Theory Division

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 in the infrared region as well as at strong lasers in the complementary short-wavelength domain up to the extreme ultraviolet. A strong x-ray free-electron laser (XFEL) is being developed at DESY (Hamburg). Moreover, experimentalists are able to produce well-controlled ultra-short 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 a few cycle laser pulses, attosecond pulses of XUV radiation, production of coherent ultraviolet radiation via self-amplified spontaneous radiation in FEL (FLASH facility) 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 and thin matter layers to vacuum including quantum fluctuations. In the center of interest are the relativistic regimes of interaction. In particular, our attention is focused on vacuum laser acceleration of particles such as electrons and ions, 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, or even via vacuum quantum fluctuations in strong fields as well as radiation reaction effects. As regards atomic systems, much emphasis is placed on the extension of the ionization-recollision dynamics to the relativistic domain because this will allow HHG in a short wavelength domain and may be the initiation of nuclear reactions with a single ion. To this purpose we consider different setups for the suppression of the magnetically induced drift in the relativistic regimes of the above-threshold ionization (ATI) and HHG processes. For example, we have employed various combinations of fields or particular atomic systems such as pre-accelerated highly charged ions, exotic atomic systems (positronium) or antisymmetric molecular orbitals. What appears particularly promising for the suppression of the relativistic drift is the use of attosecond pulses. We have also investigated the influence of non-dipole effects on the stabilization phenomenon, particularly in two-electron atoms or in the case of excited atomic states. Recently, we have been involved in the investigation of the distinct features of strong field atomic phenomena in the short-wavelength (UV, XUV) domain.

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 . Thus,  the harmonic generation in vacuum during the collision of two counterpropagating laser beams has been investigated. Although the experimental observation of HHG in vacuum is a matter of the remote future, a particular case of it, i.e. elastic photon-photon scattering, is quite feasible for experimental observation as our analysis shows. 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. We have shown how the presence of a cold plasma, due to the unique behavior of plasma near the threshold of transparency, can enhance vacuum polarization effects even more than one order of magnitude. Recently, we have shown the observability of non-perturbative vacuum-polarization effects in proton-laser collisions. 

Some recent and current projects:

Atomic quantum dynamics beyond the dipole approximation

Fig. 1: In the relativistic regime magnetic field effects induce a drift of the electron in the laser propagation direction. This prevents the electron from revisiting the ionic core, which diminishes the effectiveness of rescattering.

Fig. 2: Harmonic emission rate in laser propagation direction in the relativistic regime with laser intensity of 1.8x1017 W/cm2 and angular frequency 0.05 a.u. The model potential is adapted to Be3+ with ionization potential Ip=8 a.u..: (grey) within the dipole approximation, (black) with leading nondipole corrections, (blue) with leading nondipole and relativistic corrections, (red) with respect to the Klein-Gordon equation.  Reprinted from [11]. Copyright 2007 by APS.

Fig. 3: Schematic diagram displaying positronium dynamics in an intense laser field. The bound system, depicted by the density of its wave function, may be ionized in the laser field. Once free, both electron e- and positron e+ could be described as classical particles. Their trajectories are shown by the solid lines. Reprinted from [7]. Copyright 2007 by APS.

Fig. 4: The wave packet evolution for orthogonal alignment of the molecule: (a) within the dipole approximation, (b) without the dipole approximation. Within the dipole approximation: antisymmetry conserved, large wave packet spreading in laser polarization direction in (a). Antisymmetry broken beyond the dipole approximation, concentration of density in the vicinity of the former nodal line in (b).  Reprinted from [8]. Copyright 2006 by APS.

With regard to infrared laser intensities exceeding 10¹⁵ W/cm², the influence of the magnetic field on electron dynamics becomes non-negligible. Therefore, a theoretical approach beyond the dipole approximation is needed [1]. Consequently, in [2-10] the laser-atom interaction beyond approx. 10¹⁵ W/cm² has been investigated on the basis of the Schrödinger equation in nondipole approximation.

