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Theory Division

Theoretical Quantum Dynamics and Quantum Electrodynamics


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Relativistic and Ultrashort Quantum Dynamics

Since the invention of laser light amplification with chirped pulses, extremely short and strong laser fields have been generated with ever-increasing intensities. While current lasers reach up to 1022 W/cm2, the European Extreme-light-infrastructure (ELI) project aims at much higher intensities. A free-electron laser for a strong XUV radiation (FLASH facility) has been developed at DESY (Hamburg) where now a new x-ray free-electron laser (XFEL) is under construction. Another XFEL \96 LCLS (Linac Coherent Light Source) operates in SLAC (Stanford, USA). Moreover, experimentalists are able to produce well-controlled ultrashort and tailored laser pulses which offer efficient control over the bound electron dynamics, thus paving the way for attosecond spectroscopy. Thus, there is a bright outlook for the investigation of strong laser radiation interacting with matter, strong field physics. Among recent achievements are the production of few cycle laser pulses, attosecond pulses of XUV radiation, production of coherent ultraviolet radiation via self-amplified spontaneous emission in FEL as well as monoenergetic GeV electrons and MeV ions via laser-plasma interaction.

We investigate the interaction of strong laser radiation with matter. The systems under consideration range from free electrons, single atoms/ions, few-atom ensembles, thin matter layers and plasmas up to vacuum with quantum fluctuations. In the center of interest are the relativistic regimes of interaction. In particular, our attention is focused on nonlinear ionization dynamics in strong fields and on radiative effects in strong fields, such as high-order harmonic generation (HHG) via free electrons, atomic systems, and plasmas.

As regards atomic systems, we have investigated, in particular, the role of relativistic effects during the under-the-barrier dynamics, and its implication for the tunneling time and spin effects. An interesting finding was that even during the short time of tunneling, relativistic effects arise imprinting their signatures on the electron dynamics. Thus, the relativistic effects allow to map the, so-called, tunneling time (Wigner time) into the shift in space distribution of the ionized electron wave packet and the, so-called, tunneling formation time (Keldysh time) into the shift of the electron momentum distribution.

We have investigated the ways for extension of the ionization-recollision dynamics to the relativistic domain as a pathway to radiation sources in a hard x-ray domain via HHG. To this purpose we consider different setups for the suppression of the magnetically induced drift in the relativistic regimes of HHG .

The role of Coulomb field effects in strong field physics is also investigated. We have explained the origin of the low-energy structure in the photoelectron energy distribution at above threshold-ionization (ATI) in mid-infrared laser fields. It appears that the low-energy structure arises due to Coulomb focusing because of multiple forward scattering of the ionized electron by the parent ion. A surprising fact was that the high-order scattering events have a nonperturbative comparable contribution to the total Coulomb focusing and persist up to high ellipticity values of the driving laser field.

During the interaction of a strong laser radiation with electrons, the radiation reaction can play an important role in the relativistic regime. Unlike in the nonrelativistic case, a situation can occur in the ultrarelativistic regime in which the radiation reaction force becomes comparable with the Lorentz force in the laboratory frame while being much smaller in the instantaneous rest frame of the electron. This is the so-called radiation dominated regime in which the electron dynamics and its radiation are supposed to be significantly modified due to the radiation reaction. We have investigated the signatures of radiation reaction in the classical and quantum regimes of radiation in a strong laser field. Moreover, we have shown that the radiation reaction can significantly enhance the Raman scattering of a strong laser radiation in plasma due to the induced nonlinear mixing of the Stocks and anti-Stocks sidebands.

For the high energy domain of laser physics, we have proposed the concept of a laser-driven high-energy electron-positron collider which employs a bunch of positronium atoms. Ultraintense laser pulses are applied to combine in one single-femtosecond stage the electron and positron acceleration and their microscopic coherent collision in the GeV regime. We have shown that such coherent collisions yield a largely enhanced luminosity compared to conventional incoherent colliders, so that particle physics reactions with high-power lasers become possible. As an example, the feasibility of muon pair production from a positronium gas in a strong laser field has been investigated.

Super-strong laser fields offer unique possibilities for the investigation of the quantum vacuum. Different effects of vacuum QED nonlinearities induced by strong laser fields have been considered. The experimental observation of elastic photon-photon scattering is quite feasible for experimental observation as our analysis shows. Using a setup of multiple crossed superstrong laser beams the photon-photon scattering rate can be significantly enhanced due to Bragg interference. We have investigated the role of diffraction on the birefringence effects arising from the nonlinear interaction of an x-ray probe with a tightly focused standing laser wave. It turned out that the diffraction is unavoidable in a conventional experimental setup decreasing the birefringence effect by an order of magnitude. Nevertheless, the latter will, in principle, be technically measurable in the near future.

Research directions:

Atomic quantum dynamics in the relativistic regime



Relativistic features of the under-the-barrier dynamics in laser-induced ionization

We have investigated the role of relativistic effects during the under-the-barrier dynamics in laser-induced ionization. An interesting finding was that even during this short time before the release of the electron from the bound state, relativistic effects arise imprinting their signature on the electron dynamics and characteristics.\A0 An intuitive picture for the relativistic tunneling regime is developed demonstrating that the tunneling picture applies also in the relativistic case by introducing position dependent energy levels. The quantum dynamics in the classically forbidden region features two time scales, the typical time that characterizes the probability density\92s decay of the ionizing electron under the barrier (Keldysh time) and the time interval which the electron wave packet spends inside the barrier (Eisenbud-Wigner-Smith tunneling time). In the relativistic regime, an electron momentum shift as well as a spatial shift of the ionized electron wave packet along the laser propagation direction arise during the under-the-barrier motion which are caused by the laser magnetic field induced Lorentz force. The momentum shift is proportional to the Keldysh time, while the wave-packet\92s spatial drift is proportional to the Eisenbud-Wigner-Smith time. The signature of the momentum shift is shown to be present in the ionization spectrum at the detector and, therefore, observable experimentally, see Fig. 1. In contrast, the signature of the Eisenbud-Wigner-Smith time delay disappears at far distances for pure tunneling dynamics [1,2].


