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Theory Division

Theoretical Quantum Dynamics and Quantum Electrodynamics

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High-Energy Quantum Electrodynamics


Quantum electrodynamics (QED) was formulated in its final form by Feynman, Schwinger, and Tomonaga, and it is the most successful among the physical theories in terms of agreement with experimental data. The validity of QED has been thoroughly scrutinized with great precision at high energies by means of particle accelerators and in bound atomic systems by means of highly-charged ions. Modern high-power laser facilities are becoming an alternative tool to test QED in the highly nonlinear regime and in the so-called strong-field sector (see the review A. Di Piazza et al., Rev. Mod. Phys. 84, 1177 (2012)).

In the strong-field QED regime elementary QED processes occur in the presence of electromagnetic fields with amplitudes effectively of the order of Ecr=1.3 × 1016 V/cm and Bcr=4.4 × 1013 G. A laser beam with an electric (magnetic) field amplitude of the order of Ecr (Bcr) would have a peak intensity of Icr=4.6 × 1029 W/cm2. In the presence of such strong electromagnetic fields fascinating phenomena take place and become sizable. Among them we quote spontaneous electron-positron pair production from vacuum, harmonic generation in the collision of two strong laser beams in vacuum, and the fact that the vacuum becomes a birefringent and dichroic medium. All these phenomena have their ultimate origin in the pure quantum interaction among electromagnetic fields also in vacuum, this interaction being mediated mainly by virtual electron-positron pairs (vacuum fluctuations, Fig. 1). Indeed, unlike Maxwell's equations in vacuum, the quantum equations describing the evolution of electromagnetic fields in vacuum are highly nonlinear.

Fig. 1. Cartoon view of the quantum vacuum where particle-antiparticle pairs (vacuum fluctuations) are continuously created and annihilated.

In this context high-power optical laser facilities that are already or soon available represent a unique tool to investigate experimentally the predictions of QED. Multiterawatt laser systems have already allowed for the generation of intensities beyond 1022 W/cm2. Numerous Petawatt laser facilities are either operating or under construction aiming at laser intensities of the order of 1023 W/cm2. Moreover, intensities of the order of 1024-1025 W/cm2 are envisaged at the Extreme Light Infrastructure (ELI) and at the Exawatt Center for Extreme Light Studies (XCELS). Although even the intensities available at future laser facilities will lie well below the critical intensity, the strong-field QED regime can be effectively entered by employing ultrarelativistic electron beams or photon beams with energies much larger than the electron rest energy. The reason is that the Lorentz- and gauge-invariant parameter controlling quantum nonlinear effects in the laser intensity is related to the laser intensity which charged particles involved in the reaction experience in their rest frame. Now, available accelerator facilities and the recent accelerator technique known as laser wake-field acceleration provide electron beams with energies beyond the GeV threshold. Thus, it can be easily ascertained that present-day electron and laser technology already allow in principle to enter the strong-field QED regime.

We investigate some main aspects of QED in the presence of intense laser fields:

  1. The nonlinear dielectric properties of the quantum vacuum. QED predicts that the presence of a strong electromagnetic field in the vacuum changes the dielectric properties of the vacuum itself, rendering it a birefringent medium. In order to reveal these properties strong electromagnetic fields are needed of the order of the critical fields. Alternatively, ultra-high precision polarimetry can be employed to reveal tiny modifications in the polarization of probe fields propagating through regions where a strong background field is present.
  2. High-energy QED processes in intense laser fields. QED processes where high-energy electron or photon beams collide with intense laser fields are of great interest both theoretically and experimentally. The corresponding probabilities show a very complex and highly nonlinear dependence on the laser field parameters giving the possibility of testing QED with high accuracy. Among these processes we have studied single and double nonlinear Compton scattering and nonlinear Breit-Wheeler pair production.
  3. Radiation-reaction effects in intense laser fields. In the presence of intense laser fields, the interaction of an electron with its own electromagnetic field can change significantly the electron dynamics (radiation reaction). We have recently studied the problem of radiation reaction both classically (Landau-Lifshitz equation) and quantum mechanically (the quantum origin of radiation reaction) and we have put forward experimental setups potentially allowing for measuring such effects.
  4. Electromagnetic cascades. If two or more laser beams collide in a region where bound or free electrons are present, the latter are violently accelerated emitting high-energy photons. These in turn interact with the laser beams transforming into electron-positron pairs, which again can emit high-energy photons. If the intensities of the colliding lasers is initially sufficiently high (indicatively of the order of 1024 W/cm2 or higher), an avalanche or cascade of electrons, positrons, and photons is initiated. Among other reasons, the study of such electromagnetic cascades is attracting a lot of attention because of the complex dynamics of the generated particles and its importance for reproducing at lower energy and density scales in the laboratory the conditions present in some astrophysical scenarios like supernova explosions.