The quantum-mechanical evolution of the electron wave-packet in a laser field of arbitrary shape is investigated in [2] in the weakly relativistic regime. Analytical expressions for the width of a wave packet and its orientation in a strong laser field are established, thus giving an intuitive understanding of the wave packet dynamics when the laser magnetic field starts to play a role. The significance of quantum effects for a free electron in a laser field is demonstrated in [3] by looking for negativities in the Wigner function of the system.  The influence of nondipole effects on the stabilization phenomenon in the weakly relativistic regime is investigated for two-electron atoms [4] as well as in the case of excited atomic states [5].

Relativistic effects are especially dramatic for ATI and HHG processes. 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 (see Fig. 1). That is why the HHG frequencies cannot be increased by a straightforward increase of the laser intensity. 

Relativistic signatures in high-order above-threshold ionization have been investigated in [6,11]. In [6] we have identified a weakly relativistic regime where the rescattering process, though reduced, still plays a role and where at the same time relativistic signatures are clearly exhibited. In [11] a fully relativistic treatment of ATI and HHG has been performed via the Klein-Gordon equation. The following relativistic signatures in the ATI/HHG spectra have been observed (see Fig.2). First, in the relativistic regime the plateau is lowered. Second, the strongly fluctuating spectrum resulting, within the dipole approximation, from the interference of at least the two shortest trajectories of the ionized electron, vanishes in the relativistic case. The multiplateau structure also vanishes. Further, the plateau is no longer flat, but has a curving structure that rises with increasing laser intensity. The angular distribution of the emitted electrons is significantly modified, tilting in the laser propagation direction. Moreover, the electron energy cutoff is slightly increased. Note that the inclusion of the relativistic mass shift reduces the cutoff, while the nondipole effects enhance it.

Various methods for counteracting the relativistic drift have been proposed. To this end, the properties of the atomic system can be modified by using highly charged ions which move relativistically against the laser propagation direction (see below), by employing  positronium atoms instead of ordinary atoms [7], or antisymmetric molecular orbitals [8,9]. In the first case, the drift of the ionized electron is reduced by the increase of the laser frequency in the system's center of mass. In the second, the electron's relativistic drift is compensated by an equally strong drift of the positron (see Fig.3). In the last case, however, the compensation is due to the initial momentum of the tunneling electron from the antisymmetric orbital. On the other hand, the laser field can be modified, assisting it by a strong magnetic field [10], by using two consecutive laser pulses or strong attosecond pulses (see the next Section).

An interesting possibility to counteract the relativistic drift in the weakly relativistic regime, based on the use of antisymmetric molecular orbitals, is shown in [8,9]. HHG of a diatomic molecule is investigated by solving the Schrödinger equation numerically, beyond the dipole approximation. Due to symmetry, the momentum distribution of the antisymmetric molecular orbital has two peaks at the nonzero momentum component along the axis of the molecule. Therefore, in a strong laser field the electron tunnels out from the molecule with nonzero momentum along the molecular axis. If the axis is oriented in the laser propagation direction, the initial momentum of the ionized electron will counteract the relativistic drift, allowing for rescattering and, thus, considerably increasing the harmonic signal (see Fig. 4).

Fig. 1: The ionized electron trajectory in the sinusoidal pulse is shown as a dashed line, in the tailored pulse as a solid line.

Fig. 2: The HHG rate, calculated with the gauge invariant strong field approximation [2], (a) within the dipole approximation and a sinusoidal field, (b) with respect to the Klein-Gordon equation and a sinusoidal field, (c) with respect to the Klein-Gordon equation and the APT. The laser  intensity is 5x1019W/cm2, the frequency 0.1 a.u., and  Ip=62 a.u..

Fig. 3: HHG spectra in a laser field with E=2.1 a.u. and a frequency of 0.05 a.u. in Ar7+ Ip=5.29 a.u: (gray) dipole approximation, (red) Klein-Gordon equation, and (black) additionally assisted by an APT with Ea=0.02 a.u.,  carrier frequency 8.5 a.u., pulse length 4 a.u., and a phase delay of -1.2 rad with respect to the laser wave. Reprinted from [6]. Copyright 2008 by OSA.