[1] M. Klaiber, E. Yakaboylu, H. Bauke, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Rev. Lett. 110, 153004 (2013); arXiv:1205.2004v1 [physics.atom-ph]

[2] E. Yakaboylu, M. Klaiber, H. Bauke, K. Z. Hatsagortsyan and C. H. Keitel,\A0 arXiv:1309.0610 [quant-ph]



Fig. 1. Density plot of the asymptotic spin averaged electron momentum distribution at the detector. The distribution\92s maximum (white cross) is shifted from the nonrelativistic limit (white line). Copyright 2013 by APS.


Fig. 2. Numerical simulation of the electron wave function in a soft-core potential via the Dirac equation. The density plot shows the electron density at the moment when maximal laser field strength is attained. The solid black line indicates the maximum of the density in the laser propagation direction while the dashed line corresponds to the most probable trajectory resulting from the quasiclassical description. Solid green lines correspond to the border of the classical forbidden region. White arrows and the cross indicate the directions of the laser\92s electromagnetic fields and its propagation direction. The inset shows a scale-up of the region close to the tunnel exit. Copyright 2013 by APS.




Spin dynamics in relativistic ionization


Spin effects arise during the relativistic tunneling ionization process [3]. We have investigated the spin-resolved ionization dynamics employing the relativistic Coulomb corrected dressed strong field approximation [4,5], and taking into account the laser field driven electron spin dynamics in the bound state.\A0 Even if an electron is very tightly bound to an ionic core, it may still be crucially affected by a laser field of moderate intensity. Spin effects in the tunneling regime of ionization are built up in three steps: spin precession in the bound state, spin rotation during tunneling, and spin precession during the electron motion in the continuum, see Fig. 3. The magnitude and scaling of the spin-flip and spin asymmetry effects at ionization are reduced, when the electron spin dynamics in the bound state is taken into account.\A0 However, with super-strong laser fields a large spin-flip effect is measurable when employing highly charged ions, initially polarized along the laser propagation direction. The anticipated spin-flip effect is expected to be measurable with modern laser techniques combined with an ion storage facility.

Such measurements require an initially polarized target of ions and detection of the photoelectron polarization via Mott polarimetry. The electron spin flip can also be revealed via measurement of ion parameters, relating the angular momentum change of the ion during ionization to the electron spin change. The spin flip will be indicated by a non-vanishing signal for the difference in the ion angular distribution when appliying of left versus right circularly polarized laser fields.

In a different study the spin oscillations of a bunch of electrons have been investigated as a function of the interaction time with an intense laser field (see fig. 8 in the Theory Division webpage). For intensities of the order of 1018 W/cm2 the magnetic component is weak enough that quantum features (so-called collapses and revivals) become measurable in the dynamics [6].


[3] M. Klaiber, E. Yakaboylu, H. Bauke, C. M\FCller, G. G. Paulus and K. Z. Hatsagortsyan,\A0 arXiv:1305.5379 [physics.atom-ph]


[4] M. Klaiber, E. Yakaboylu, and K. Z. Hatsagortsyan, Phys. Rev. A 87, 023418 (2013);\A0\A0 arXiv:1301.5764 [physics.atom-ph]


[5] M. Klaiber, E. Yakaboylu, and K. Z. Hatsagortsyan, Phys. Rev. A 87, 023417 (2013);\A0 arXiv:1301.5761 [physics.atom-ph]


[6] O. D. Skoromnik, I. D. Feranchuk, C. H. Keitel, Phys. Rev A 87, 052107 (2013)



Fig. 3. The qualitatively different behavior of spin effects within the S-SFA (left column) and the D-SFA (right column). The initial spin is along the laser\92s magnetic field (top row). The spin is along the laser propagation direction with weak fields. (middle row), and with strong fields\A0 (bottom row); red wiggled arrow, blue dotted arrow and green solid arrow indicate the initial spin, the spin after the tunneling, and the final spin, respectively.




Relativistic high-order harmonic generation


Relativistic effects are very dramatic for ATI and HHG processes based on the three-step process (tunneling ionization-excursion in the laser field-recollsion with the atomic core). The laser magnetic field induces a drift of the ionized electron in the laser propagation direction which severely suppresses the probability of the electron to revisit the ionic core and, consequently, the yield of ATI electrons or harmonic photons [6]. That is why the HHG frequencies cannot be increased by a straightforward increase of the laser intensity. To this purpose we have considered different setups for the suppression of the magnetically induced drift in the relativistic regimes of HHG. For example, we have employed various combinations of fields or particular atomic systems such as pre-accelerated highly charged ions, exotic atomic systems (positronium). What appears particularly promising for the suppression of the relativistic drift is the use of a HHG scheme with counter-propagating attosecond pulse trains where a special method for phase-matching has been developed.