People:

June 2019, from left to right: Maitreyi Sangal, Archana Sampath, Tobias Wistisen, Tobias Podszus, Antonino Di Piazza, Sergey Bragin, Matteo Tamburini.

Open positions:

  • No available positions at the moment. Excellent candidates are invited to directly contact Antonino Di Piazza at any time.

Selected projects

Improved local-constant-field approximation for strong-field QED codes

The local-constant field approximation (LCFA) is a powerful and thus widely-used approximation in strong-field QED because it allows to compute transition probabilities in arbitrary background electromagnetic fields starting from the corresponding expressions in a constant crossed field, i.e., a constant electromagnetic field with electric and magnetic fields having the same amplitude and being perpendicular to each other. The LCFA is particularly suitable for strong-field QED in intense laser fields because it is applicable, generally speaking, when the laser field is able to accelerate an electron to untrarelativistic energies already in a single cycle [1]. This condition should imply that a generic strong-field QED process is formed over a distance, called formation length, which is much smaller than the laser wavelength, such that the laser field can be approximated as constant in order to compute the probability of the process itself.
However, we have realized that this condition is insufficient to guarantee the validity of the LCFA for the process of photon emission (nonlinear Compton scattering) in the infrared part of the emission spectrum [2]. The intuitive explanation of this failure is that low-energy photons have relatively large wavelengths such that it is not surprising that at a certain point the background field can no longer be approximated as being constant on the photon formation length.
As a related result, we have found a local method to capture the nonlocal features of the radiation probability for low-energy photons in a way that it can still be efficiently implemented in numerical codes [3]. The virtue of this method can be seen in Fig. 1, which shows photon emission spectra according to a full quantum calculation (solid red curve), to the LCFA (dotted, black curve), and to the method developed in Ref. [3] (dashed blue curve).
     

Fig. 1. Photon mission probability spectra as functions of the photon light-cone energy in units of the emitting electron light-cone energy. The solid, red curve corresponds to the full quantum calculation, the dotted, black curve corresponds to the LCFA, and the dashed blue curve to the method developed in Ref. [3]. Figure adapted from Ref. [3], copyright of the American Physical Society.


[1] V. I. Ritus, J. Sov. Laser Res. 6, 497 (1985).
[2] A. Di Piazza, M. Tamburini, S. Meuren, and C. H. Keitel, Implementing nonlinear Compton scattering beyond the local-constant-field approximation, Phys. Rev. A 98, 012134 (2018).
[3] A. Di Piazza, M. Tamburini, S. Meuren, and C. H. Keitel, Improved local-constant-field approximation for strong-field QED codes, Phys. Rev. A 99, 022125 (2019).

Quantum limitation to the coherent emission of accelerated charges

In classical electrodynamics, accelerated charges emit electromagnetic radiation. By virtue of the superposition principle, the total emitted electromagnetic field is the sum of the fields emitted by each charge [1], whereas the total radiated energy is quadratic in the amplitude of the emitted electromagnetic field. It can be shown that if accelerated charged particles move along sufficiently close trajectories, the total energy emitted scales quadratically with their number.
The process of emission of radiation by charged particles, accelerated by an intense laser field, can be studied at the quantum level within strong-field QED [2]. In this framework the "quantum nonlinearity parameter" generally describes the importance of quantum effects like the photon recoil [2]. In the case of a single emitting charge, when the quantum nonlinearity parameter is much smaller than unity, the predictions of strong-field QED for the emitted energy spectrum agree with the classical ones. One may naively expect the same to hold true for the process of emission of radiation by multiple charges.
By investigating the emission by a two-electron wavepacket in the presence of an electromagnetic plane wave within strong-field QED, we have shown [3] that quantum effects deteriorate the coherence predicted by classical electrodynamics even if the typical quantum nonlinearity parameter of the system is much smaller than unity (see Fig. 1). We explain this result by observing that coherence effects are also controlled by a new quantum parameter which relates the recoil undergone by the electron in emitting a photon to the width of its wave packet in momentum space [3].
     

Fig. 1. Classical (dash-dotted red curve) and quantum (solid black curve) emitted energy spectra by two electrons for numerical parameters given in Ref. [3]. The two electrons have the same initial (average) energy and the corresponding (average) quantum nonlinearity parameter is 0.02 (see Ref. [3] for additional details). Figure from Ref. [3], copyright of the American Physical Society.