Fig. 4:  The harmonic emission rate as a function of the harmonic energy for counterpropagating APTs with various time delays compared with the spectrum in the DA (pulse delay 10 fs) (gray). The pulse delays for the corresponding spectrum are indicated in units of fs; further, the driving laser field is E=325 a.u. and Ip=150 a.u. Ar16+). Reprinted from [3]. Copyright 2007 by APS.

Fig. 5: Temporal shape of the generated pulse with a Gaussian window at 805 keV with a FWHM of 4 keV. The FWHM of the pulse is approximately 900 zs. The laser field is E=88 au. , Ip=63 a.u., with a time delay between the pulses of  0.9 fs

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) (see Fig.1) [1]. 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.

In Fig.2 the HHG spectrum with a strongly relativistic APT calculated via the Klein-Gordon equation is compared with two calculations for the harmonic emission by using a sinusoidal field. While the first calculation is based on the dipole approximation as a reference for the ideal ionization-rescattering process, the second one is fully relativistic. From the HHG spectrum one can see that cutoff energies of 1 MeV are attainable with a probability not smaller than that in the dipole approximation. This will permit HHG in the hard x-ray domain and the initiation of nuclear reactions with single ions.  The optimal conditions for high-energy rescattering are found in relation to the duration of the attosecond pulse, to the time delay between pulses in the APT, as well as to the spectral content of the APT [3]. The stability of the harmonic yield vs. deformations of the tailored pulse shape is shown. The proposed method renders energies of more than 1 MeV possible for the rescattering atomic electron.

We also propose other recollision schemes based on two consecutive counter-propagating laser pulses [4] and in another project including also a magnetic field [5]. The drift velocity occuring in strong laser fields is employed to drive electrons first away from the core and then back for recollisions. The intensities of the pulses are chosen so as to enable recollisions with the maximal kinetic energy which laser-driven electrons can reach in a propagating laser field.

Recently, we have discussed two new methods for realizing efficient rescattering of the ionized electron in the relativistic regimes employing attosecond pulses, the  recollision energy spanning from keV to MeV range [6,7]. In an energy domain of several tens of keV, the recollisions are achieved via assisting the infrared strong laser field with a weak attosecond pulse train of XUV photons (see Fig.3). Due to one-photon ionization by the x-ray pulse, the ionized electron obtains a starting momentum that compensates the relativistic drift which is induced by the laser magnetic field, and allows the electron to efficiently emit harmonic radiation upon recombination with the atomic core in the relativistic regime. This allows for emission of coherent hard x-rays of up to 40 keV energy. For recollisions in the MeV domain (see Fig.4), strong counter-propagating attosecond pulse trains are employed. In this setup, recollisions of the ionized electrons can be achieved in the highly relativistic regime via a reversal of the commonly deteriorating drift and without instability of the electron dynamics such as in a standing laser wave. HHG yield is enhanced by several orders of magnitude in comparison to the case of a single laser wave. This way hard x-ray and gamma-ray harmonics and extremely short pulses can be feasible.

HHG in the relativistic regime can be employed to obtain zeptosecond pulses of gamma-rays. Thus, our calculations show that coherent attosecond gamma-rays in the 10 MeV energy range as well as coherent zeptosecond gamma-ray pulses of MeV photon energy for time-resolved nuclear spectroscopy become feasible [8]  (see Fig.5).

Relativistic dynamics in ultra-intense laser fields

Fig. 1: Snapshots of the electron wave packet taken at four different times t from 0.00055 a.u. to 0.04675 a.u.; the contour lines indicate the logarithm of the probability density as shown by the labels in (a). In (d) the thick line which starts at the origin, marks the trajectory of the position operator's expectation value. The marker '+' indicates the position of the ionic core with the charge Z=8. The Lorentz transformed amplitude and frequency of the laser field are 4.334x 105 a.u. and 1208.5 a.u., respectively. Reprinted from [2]. Copyright 2003 by APS.