The conventional method of HHG uses sinusoidal laser fields. Is it possible to modify the laser pulse in such a way that it would allow efficient rescattering in the strongly relativistic regime and thus allow HHG in the hard x-ray domain? We have shown that this is feasible by employing strong laser pulses tailored as an attosecond pulse train (APT) [7]. The temporal tailoring of the laser pulse is intended to concentrate the ionizing and accelerating laser forces in short time intervals within the laser period, maintaining the average intensity of the pulse constant. This is due to the fact that in the tailored laser pulse, fragments are avoided in the electron trajectory, in contrast to the sinusoidal laser pulse where the electron acceleration is compensated by deceleration without a net energy gain by the electron, while the electron nevertheless continues to drift in the laser propagation direction. By the tailoring, the time span when the electron moves with relativistic velocity is decreased and a shorter drift in the laser propagation direction is obtained, leading to an increase of the recombination probability.

We have shown in [8] that strong counter-propagating\A0 APT employed as a driving field for HHG can be very useful to suppress the relativistic drift. This is achieved by reverting the relativistic drift of the ionized electron during the motion in APTs (see Fig. 4). The electron dynamics in APTs is the following. The electron is ionized by one APT, driven by this pulse up to the end of the pulse, then taken by the second counter-propagating pulse that realizes the rescattering with the atomic core.\A0 The second pulse induces a reverted drift toward the atomic core. The drift in the different pulses actually cancel each other out, due to which the initial momentum of the recolliding electron in the laser propagation direction can be small.\A0 As a result, the ionization probability and the HHG yield are enhanced. This concerns HHG from a single atom. Can coherent phase matched HHG emission is realized by this setup from a macroscopic gas target? Our investigation shows that this is the case.\A0 The propagation of the harmonics through the medium and the scaling of HHG into the multi-keV regime are investigated in [9]. We show that the phase-mismatch caused by the free electron background can be compensated by an additional phase of the emitted harmonics specific to the considered setup which depends on the delay time between the pulse trains. This renders feasible the phase-matched emission of harmonics with photon energies of several tens of keV\A0 from an underdense plasma.


In the relativistic regime the XUV or x-ray assistance can be employed to overcome the relativistic drift motion [10,11]. Thereby, the XUV frequency has to exceed the ionization energy to liberate the electron with a single photon and to deliver a significant initial momentum to the freed electron, see Fig. 5. This way the electron can obtain sufficient momentum in the direction opposite to the laser propagation direction to compensate for subsequent drift motion and return to the atomic core, recombine, and emit harmonics after the excursion in the relativistically strong laser field. The medium is a gas of multiply charged ions with ionization energy large enough to withstand the strong optical laser field. We have shown the feasibility of phase-matched emission and the macroscopic yield of harmonics in the relativistic regime of the x-ray assisted HHG setup in a strong IR laser field [11]. Generally, the efficiency of HHG is rather small even in the nonrelativistic regime due to the wave packet spreading. In the relativistic regime, the single-atom HHG emission rate continues to decrease even when the relativistic drift is compensated. Thus, a large phase-matching volume is crucial in order to achieve a significant HHG yield. Furthermore, the large ponderomotive potential is likely to result in rapid phase changes if ions emit under different conditions. For generating relativistic harmonics both challenges have to be met: circumventing the drift and having the setup stable against phase changes. This setup overcomes both issues and renders a measurable HHG yield in the relativistic regime possible, see Fig.6.


We have proposed a method for engineering the HHG phase which is achieved by shaping a laser pulse and employing XUV light or x rays for ionization [12]. This renders the production of bandwidth-limited attosecond pulses possible while avoiding the use of filters for chirp compensation. By adding the first 8 Fourier components to a sinusoidal field of 1016\A0 W/cm2 , the bandwidth-limited emission of 8 as is shown to be possible from a Li2+\A0 gas. The scheme is extendable to the zs-scale. In a similar way one can engineer the continuum fraction of the electron wave packet in HHG such that a quasi-monochromatic recollision with the atomic core is rendered possible even for parts of the wave packet that were launched to the continuum at different laser phases [13]. Because of this, the HHG spectrum is shown to be enhanced in a specified controllable spectral window.\A0


Fig. 4. The HHG setup with counter-propagating APTs. (a) Single-atom perspective: The classical trajectory of a rescattered electron of a single atom in the gas target. After ionization by pulse 1, the ejected electron is driven in the same pulse (light blue), propagates freely after the pulse has left (gray dashed) and is driven back to the ion by the second laser pulse (dark blue); (b) macroscopic perspective: the atoms are denoted as small dots. The medium is divided into different zones separated by the dashed lines and indicated by A, B, C. . . . The harmonics emitted from the green (dark) area propagate along the red (wavy) arrow. HHG from shaded (light green) areas is damped. The two driving APTs are shown in different blue color shades at the moment of overlap. The arrows indicate the propagation direction. Copyright 2012 by APS.



Fig. 5. Geometry of the HHG process for a collinear alignment of the x-ray and laser field. The co-propagating x-ray field (orange) has a frequency above the ionization energy to achieve drift compensation. The weak IR field (brown) is employed to accomplish phase matching. Copyright 2012 by APS.