[1] J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1999).
[2] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012).
[3] A. Angioi and A. Di Piazza, Quantum limitation to the coherent emission of accelerated charges, Phys. Rev. Lett. 121, 010402 (2018).

Experimental evidence of quantum radiation reaction in aligned crystals

In classical electrodynamics there is a century old outstanding problem, the so-called radiation reaction problem [1]. When a charge, an electron, for definiteness, is accelerated, it emits radiation, and this should be taken into account in the description of the subsequent motion of the electron. The original attempt at a solution of this problem, resulted in the so-called Lorentz-Abraham-Dirac equation, which is a 3rd order differential equation and can be shown to conflict with the principle of causality. More recently this has been amended by a scheme of reduction of order, leading to the so-called Landau-Lifshitz equation, which is free of these problems [2].
The quantum mechanical picture of the problem of radiation reaction under certain approximations is that of an electron consecutively emitting several photons. For radiation reaction effects to be treated quantum mechanically, the electron should be driven by strong electromagnetic fields, such that the quantum nonlinearity parameter &chi, i.e., the ratio of the field acting on the charge in the electron's instantaneous rest-frame and the Schwinger critical electric (magnetic) field strength Ecr (Bcr), is of the order of unity or larger.
In the experiment described in [3] we achieved such strong fields by channeling positrons with energy of 180 GeV through a Silicon crystal. In addition, the crystal is thick enough that on the order of 20 photons are emitted per incoming positron. Now, the calculation of the emission spectrum corresponding to several photons is in general a formidable task. Therefore one must rely on an approximation, the so-called constant crossed field approximation, where the emission of many photons can be calculated knowing only the emission spectrum of the single photon emission process. In our experiment, however, the conditions for this approximation to be applicable are not completely fulfilled, and we see some deviations between theory and experiment due to this in the case of a crystal with 3.8-mm thickness (see Fig. 1a). The results in the 10.0-mm case theory and experiments agree to a better degree (see Fig. 1b). In both cases, we refer to the theoretical results obtained by treating quantum mechanically the emission of several photons (red lines), whereas the theoretical models corresponding to the other curves are described in detail in [3].
     

Fig. 1. Experimental and simulated power spectra. Background subtracted power spectra in the aligned case for two crystal thicknesses: 3.8 mm (a) and 10.0 mm (b). The experimental data are compared to the four different theoretical models, described in [3], after being translated through the simulation of the experimental setup. The error bars are due only to the statistical counting error in each bin. Figure from Ref. [3], copyright of the Nature Publishing Group.


[1] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012).
[2] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Elsevier, Oxford, 1975).
[3] T. N. Wistisen, A. Di Piazza, Helge V. Knudsen, and U. I. Uggerhøj, Experimental evidence of quantum radiation reaction in aligned crystals, Nat. Commun. 9, 795 (2018).

High-energy vacuum birefringence and dichroism in an ultrastrong laser field

In the realm of classical electrodynamics the electromagnetic field experiences no self-interaction in vacuum. According to QED, however, a finite photon-photon coupling is induced by the presence of virtual charged particles in the vacuum [1]. For low-frequency electromagnetic fields such vacuum polarization effects are described by the Euler-Heisenberg Lagrangian density [2]. The Euler-Heisenberg Lagrangian density predicts that the vacuum resembles a birefringent medium. Despite having been predicted long time ago, vacuum birefringence has not been observed in a laboratory experiment yet, due to the smallness of the photon-photon coupling.
As the light-by-light scattering cross section attains its maximum at the pair-production threshold [2], it is natural to consider high-energy photons to probe vacuum birefringence. In [3] we have derived how a generally polarized probe photon beam is influenced by both vacuum birefringence and dichroism. Furthermore, we have considered an experimental scheme to measure these effects in the high-energy regime, where the Euler-Heisenberg approach breaks down. The scheme is based on Compton backscattering to produce polarized gamma photons [2] and exploits pair production in matter to determine the polarization state of the probe photon after it has interacted with a strong laser pulse.
By analyzing the consecutive stages of this type of experiment, we have shown that for vacuum birefringence the required measurement time is reduced by two orders of magnitude if a circularly polarized probe photon beam is employed. Assuming conservative experimental parameters, we demonstrate that the verification of the strong-field QED prediction for vacuum birefringence is feasible with an average statistical significance of 5σ on the time scale of several days at upcoming 10-PW laser facilities. We also show that vacuum dichroism and anomalous dispersion in vacuum (see Fig. 1) could be accessible at these facilities.
     

