i t and i ω are the pulse durations and frequency of the first and the second wave; x e , y e , z e are unit vectors directed along the x , y and z axes.
It is well-known fact that in a frame of the dipole approximation ( 0 k = ) and without taking into account terms of the order / << 1 v c the particle interaction with plane-wave field does not affect on the particle relative motion in center-of-mass system. Thereby, we consider particle motion in the laser field beyond to the dipole approximation and an accuracy of quantities of order / << 1 v c ( v is the relative velocity).
Newton equations for motion of two identically charged particles with the mass m and charge e ( 1 2 e e e = = ) in the pulsed field of two mutually perpendicular laser waves (1) -(6) are determined by following expressions: (8) where ( ) Hereafter, wave frequencies are same: Subsequent consideration we carry out in the center-of-mass system: There are following equations for particle relative motion in the center-of-mass system: (10) 1   1  1  2sin  sin  cos  cos  ,  2   1  2sin  sin  cos  ,  2   1 cos , Note that equations for relative motion (10) - (12) are written beyond to the dipole approximation ( ) 0 k ≠ and an accuracy of term of order / << 1 v c . Note, small influence of external strong pulsed laser field on radius-vector of the center-of-mass motion was shown in the work [9]. Thereby, study of relative motion of electrons (positrons) makes sense to be done only. Equations (10) - (11) have to be written in the dimensionless form: 1   sin sin  cos  cos ,  2  2   sin sin  cos  ,  2  2 cos , 2 where, Here, ( )  ξ is the radius-vector of the relative distance between electrons (positrons) in units of the wavelength, the parameters 1,2 η are numerically equal to ratio of an oscillation velocity of a particle in the peak of pulse of the first or second wave to the velocity of light c (hereinafter, we consider parameters 1,2 η as oscillation velocities); the parameter β is numerically equal to ratio of the energy of Coulomb interaction of particles with reduced mass µ at the wavelength to the particle rest energy. The pulse duration exceeds considerably the period of wave rapid oscillation ( ) for a majority of modern pulsed lasers: Consequently, the relative distance between particles should be averaged over the period of wave rapid oscillation: We emphasize that in the expression for the effective force F (13) the first summand corresponds to the Coulomb repulsion of like-charged particles, and the second summand corresponds to the interaction of charged particles with the external laser field. It is important to note that for electrons and positrons this interaction has a different sign (see the sign before the second summand in expression (13)). Because of this, in the external laser field, the symmetry disappears in the interaction of particles of the same charge, but of different sign. We note that expressions (13), (14) consider interaction with the Coulomb field and the pulsed-wave field strictly, and don't have the analytical solution. For subsequent analysis we will study all equations numerically. Electrons (positrons) initial relative coordinates and velocities are the following: The interaction time is

Anomalous repulsion
Here we consider the case when the oscillation velocity of the first wave is greater than the initial velocity of particles (     Thus, for identical initial conditions and wave parameters, if the oscillation velocity of second wave is much larger than the initial velocity of particles and phase shifts of the waves have different signs, the effective interaction of two electrons and two positrons is appeared in their anomalous repulsion. In this case, the electrons can scatter over distances exceeding the corresponding values without a laser field up to two orders of magnitude, and positrons -up to three orders of magnitude. This indicates a significant asymmetry in the effective interaction of particles and antiparticles in strong pulsed laser fields. This asymmetry is associated with a positive sign of positrons charge, which leads to an increase in the effective repulsive force in comparison with electrons (see equations (13)).

The effective slowing-down of particles
In contrast to the previous section, we consider the case, when the oscillation velocity 1 η has to be close

Conclusion
The study of the interaction of two electrons and two positrons in strong laser fields has shown a significant asymmetry in the effective interaction of particles and antiparticles. This asymmetry essentially depends on the magnitude and sign of phase shifts of waves relative to each other and intensities of both waves. 1. If the phase shifts of the waves have different signs (for the first wave the phase shift is negative, and for the second wave -positive), electrons and positrons anomalous strongly repulse. Moreover, at the maximum intensity of the second wave, it is possible to select phase shifts of waves when the anomalous repulsion of particles becomes maximal, exceeding the averaged scatter distances of particles without a field: for electrons by two orders of magnitude, and for positrons by three orders of magnitude (see figure 2 and figure 3). 2. If the phase shifts of both waves have the same sign (for the first and second waves the phase shift is positive), then the interaction of the particles varies significantly. If the velocities of the wave oscillations do not differ much from each other (