Fig. 6. (a) Single-atom emission probability for E = 2.5 a.u., Ip,x = 8 a.u., ωx = 14 a.u. and Ex = 0.65 a.u. (black) and a conventional laser field (Ex = 0) with E0 = 2.5 a.u. and Ip,t = 4.8 a.u. (dashed black) and the same configuration in the DA (gray). For the second configuration, Ip,t is chosen such that the average tunnel-ionization rate is the same as the single-photon ionization rate in the case before. (b) Separate HHG yields of the three contributing quasiclassical trajectories for the discussed setup [blue line in (a)]. The dotted red contribution (long trajectory) is suppressed because the drift for the trajectory is not completely compensated (as discussed in Sec. III B2). The short trajectory contributions [solid lines in blue (gray thick line) and orange (gray thin line)] are nearly identical and separate only at the cutoff region where the saddle point approximation breaks down. Copyright 2012 by APS.





[6] M. C. Kohler, T. Pfeifer,\A0 K. Z. Hatsagortsyan, and C. H. Keitel, Frontiers of atomic high-harmonic generation (review)

Adv. Atom. Mol. Opti. Phys. 61, 159 (2012) arXiv:1201.5094v1 [physics.atom-ph]


[7] M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Rev. A 75, 063413 (2007).


[8] K. Z. Hatsagortsyan, M. Klaiber, C. M\FCller, M. C. Kohler, and C. H. Keitel, Opt. Soc. Am. B 25, 92 ( 2008).


[9] M. C. Kohler, M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, EPL 94, 14002 (2011);\A0 arXiv:1008.0511v1 [physics.atom-ph]


[10] M. Klaiber, K. Z. Hatsagortsyan, C. M\FCller, and C. H. Keitel, Optics Letters\A0 33, 411 (2008);\A0 arXiv:0708.3360


[11] M. C. Kohler, K. Z. Hatsagortsyan, Phys. Rev. A 85, 023819 (2012); arXiv:1111.4113 [physics.atom-ph]


[12] M. C. Kohler, C. H. Keitel, and K. Z. Hatsagortsyan,

Optics Express, 19, 4411 (2011); arXiv:1101.5885v1 [physics.atom-ph]. Included in research highlights of Nature Photonics 5, 250 (2011)


[13] M. C. Kohler and K. Z. Hatsagortsyan, JOSA B 30, 57 (2013).




Nontunneling harmonics



Coherent x-ray generation from below-threshold harmonics


The possibility of x-ray emission employing below-threshold-harmonic (BTH) generation in the nontunneling regime is considered in [14]. The interaction of a tightly bound valence electron in a highly charged ion with intense XUV laser radiation is investigated in the weakly relativistic regime by numerical solution of the two dimensional relativistically corrected Schroedinger equation. Highly charged with a large ionization potential are applied because in the BTH regime the HHG emission frequency is limited by the ionization potential. A high density of ions is necessary for sizable HHG yield, which can be realized using underdense plasma.

However, in this case a large free-electron background will exist, hindering the realization of phase matching for the emitted x rays with the driving infrared laser field. To weaken the phase-matching problem, we employ a strong XUV field to drive the harmonic generation process, as at higher frequencies the plasma refractive index is closer to one. High frequencies are also necessary to avoid the fully relativistic regime because the laser magnetic-field-induced drift also has consequences for BTH, see Fig. 7. The harmonics below the ionization energy of the tightly bound system are found to be emitted with much higher probability than the standard plateau harmonics of loosely bound systems in the tunneling ionization regime for the same photon energy. This paves a path toward coherent hard x rays.


Attosecond pulses at keV photon energies from high-order harmonic generation with core electrons


High-order harmonic generation (HHG) in simultaneous intense near-infrared (NIR) laser light and brilliant x rays above an inner-shell absorption edge is examined in [15]. A tightly bound inner-shell electron is transferred into the continuum. Then, NIR light takes over and drives the liberated electron through the continuum until it eventually returns to the cation leading in some cases to recombination and emission of a high-order harmonic photon that is upshifted by the x-ray photon energy, see Fig. 8. We have applied this scenario to 1s electrons of neon atoms. The boosted harmonic light is used to generate a single attosecond pulse in the keV regime. This opens prospects for x-ray HHG spectroscopy for time-resolving core electrons atosecond dynamics.




Fig. 7. The relativistic and nondipole effects on the photon emission of an electron in the soft-core potential of Ip =112.2 a.u. exposed to a laser field with an intensity of\A0\A0 1020W/cm2 and photon energy of ω = 3 a.u. (a) and ω = 1 a.u. (b). Copyright 2011 by APS.



Fig. 8. Schematic of the modified three-step model of HHG with core electrons ionized by x rays. (a) X-ray absorption ejects a core electron (b) which is subsequently driven through the continuum by the NIR light; (c) upon returning to the parent ion it may recombine with the core hol e and release its excess energy in terms of a high-order harmonic of the NIR laser upshifted by the x-ray photon energy. The NIR laser has only a noticeable influence on valence and continuum electrons and barely influences core electrons. The converse holds true for the interaction with the x rays. Copyright 2013 by APS.


[14] Ji-Cai Liu, M. C. Kohler, K. Z. Hatsagortsyan and C. H. Keitel, Phys. Rev. A 84, 063817 (2011).