Fig. 1. Relative phase shift of the probe photon polarization components along and perpendicular to the intense laser polarization, after the propagation through the intense laser pulse. The quantum nonlinearity parameter χ characterizes the center-of-momentum energy of the collision, the classical intensity parameter ξ characterizes the strength of the intense laser field, N is the number of cycles of the intense laser pulse. For each of the three laser facilities gamma photons with energy 0.1 GeV (left point), 0.5 GeV (central point), and 1 GeV (right point) are indicated. Note that the decrease in the relative phase shift for χ ≳ 2.5 characterizes the anomalous dispersion of the vacuum. Figure from Ref. [3], copyright of the American Physical Society.


[1] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012).
[2] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Elsevier, Oxford, 1982).
[3] S. Bragin, S. Meuren, C. H. Keitel, and A. Di Piazza, High-energy vacuum birefringence and dichroism in an ultrastrong laser field, Phys. Rev. Lett. 119, 250403 (2017).

Nonlinear Breit-Wheeler pair production in a tightly focused laser beam

The electric field strength where such nonlinear QED effects become sizable identifies the "strong-field QED" regime and is given by the so-called Schwinger field: Ecr=1.3 × 1016 V/cm. Due to the extremely large value Ecr, present and upcoming lasers have to be tightly focused in space (and in time) in order to aim at values comparable with Ecr. Nonetheless, the value of Ecr exceeds by about four orders of magnitudes presently available laser-field amplitudes [1,2]. However, the effective field at which a QED process occurs is that experienced by participating charged particles in their rest frame. Thus, by employing ultrarelativistic electron (positron) beams, the strong-field QED regime can effectively be probed also nowadays in principle.
Now, all systematic approaches to investigate analytically strong-field QED processes rely on approximating the laser beam as a plane wave, which allows for solving the Dirac equation exactly but which cannot account for laser spatial focusing effects. In [3] we have realized that in order to enter the strong-field QED regime at present and upcoming laser facilities, the electrons have to be so highly relativistic that the Wentzel-Kramers-Brillouin (WKB) approximation can be employed (at the next-to-the-leading order) to solve analytically the Dirac equation in the presence of a background laser field practically of arbitrary space-time shape. The electron wave functions obtained in this way open the possibility of investigating analytically and in a systematic way strong-field QED processes in the presence of a tightly focused laser beam of complex and realistic space-time shape by employing the so-called Furry picture. Indeed, we have already determined analytically the energy spectrum and the angular distribution of the electron-positron pairs produced in the collision of a photon bunch with an intense and tightly-focused laser beam (nonlinear Breit-Wheeler pair production) [4]. As a byproduct, by means of a numerical implementation of the analytical results, we have proven that the inclusion of the laser tight focusing is essential for a correct quantitative estimate of the number of created pairs (see Fig. 1).
     

Fig. 1. Angular resolved positron energy distribution produced via nonlinear Breit-Wheeler pair production in a focused Gaussian beam (black continuous curves) and in a plane wave (red dashed curves) at different values of the observation polar angles with respect to the direction of propagation of the incoming photon (the azimuthal angle is zero in all cases). Figure from Ref. [4], copyright of the American Physical Society.


[1] V. I. Ritus, J. Sov. Laser Res. 6, 497 (1985).
[2] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012).
[3] A. Di Piazza, Phys. Rev. Lett. 113, 040402 (2014).
[4] A. Di Piazza, Nonlinear Breit-Wheeler pair production in a tightly focused laser beam, Phys. Rev. Lett. 117, 213201 (2016).

Nonlinear single Compton scattering of an electron wave packet

According to QED, an electron interacting with an intense laser field can emit a photon, while exchanging many photons with the laser field itself [1]. This process is indicated as nonlinear single Compton scattering.
Due to the progress in the technology of ultrashort few-cycles laser pulses [2], there are already many envisaged laser facilities that will soon allow testing QED in the high-intensity regime and probe nonlinear single Compton scattering.
In particular, motivated by the fact that electrons are produced in general as wave packets and that they are localized to some extent in beams, we focused on the effect that a certain energy width of the initial wave packet has on the emitted energy spectrum of nonlinear single Compton scattering [3]. We have shown that due to energy-momentum conservation laws and on-shell conditions one does not find quantum interference in the spectrum of the different momentum components of the wave packets, such that the spectrum itself is indeed only the average of the spectra corresponding to electrons having initially different but definite momenta. Moreover, we have analyzed to which extent the indeterminacy of the initial momentum of the electron alters the photon spectra. The most typical effect of the indetermination of the initial state is to wash out the highly oscillating structure of the spectrum obtained for electrons with definite momentum, as well as the broadening the angular emission range. At a given relative indeterminacy, we have shown that the one on the electron energy affects the photon spectra more significantly than that on the laser photon energy.
     