Coherent collisions by laser-driven positronium and laser-enhanced annililation

Fig. 2: Feynman diagrams displaying gamma-photon emission via electron-positron annihilation by a) multiphoton annihilation in super strong laser fields with emission of 106 laser photons along with one gamma -quantum; b) annihilation in a laser field 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 to the emitted photons. Reprinted from [1]. Copyright 2004 by APS.

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. 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. 2). 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.

Vacuum QED nonlinearities in strong laser fields

Fig. 1: The Feynman diagram for high harmonic generation in the crossed laser beams in vacuum.

Fig. 2: The harmonic spectrum in the standing laser wave with the electrical field EL=2x1017V/cm.  Reprinted from [1]. Copyright 2005 by APS.

Fig. 3: The change of x-ray polarization due to vacuum nonlinearity during its propagation through tightly focused laser beam.

Fig. 4: The polarization rotation angle  and the ellipticity as functions of the observation distance yd. Reprinted from [2]. Copyright 2006 by APS.

Fig. 5: The Feynman diagram corresponding to the process of laser-photon merging induced by VPEs in a proton field.

Harmonic generation in vacuum

Due to the presence of charged virtual particles, two strong counterpropagating laser fields can give rise to high-order harmonic generation in vacuum [1]. The n^th harmonic is emitted when n-photons from each laser wave are absorbed by the virtual pair and only two high-energy photons are emitted (see Fig.1). At superstrong laser intensities of the order of the critical one, the spectrum of vacuum high-harmonic generation shows plateau and cutoff structures (see Fig.2) well-known from the atomic high-harmonic generation phenomenon. At lower laser intensities, mainly the first harmonic is generated, i.e. two laser photons are absorbed by a virtual pair and are emitted, in general, with different propagation directions. This is, in fact, elastic photon-photon scattering. As we estimated in [1], the laser-assisted photon-photon scattering via the three laser beams collision may be observed experimentally when employing petawatt-class lasers which will be available in the near future.

Diffraction effects in laser-laser collision

In [2] we have approached the question of how the spatial focusing of a strong optical standing wave in vacuum modifies the vacuum polarization effects on the propagation of an x-ray beam. Since the strong laser beam is tightly focused, the typical length of the interaction region between the strong standing wave and the x-ray probe is of the order of 1 µm. This leads to diffraction effects that result in a modification of the probe polarization change due to vacuum polarization. This is schematically shown in Fig.3.

If initially the probe is linearly polarized, then, after the interaction, it becomes elliptically polarized and the main axis of the ellipse rotated in comparison to the initial polarization direction. Both the ellipticity and the rotation of polarization decrease at far observation distances where the deteriorating role of diffraction increases (see Fig.4). The diffraction effects are significant quantitatively and qualitatively. In fact, without diffraction effects the rotation of the polarization disappears and the ellipticity remains constant in the case considered in Fig.4, equal to 4x10^-4 rad. In real experimental conditions, however, the effect is more than ten times smaller. Nevertheless, in [2] it is shown that, in principle, values of the ellipticity and the polarization rotation angle like those in Fig. 1 are measurable.

Enhancement of vacuum polarization in plasma

We have shown in [3] that an enhancement of vacuum polarization effects in plasma can be obtained if a strong laser pulse propagates through plasma near the threshold of the plasma transparency. In the proximity of this singular point, the plasma refractive index tends to zero, the field increases and conditions arise under which the vacuum refractive index becomes more visible. We have proposed an experimental setup for the observation of this effect .

Photon splitting in a laser field

Photon splitting in a laser field due to vacuum polarization is considered in [4]. Using an operator technique, we derive the amplitudes for arbitrary strength, spectral content, and polarization of the laser field. The case of a monochromatic circularly polarized laser field is studied in detail, and the amplitudes are obtained as threefold integrals. The asymptotic behavior of the amplitudes for various limits of interest is investigated also in the case of a linearly polarized laser field. On the basis of the results obtained, the possibility of experimental observation of the process is discussed.

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