[15] C. Buth, Feng He, J. Ullrich, C. H. Keitel, and K. Z. Hatsagortsyan, Phys. Rev. A 88, 033848 (2013); arXiv:1203.4127v1 [physics.atom-ph]





Coulomb focusing effects in strong-field processes



Low-energy structures


The recent experiments of DiMauro group on the photoionization of atoms and molecules in strong mid-infrared laser fields reveal a previously unexpected characteristic spike-like low-energy structure (LES) in the energy distribution of electrons emitted along the laser polarization direction, see Fig. 9. These observations manifest a striking contrast to the prediction of the SFA or the SFA with Coulomb corrections and point to a lack of complete understanding of strong field physics. Varying the laser polarization from linear to circular, LES is significantly reduced. The latter indicates that rescattering is playing an essential role in this process. However, many questions arise: How exactly does the LES arise? Why does it have a peaked structure? Why is the effect of rescattering more pronounced in mid-infrared laser fields? We have investigated and identified the mechanism of LES using the classical-trajectory Monte Carlo method with tunneling and the Coulomb field of the atomic core fully taken into account [16]. With a qualitative theoretical estimation for the Coulomb field effects: initial Coulomb focusing (CF), multiple forward scattering, and asymptotic CF, we have quantified their relative role in the electron dynamics and conclude that (1) the behavior of the transverse (with respect to the laser polarization direction) momentum change of the electron due to Coulomb field effects with respect to the ionization phase is the key for understanding of the LES, see Fig. 10, and (2) at mid-infrared wavelengths, multiple scattering of the ionized electron plays a significant nonperturbative role [17].


Coulomb focusing in an elliptically polarized laser field


It appears that the Coulomb focusing has a significant role also in a laser field of elliptical polarization, although it is known that the rescattering effect is reduced in this regime. We have investigated the role of Coulomb focusing in above-threshold ionization in a mid-infrared laser field of elliptical polarization [18[. We have shown that multiple forward scattering of the ionized electron by the atomic core has the dominated contribution in Coulomb focusing up to moderate ellipticity values, see Fig. 11. The multiple forward scattering causes squeezing of the transverse momentum-space volume, which is the main factor influencing the normalized yield at moderate ellipticities. It is responsible for the peculiar energy scaling of the ionization normalized yield along the major polarization axis, and for the creation of a characteristic low-energy structure in the photoelectron spectrum. At large ellipticities, the main CF effect is due to the initial Coulomb disturbance at the exit of the ionization tunnel. The initial Coulomb disturbance, as our estimates show, enhances the ionization yield. This is because the electrons are tunneled out at larger laser fields or with smaller initial transverse momentum when the initial CF is taken into account for the electron drifting along the main polarization axis. The enhancement factor is shown to be sharply pronounced at intermediate ellipticities when both of the above-mentioned enhancement mechanisms contribute. In this region of ellipticity, the yield is enhanced by an order of magnitude due to the CF.


[16] Chengpu Liu and K. Z. Hatsagortsyan, Phys. Rev. Lett. 105, 113003 (2010);\A0 arXiv:1007.5173v1 [physics.atom-ph].


[17] C. Liu and K. Z. Hatsagortsyan, J. Phys. B 44, 095402 (2011);\A0\A0 arXiv:1011.1810v1 [physics.atom-ph]


[18] C. Liu and K. Z. Hatsagortsyan, Phys. Rev. A 85, 023413 (2012) ; arXiv:1109.5645v2 [physics.atom-ph]



Fig. 9. Photoelectron spectra: (a) The experimental result (squares) for a xenon atom in a laser field with peak intensity 3.2x1013W/cm2, wavelength 2.3μm as and the CTMC simulation (circles). (b) CTMC simulations\A0 exact (circles); with totally neglecting the Coulomb potential (NCP) (stars); NCP only for the electron longitudinal momenta (squares) and NCP only for the transverse momenta (triangles). The high-energy limit of LES defined by the break in slope is indicated with an arrow. Copyright 2010 by APS.



Fig. 10. The distribution of electrons in the LES in phase space (the initial transverse momentum versus the ionization phase within the electron energy interval (0,20) eV, with color coded probability. Irregular points are shown by triangles. The circles indicate the maximum probability for each phase. The laser field is maximal at the ionization phase π/2. Copyright 2010 by APS.



Fig. 11. The photoelectron normalized yield emitted along the major polarization axis vs ellipticity and photoelectron energy: (a) analytical estimate of the yield in the PPT model, (b) numerical simulations without taking into account Coulomb field effects, and (c) numerical simulations with Coulomb field effects. The laser intensity is\A0 9 \D7 1013 W/cm-2 and the wavelength is 2μm. The target atom is hydrogen. Copyright 2010 by APS.




Radiation reaction effects in strong laser fields



Radiation-reaction-force-induced nonlinear mixing of Raman sidebands of an ultraintense laser pulse in plasma


When an electron moves in an external field it may radiate losing in this way energy and momentum. This effect is known as radiation reaction [19]. During the interaction of strong laser radiation with electrons, the radiation reaction can play an important role in the relativistic regime. In our recent work [20] we found a surprising counter-intuitive effect of the radiation reaction for Raman scattering of a strong radiation in plasma. Usually, one relates the radiation reaction effect to a resulting damping and could expect that this would cause decreasing\A0 the growth rate of the Raman scattering. In contrast to that our calculation shows a significant increase of the forward Raman scattering (FRS) growth rate, see Fig. 12. Our calculation is based on the solution of the Landau-Lifshitz classical equation of motion taking into account the radiation reaction force perturbatively. The reason for this unexpected effect of radiation reaction is that the radiation reaction force causes a phase shift in the nonlinear current densities that drive the two Raman sidebands (anti-Stokes and Stokes waves). Because of the latter\A0\A0 the nonlinear mixing of two sidebands of the FRS becomes possible mediated by the radiation reaction force. This mixing results in a strong enhancement in the growth of the forward Raman scattering instability. In the absence of the radiation reaction force, nonlinear currents that drive the Stokes and the anti-Stokes modes have opposite polarizations. Consequently, the phase shift induced by the radiation is opposite for these modes. This results in the interaction between the nonlinear current terms, culminating into phase shift accumulation. We term the nonlinear mixing of the two modes due to the radiation reaction force as the manifestation of this accumulation of phase shifts, and it leads to the enhanced growth rate of the FRS instability. One can also intuitively argue that this growth enhancement occurring due to the availability of an additional channel of radiation-reaction-force-induced laser energy decay and its efficient utilization by both the Stokes and the anti-Stokes modes. In other words, due to nonlinear mixing of the two Raman sidebands, they assist each other to extract the energy from the propagating laser wave.