Fig. 1. A typical collection of emission spectra along the negative z-direction for electrons colliding head-on (or almost head-on) with a very intense pulse propagating along the positive z-direction. The different spectra are obtained by setting to zero the initial electron momentum along the electric field (lower panel) or along the magnetic field (upper panel) of the laser beam and varying the other component. Figure from Ref. [3], copyright of the American Physical Society.


[1] V. I. Ritus, J. Sov. Laser Res. 6, 497 (1985).
[2] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012).
[3] A. Angioi, F. Mackenroth, and A. Di Piazza, Phys. Rev. A 93, 052102 (2016) (Editors' Suggestion).

High-energy recollision processes of laser-generated electron-positron pairs

In quantum field theory in vacuum, Feynman diagrams with particle loops correspond to quantum fluctuations, with the extension, for example, of an electron-positron loop being limited to the Compton space-time scale by the Heisenberg uncertainty principle [1]. However, the situation changes profoundly in the presence of a strong laser field [2]. In fact, above the threshold for real electron-positron pair production the laser field can transfer enough energy to the electron and the positron in the loop to significantly increase the space-time extension of the loop itself. Correspondingly, the intermediate electron and positron can be accelerated over a macroscopic distance (i.e. of the order of the laser wavelength) rather than over a microscopic one (i.e. of the order of the Compton wavelength), and gain an energy corresponding to many laser photons. A careful analysis of the trajectory of an electron (positron) inside a linearly polarized plane-wave field reveals that for certain initial conditions the high-energy electron and positron can recollide and annihilate providing energy and momentum for secondary reactions as in an "electron-positron vacuum collider".
The simplest Feynman diagram which contains an electron-positron loop is the leading-order contribution to the polarization operator [1]. The complete evaluation of the square of the polarization operator (see the gray curve in Fig. 2) shows that both vacuum fluctuation-type processes (yellow curve), corresponding to the creation and the annihilation of the electron-positron pair within the same microscopic formation region, and recollision-type processes (red curve), corresponding to the creation and the annihilation of the electron-positron pair being separated by a macroscopic distance of the order of the laser wavelength, contribute to the probability of absorbing n laser photons [2]. The recollision contribution is responsible of the existence of a large plateau region in the spectrum, in close analogy to high-harmonic generation in atomic physics.
     

Fig. 1. Probability for the absorption of n laser photons by an electron-positron loop (in arbitrary units). Yellow curve: vacuum-fluctuation-type contribution, red curve: recollision-type contribution, gray curve: full numerical evaluation. Figure from Ref. [2], copyright of the American Physical Society.


[1] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Elsevier, Oxford, 1982).
[2] S. Meuren, K. Z. Hatsagortsyan, C. H. Keitel and A. Di Piazza, Phys. Rev. Lett. 114, 143201 (2015).

Generation of neutral and high-density electron-positron pair plasmas in the laboratory

Electron-positron plasmas are emitted as ultrarelativistic jets in different astrophysical scenarios under extreme conditions, like during gamma-ray bursts. In collaboration with Dr. Gianluca Sarri and Prof. Matt Zepf, from the Queen's University Belfast, we have generated such a unique state of matter in the laboratory [1] in the collision of an ultrarelativistic electron beam with a Lead solid target. As a consequence of the complex interaction of the electron beam with the nuclei and the electrons in the target, an ultra-relativistic electron-positron bunch was observed on the rear side of the solid target, with a fraction of electrons and positrons depending on the target thickness (see Fig. 1). The density of the bunch was found to be sufficiently high that its skin-depth resulted smaller than the bunch transverse size, allowing for collective, i.e., plasma effects.
We have identified the main mechanisms responsible for the production of the electron-positron bunch and described its formation and evolution inside the solid target. A simple model has been put forward, which, among all possible interactions occurring inside the solid target, includes only two fundamental quantum electrodynamical processes: 1) bremsstrahlung of electrons and positrons, and 2) electron-positron photoproduction of photons, both occurring in the presence of the screened electromagnetic field of the solid target atomic nuclei. Analytical estimations and numerical integrations of the corresponding kinetic equations agree extremely well with the experimental results on the relative population of electrons and positrons in the generated beam (see in particular the blue dots and the green dashed line in Fig. 1c). Absolute electron and positron yields were also very well predicted by the model apart from an overall factor of the order of unity. In order to reproduce theoretically also more detailed features of the experimental results, Dr. Gianluca Sarri has employed the available fully integrated particle physics Monte-Carlo simulation code FLUKA (see in particular the red crosses in Fig. 1), which among others also includes electron-electron and electron-positron interactions, atomic scattering and other breaking mechanisms, together with high-energy processes like muon--anti-muon pair production.
     