The radiation reaction force strongly enhances the growth of the FRS only when both the Stokes and the anti-Stokes modes are the resonant modes of the plasma. The growths of the FRS with only the resonant Stokes wave excitation and the backwards Raman scattering (BRS) are also enhanced by the inclusion of the radiation reaction force, although the enhancement is\A0 minor\A0 due to the absence of the radiation-reaction-force-induced nonlinear mixing of the anti-Stokes and the Stokes modes. Thus, the radiation reaction force appears to strongly enhance the growth of the SRS involving four-wave decay interaction. These results are important for the ELI Project, as the ultraintense laser pulses are expected to create dense plasma by strongly ionizing the ambient air and also by producing the electron-positron pairs. The subsequent interaction of this plasma with the laser pulse can lead to the onset of parametric instabilities leading to significant change in the frequency spectra and shapes of these extremely intense short laser pulses due to the radiation reaction force. Moreover, contrary to nonlinear Compton scattering of a counter-propagating relativistic electron in a strong laser field aiming to discern the signatures of the radiation reaction force on the spectra of high-energy gamma-ray photons [19], enhanced FRS due to the radiation reaction force provides an alternative way to detect the radiation reaction effects in the spectra of low-energy optical photons.


Radiation dominated regime


Unlike in the nonrelativistic case, a situation can occur in the ultrarelativistic regime in which the radiation reaction force becomes comparable with the Lorentz force in the laboratory frame while being much smaller in the instantaneous rest frame of the electron. This is the so-called radiation dominated regime in which the electron dynamics and its radiation are supposed to be significantly modified due to the radiation reaction. In the classical regime of interaction we have found a strong signatures of radiation reaction below the Radiation-Dominated Regime [21]. The influence of radiation reaction on multiphoton Thomson scattering by an electron colliding head-on with a strong laser beam is investigated in a new regime, in which the momentum transferred on average to the electron by the laser pulse approximately compensates the one initially prepared. This equilibrium is shown to be far more sensitive to the influence of radiation reaction than previously studied scenarios, see Fig. 13. As a consequence, the radiation reaction can be experimentally observed with currently available laser systems.


Quantum radiation reaction effects in multiphoton Compton scattering are investigated in [22] in the realm of quantum electrodynamics. We identify the quantum radiation reaction with the multiple photon recoils experienced by the laser-driven electron due to consecutive incoherent photon emissions. After determining a quantum radiation dominated regime, we demonstrate how in this regime quantum signatures of the radiation reaction strongly affect multiphoton Compton scattering spectra and that they could be measurable in principle with presently available laser technology, see Fig. 14.




Fig. 12. Normalized growth rate\A0 of the Forward Raman scattering as a function of the normalized plasma density\A0 and normalized pump laser amplitude (a) including the radiation reaction force and (b) without the radiation reaction force. The normalized growth rate is plotted on a log10 scale. Copyright 2013 by APS.



Fig. 13. Angle resolved spectral energy emitted by the electron without (a) and with (b) radiation reaction. The initial electron energy is 40 MeV, the laser field intensity\A0 5x1022 W/cm2, the wavelength 0.8 \B5m, the pulse duration 27 fs and the waist size 2.5\A0 \B5m. Copyright 2009 by APS.



Fig. 14. Multiphoton Compton spectra calculated quantum mechanically with (solid, black line) and without (long dashed, red line) the RR. For the sake of comparison, the corresponding classical spectra with (short dashed, blue line) and without (dotted, magenta line) the RR are also shown. The inset shows a zoom of the spectral region where different curves cross. The electron energy is 1 GeV,the infrared laser intensity is 5x 1022 W/cm2. Copyright 2010 by APS.



Photoemission of a single-electron wave packet

in a strong laser field


We have studied the amount of light that an electron scatters out the side of a laser and showed that even when it spreads to the scale of the wavelength of the driving laser field, it cannot be treated as an extended classical charge distribution, but rather behaves as pointlike emitter carrying information on its initial quantum state [23,24].



[19] A. Di Piazza, C. M\FCller, K. Z. Hatsagortsyan, and C.H. Keitel, Rev. Mod. Phys. 84, 1177 (2012);\A0 arXiv:1111.3886v1 [hep-ph]


[20] N. Kumar, K. Z. Hatsagortsyan, C. H. Keitel,

Phys. Rev. Lett. 111, 105001 (2013);\A0\A0 arXiv:1307.3939 [physics.plasm-ph]


[21] A. Di Piazza, K. Z. Hatsagortsyan and C. H. Keitel,

Phys. Rev. Lett. 102, 254802 (2009);\A0 arXiv:0810.1703 [physics.class-ph]


[22] A. Di Piazza, K. Z. Hatsagortsyan, and C.H. Keitel,

Phys. Rev. Lett. 105, 220403 (2010); arXiv:1007.4914v1 [hep-ph]


[23] J. Peatross, C. M\FCller, K. Z. Hatsagortsyan and C. H. Keitel, Phys. Rev. Lett. 100, 153601 (2008); arXiv:0712.0259 [quant-ph]


[24] J. P. Corson, J. Peatross, C. M\FCller, and K. Z. Hatsagortsyan, Phys. Rev. A 84, 053831 (2011).



Nonlinear QED effects in strong laser fields



Bragg scattering of light in vacuum structured

by strong periodic fields


Super-strong laser fields offer unique possibilities for the investigation of the quantum vacuum. Different effects of vacuum QED nonlinearities induced by strong laser fields have been considered. Elastic photon-photon scattering is a process quite feasible for experimental observation. We have shown that using a setup of multiple crossed superstrong laser beams the photon-photon scattering rate can be significantly enhanced due to Bragg interference [25]. The Bragg interference arising at a specific impinging direction of the probe wave concentrates the scattered light in specular directions, see Fig. 15. The interference maxima are enhanced with respect to the usual vacuum polarization effect proportional to the square of the number of modulation periods within the interaction region. The enhancement is maintained also in the total probability of the scattering, integrated by the scattering angle. The Bragg scattering can be employed to detect the vacuum polarization effect in a setup of multiple crossed superstrong laser beams with parameters envisaged in the future Extreme Light Infrastructure. Similar enhancement effects will exist in all types of inelastic light-by-light scattering and other processes based on spatially modulated vacuum polarization.


Pair production in laser fields oscillating

in space and time


In the extremely strong Schwinger field electron-positron pair can be created. Usually, the laser field in a node of a standing wave is approximated by an oscillating laser field. What is the role of the laser magnetic field for pair production process? When the pair production coherence length is on order of or larger than the laser wavelength, the laser magnetic field effect cannot be neglected. This effect we investigate in [26] where the production of electron-positron pairs from vacuum by counter-propagating laser beams of linear polarization has been studied numerically solving Dirac equation. In contrast with the usual approximate approach, the spatial dependence and magnetic component of the laser field are taken into account. We show that the latter strongly affects the creation process at high laser frequency: the production probability is reduced, the kinematics is fundamentally modified, the resonant Rabi-oscillation pattern is distorted, and the resonance positions are shifted, multiplied, and split, se Fig. 16. The narrow peak splitting of the resonant pair production probability could serve as a sensitive probe of the quasienergy band structure and, generally, of QED in superstrong spatially and temporally inhomogeneous fields.


Streaking at high energies with electrons

and positrons (SHEEP)


We proposed a detection scheme for characterizing high-energy γ-ray pulses down to the zeptosecond timescale, employing the pair production process in strong laser fields [27]. In contrast to existing attosecond metrology techniques, our method is not limited by atomic shell physics and therefore capable of breaking the MeV photon energy and attosecond timescale barriers. It is inspired by attosecond streak imaging, but builds upon the high-energy process of electron\96positron pair production in vacuum through the collision of a test pulse with an intense laser pulse, see Fig. 17. The scheme is shown to be feasible in the upcoming Extreme Light Infrastructure laser facility where the required three beams can be available: strong infrared beam, x-ray beam and combined with the γ -ray test beam.\A0


Microscopic laser-driven high-energy colliders


We have proposed a concept of a laser-guided electron-positron collider in the femtosecond and high-energy regime [28, 29]. Ultra-intense laser pulses are employed to unite in a single stage the electron and positron acceleration and their microscopic coherent collision in the GeV regime. We showed that such coherent collisions yield a very large enhancement of the luminosity as compared to conventional collider. The proposed collider consists of a gas of Ps atoms exposed to laser fields. When a Ps atom is submitted to a strong, linearly polarized laser pulse, then the electron and positron are instantaneously ionized, excurse afterwards in the laser field where they acquire energy, and finally recollide head-on-head. We call this recollision \93coherent\94, in contrast to the incoherent collisions in the beam-beam collision, since the mean impact parameter is microscopic, i.e. is of order of the positronium atomic size, see Fig. 18. The latter is true because 1) the initial coordinates of the electron and positron are confined within the range of the Bohr radius; due to the equal (by modulus) but opposite (by sign) charge-to mass ratios of the constituents, 2) the drift in the laser propagation direction for the electron is the same as for the positron, while 3) the oscillations along the transversal direction are opposite [30]. As a result, the current density of one colliding particle can be larger than the mean current density of a bunch of electrons in a conventional accelerator. Accordingly, the luminosity in these \93coherent\94 collisions can be much larger than in ordinary, \93incoherent\94 collisions, where electron and positron collide starting from random distributions.


The laser collider can be applied for muon pair creation from positronium atom. The electrons and positrons are assumed to form initially nonrelativistic electron-positron plasma or a gas of positronium atoms. Since the initial energy of the particles is far below to the muon pair creation threshold, their annihilation into a muon pair cannot happen without the influence of the external field. However, the muon pair is not produced from vacuum by the laser field itself, but in an annihilating electron-positron collision, where the role of the laser field is to supply the required energy to the colliding particles [31], see Fig. 19. The minimum laser intensity required amounts to a few 1022 W/cm2 in the near-infrared frequency range.\A0 The muons are created with ultrarelativistic energies and emitted under narrow angles along the laser propagation direction.