Fig. 1. Comparison of experimental results and theoretical predictions of the number of electrons (part a), number of positrons (part b), and the derived fraction of positrons (part c) in the generated ultra-relativistic bunch. Figure from Ref. [1], copyright of Nature's Publishing Group.


[1] G. Sarri, K. Poder, J. M. Cole, W. Schumaker, A. Di Piazza, B. Reville, T. Dzelzainis, D. Doria, L. A. Gizzi, G. Grittani, S. Kar, C. H. Keitel, K. Krushelnick, S. Kuschel, S. P. D. Mangles, Z. Najmudin, N. Shukla, L. O. Silva, D. Symes, A. G. R. Thomas, M. Vargas, J. Vieira, and M. Zepf, Generation of neutral and high-density electron-positron pair plasmas in the laboratory, Nature Commun. 6, 6747 (2015).

Plasma-based generation and control of a single few-cycle high-energy ultrahigh-intensity laser pulse

A wide range of novel studies in nonlinear optics as well as the major new regimes of extreme field physics require laser pulses which simultaneously exhibit the following three key features: few-cycle duration, high-energy and ultrahigh intensity. Already in nonrelativistic atomic physics, it has been demonstrated that quantum processes can be controlled by manipulating the pulse shape of few-cycle laser pulses [1]. In order to achieve the same goal also in the ultrarelativistic regime and in the realm of strong-field QED, few-cycle laser pulses with tunable carrier-envelope phase (CEP) are required with peak intensities largely exceeding 1020 W/cm2 [2]. In [3] we put forward the concept of a laser-boosted solid-density parabolic relativistic "mirror", interacting with a superintense counterpropagating laser pulse, to generate a CEP tunable few-cycle pulse with multi-joule energy and peak intensity exceeding 1023 W/cm2. It is found that a heavy and therefore relatively slow mirror should be employed to maximize the intensity and the energy of the reflected pulse. This counterintuitive result is explained with the larger reflectivity of a heavy foil, which compensates for its lower relativistic Doppler factor. Moreover, since the counterpropagating pulse is ultrarelativistic, the foil is abruptly dispersed and only the first few cycles of the counterpropagating pulse are reflected. Our multi-dimensional particle-in-cell simulations show that a single few-cycle, multi-petawatt laser pulse with several joule of energy and with peak intensity exceeding 1023 W/cm2 can be generated already employing next-generation high-power laser systems. In addition, the carrier-envelope phase of the generated few-cycle pulse can be tuned provided that the carrier-envelope phase of the initial counterpropagating pulse is controlled.      

Fig. 1. Evolution of the electromagnetic energy density (first row, normalized units), and the electron density (second row, normalized units) showing the generation of a few-cycle reflected pulse with 5.8 fs duration, 6.8 J energy and 2.3 × 1023 W/cm2 peak intensity. Figure from Ref. [3], copyright of the American Physical Society.


[1] F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009).
[2] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012).
[3] M. Tamburini, A. Di Piazza, T. V. Liseykina, and C. H. Keitel, Phys. Rev. Lett. 113, 025005 (2014).

Stochasticity effects in quantum radiation reaction

When an electron is accelerated by an electromagnetic field, the emission of photons carrying away energy and momentum leads to a modification of the electron trajectory. In the realm of classical electrodynamics, this backreaction is called radiation reaction (RR) and is described by the so-called Landau-Lifshitz (LL) equation. Furthermore, in the ultra-relativistic, "nonlinear moderately quantum" regime, where nonlinear effects in the laser field and nonlinear QED effects are important, whereas pair production can still be neglected, RR can be described as the incoherent emission of many photons. In [2] we studied the head-on collision of an intense laser pulse with an ultrarelativistic electron beam by means of a kinetic approach. Fig. 1 shows that the emission of radiation broadens the initial electron distribution drastically for the full quantum calculations. On the other hand, the classical calculations according to the LL equation predict a strong narrowing of the electron energy spectra due to RR effects. Moreover, if the classical radiation intensity is substituted by its quantum analogue [3] in the classical equation, the energy width of the electrons would still be decreased during the interaction with the laser pulse. The peculiar difference of RR in the classical and the quantum regime can be understood by the importance of the stochastic nature of photon emission in the latter. While in classical electrodynamics the effects of stochasticity in the radiation process are negligible and, in turn, the electron dynamics can be characterized by deterministic equations, it becomes crucial in the quantum regime. Even in the case, where quantum effects are relatively small, the classical kinetic equations must be modified by an additional stochastic term inducing a spread of the electron energy distribution [2].      

Fig. 1. Comparison of the phase evolution of the electron distribution as functions of the electron energy for a 10-cycle pulse (part a)) employing the full kinetic approach (part b)), the classical radiation intensity (part c)), and the quantum radiation intensity (part d)). Figure from Ref. [2], copyright of the American Physical Society.


[1] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Elsevier, Oxford (1975).
[2] N. Neitz and A. Di Piazza, Phys. Rev. Lett. 111, 054802 (2013).
[3] V. I. Ritus, J. Sov. Laser Res. 6, 497 (1985).

Table-top laser-based source of femtosecond, collimated, ultrarelativistic positron beams

Ultra-relativistic, highly-collimated positron beams have been generated so far by employing large-scale electron accelerators. The electron beams produced by the accelerator collides with a solid target (typically gold or lead), generating on the rear side electron-positron bunches, whose oppositely-charged constituents can be later separated. The main mechanism responsible of such process is the emission of photons via bremsstrahlung by the incoming electrons deflected by the ions in the solid target and the consequent transformation of the bremsstrahlung photons into electron-positron pairs when still in the target. In collaboration with the experimental group lead by Dr. Gianluca Sarri and Prof. Matt Zepf at the Queen's University Belfast, we have generated an incoming monoenergetic electron beam via laser wake-field acceleration (see Fig. 1). After letting it interact with a thin, solid target, we succeeded in producing for the first time positron beams of short (~ 30 fs) duration, with ultra-relativistic energies (>100 MeV), and with a narrow angular distribution (~ 3 mrad) in a table-top setup. The possibility of generating such high-energy lepton beams is of central importance for astrophysics due to their similarity to jets of long gamma-ray bursts.      

Fig. 1. Table-top experimental setup for the production of short, narrow, and ultra-relativistic positron beams. Figure from Ref. [1]. Copyright of the American Physical Society.


[1] G. Sarri et al., Phys. Rev. Lett. 110, 255002 (2013)


Nonlinear double Compton scattering in the ultrarelativistic quantum regime

An electron scattered from an ultra-intense laser pulse will emit radiation which in QED is described as the emission of photons. Next to the leading order effect of the electron emitting only one single photon, it may also emit several photons, the lowest-order being the emission of two photons. This process is called nonlinear double Compton scattering. The possible detection of two-photon emission thus attracts considerable attention. It was, however, shown recently that in typical experimentally realizable scenarios, the signal of nonlinear double Compton scattering will almost always be superseded by the much larger single-photon background. Thus, its detection can only be achieved by intricate coincidence measurements. It would thus be desirable to discover a parameter regime where the strong single-photon background is suppressed and thus the detection of the two-photon signal becomes possible. We have performed this task by demonstrating an angular separation between the single- and two-photon signals [1]. This scheme takes advantage of the fact that the single photon signal is confined to a narrow emission cone [2], as well as the fact that an electron will lose energy upon the emission of a photon. Thus, by working in the full quantum regime, where the recoil do to photon emission is significant, the trajectory of the electron after the emission of a photon inside the laser field will be substantially altered. This change of the trajectory then leads to a changed angular distribution of the emitted radiation, such that any subsequently emitted second photon is likely to be emitted outside of the single-photon emission cone.      

Fig. 1. Two-photon emission spectra for observation of both photons inside the single-photon emission cone [a)] and one photon observed outside this cone [b)]. The fact that there is radiation predicted outside the emission cone is due to a significantly changed electron trajectory after the first photon emission [c)]. The threshold frequency, that the first photon has to exceed to exert enough recoil on the electron in order to facilitate emission towards the given observation direction of the second photon, is well reproduced by this picture of two smoothly joint classical trajectories [d)]. Figure from Ref. [1], copyright of the American Physical Society.


[1] F. Mackenroth and A. Di Piazza, Phys. Rev. Lett. 110, 070402 (2013).
[2] F. Mackenroth and A. Di Piazza, Phys. Rev. A 83, 032106 (2011).