Positronium lifetime in strong laser fields

Investigating positronium dynamics in intense laser fields, in addition to the coherent x-ray generation during electron-positron recombinations, we have predicted gamma-radiation of narrow bandwidth due to laser-enhanced annihilations of both particles [30]. Without an external laser field, ortho-positronium annihilates spontaneously into three photons. In the laser field, the channel that involves two gamma-quanta and one laser-photon can be enhanced by means of stimulated emission of laser photons, see Fig. 20. Due to energy-momentum conservation, in this case the bandwidth of the gamma-radiation is connected with the bandwidth of the low-frequency radiation and, therefore, is narrow. We thus obtain gamma-radiation which is enhanced in intensity and narrow in bandwidth. \A0



Enhancing the lifetime of positronium atoms

via collective radiative effects


We have proposed to harness cooperative spontaneous emission of an ensemble of Ps atoms to provide a way for controlling the annihilation dynamics. We employ a dense ensemble of (para- and ortho-) Ps atoms in which the atoms interact with each other via the common radiation field, see Fig. 21.



Fig. 15. (a) The Feynman diagram describing the photon scattering in a strong external electromagnetic field. (b) Bragg scattering of a probe laser beam by a set of focused strong laser beams. Copyright 2011 by APS.



Fig. 16. Resonant probability spectrum: Maximal value of the pair production probability during Rabi oscillation, varying the pulse length up to 200 cycles. The red crosses show the spectrum in oscillating electric field; the peak labels denote the absorbed photon number. The black triangles show the spectrum in the standing laser wave. Here, the labeling signifies the number of absorbed photons from the right-left propagating waves. A splitting occurs, as indicated by arrows for the example of the (3-2) peak. Copyright 2009 by APS.



Fig. 17. Concept of SHEEP. Electron\96positron pairs are produced through the interaction of a short test pulse with an intense anti-aligned laser field within a streaking laser pulse. The leptons acquire additional energy and momentum depending on their phase in the streaking pulse at the moment of production. Copyright 2011 by Elsevier.



Fig. 18. Schematic diagram displaying positronium dynamics in an intense laser field. The bound system depicted by the density of its wave function may ionize in the laser field. Once free, both electron and positron could be described as classical particles. Their trajectories are shown by the solid lines. The electric field accelerates both particles in opposite directions, while the Lorentz force due to the magnetic field leads to an identical drift in the propagation direction. Without an initial center-of-mass motion, these trajectories are symmetric and thus both particles overlap periodically. Copyright 2004 by APS.



Fig. 19. Muon pair creation from positronium atom in a laser field.



Fig. 20. Feynman diagrams displaying gamma-photon emission via electron-positron annihilation by (a) multiphoton annihilation in superstrong laser fields with emission of 106 laser photons along with one gamma-quantum; (b) annihilation in a laser field\A0\A0 with emission of laser photons along with two gamma-quanta; (c) annihilation of ortho-positronium with emission of three photons without laser field. Bold lines correspond to the electron (positron) Volkov states, i.e., involving a laser field; dashed lines correspond to the emitted photons. Copyright 2004 by APS.



Fig. 21. The energy levels of a Ps atom. The coherent laser drives the optical 0-2 atomic transition with a Rabi frequency, γ0 and γ1 are the single-atom spontaneous decay rates on transitions 0-1 and 1-2, respectively, whereas γ2 describes the annihilation decay rate. Copyright 2013 by APS.



Using a three-level model system (which incorporates annihilation) for a Ps atom driven by a resonant laser field, we investigate the role of collective spontaneous radiative processes on the population dynamics and its influence on the annihilation evolution of the ensemble. Two schemes are developed for the enhancement of the annihilation lifetime of the Ps ensemble. In the first scheme, the radiative decay on the 3D-2P transition is collective, other than on the 2P-1S transition, and is controlled by the density of the gas. In the second scheme, both transitions are collective, but the strength of the first is enhanced by a cavity, leading to population trapping in the 2P state and, consequently, to significant lifetime enhancement.\A0





[25] G. Yu. Kryuchkyan and K. Z. Hatsagortsyan, Phys. Rev. Lett. 107, 053604 (2011); arXiv:1102.4013 [quant-ph].


[26] M. Ruf, G. R. Mocken, C. M\FCller, K. Z. Hatsagortsyan and C. H. Keitel, Phys. Rev. Lett. 102, 080402 (2009); arXiv:0810.4047 [physics.atom-ph].


[27] A. Ipp, J. Evers, C. H. Keitel, and K. Z. Hatsagortsyan, Phys. Lett. B 702, 383 (2011) arXiv:1008.0355v2 [physics.ins-det]


[28] K. Z. Hatsagortsyan, C. M\FCller, and C. H. Keitel, Europhys. Lett. 76, 29 (2006); arXiv: 0602093 [physics]


[29] C. Liu, M. C. Kohler, K. Z. Hatsagortsyan, C. M\FCller and C. H. Keitel, New J. Phys. 11, 105045 (2009); included in IOP Select


[30] B. Henrich, K. Z. Hatsagortsyan and C. H. Keitel, Phys. Rev. Lett. 93, 013601 (2004)\A0 arXiv: hep-th/0303188


[31] C. M\FCller, K. Z. Hatsagortsyan and C. H. Keitel, Physics Letters B 659, 209 (2008); arXiv:0705.0917 [hep-ph]


[32] Ni Cui, M. Macovei, K. Z. Hatsagortsyan and C. H. Keitel, Phys. Rev. Lett. 108, 243401 (2012)\A0 arXiv:1112.1621v1 [quant-ph]



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