Peak intensity measurement of relativistic lasers via nonlinear Thomson scattering

The analysis of experiments employing a ultrarelativistic optical laser pulse requires the precise knowledge of its peak intensity. However, ultrarelativistic peak intensity measurements are especially difficult and available methods only allow for the determination of the order of magnitude of the peak intensity of such strong beams. In [1] we have proposed a new method, which allows in principle to determine the laser peak intensity of such intense pulses, peak intensities of the order of or larger than 1020 W/cm2, with an accuracy of about 10 %. The method relies on the high directionality of the electromagnetic radiation emitted by an ultrarelativistic charged particle, an electron for definiteness. This feature implies that the electron emits instantaneously along its velocity, such that the angular aperture of the energy spectrum radiated by the electron interacting with a strong laser beam is directly related to the peak intensity of the beam itself. In Fig. 1 a typical energy spectrum is shown together with the theoretical predictions for the angular aperture. The vertical dashed line indicates the position of the end of the spectrum according to the Lorentz dynamics, which neglects radiation-reaction effects [2]. Such effects are included via the Landau-Lifshitz force [2] and slightly increase the predicted value of the angular aperture of the emission spectrum. The excellent agreement between the numerical results and the analytical predictions indicates the theoretical validity of the method. Our method is valid up to intensities of the order of 1023 W/cm2, when quantum radiation-reaction effects start affecting substantially the electron dynamics [1].      

Fig. 1. Angular-resolved emitted energy spectrum (inner plot) and total energy emitted per unit solid angle (outer plot). Vertical red lines indicate our theoretical predictions for the maximal emission angle, with (solid line) and without (dashed line) radiation reaction included. Figure from Ref. [1], copyright of the Optical Society of America.


[1] O. Har-Shemesh and A. Di Piazza, Opt. Lett. 37, 1352 (2012).
[2] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Elsevier, Oxford (1975).

Quantum electron self-interaction in a strong laser field

Electrons which interact with plane-wave laser fields are normally described by Volkov states in strong-field QED calculations. These are exact solutions of the interacting Dirac equation with a classical four-potential and thus the background electromagnetic plane-wave field is taken into account exactly from the outset. However, the radiation field (which describes all modes not populated by the laser field) is a quantized field and induces radiative corrections to the electron states (which can in this case be interpreted as the interaction of the electron with its on electromagnetic field). Contrary to the vacuum case, radiative corrections in strong-field QED also affect the properties of on-shell particles. In [1] we have calculated the leading-order modifications of the Volkov-states in the fine-structure constant by solving the Schwinger-Dirac equation. We have shown that the quasi-momentum describing an electron inside a (quasi-)monochromatic laser field undergoes a pure quantum contribution due to radiative corrections. Beside this, the spin-dynamics of the electron is significantly altered due to the electron self-interaction. For a plane-wave laser field the Dirac equation predicts that the quasi-energy of the electron inside the laser-field is degenerated with respect to the spin quantum number. Similar to the Lamb-shift for electrons bound to a nucleus, non-linear QED effects remove this degeneracy. In [1] we have suggested an experiment which could measure this effect, in principle, with available technology. To this end the spin asymmetry of electrons (which did not radiate) is measured after their interaction with a short laser-pulse (see Fig. 1). According to the Dirac equation, the calculated asymmetry should be zero. A measurement of a non-zero spin asymmetry would therefore be a clear signature for non-linear quantum effects induced by the electron's interaction with its own field.

Fig. 1.Expected spin asymmetry as a function of the laser peak intensity and the laser carrier-envelope phase (CEP) for an optical laser pulse with a duration of 8 fs. The laser pulse is linearly polarized along the x-direction and collides head-on with electrons having an energy of 500 MeV and a spin-polarization along the y-direction. After the interaction, the spin of the electron is measured along the z-direction. Figure from Ref. [1], copyright of the American Physical Society.


[1] S. Meuren and A. Di Piazza, Phys. Rev. Lett. 107, 260401 (2011).

Quantum radiation reaction effects in multiphoton Compton scattering

In the realm of classical electrodynamics, when an electric charge, an electron for definiteness, is accelerated by a background electromagnetic field, it emits electromagnetic radiation and the associated energy-momentum loss alters the electron's trajectory [1]. The radiation-reaction problem is the determination of the equation of motion of an electron by including self-consistently the effects of the emitted radiation on the electron’s motion. At a more fundamental level, we have asked ourselves what is the quantum origin of radiation reaction In [2], we have answered this question by identifying quantum radiation reaction in the multiple incoherent emissions of photons by the electron driven by an external field (the case of a laser field was explicitly carried out in [2]). In general, the self-consistent inclusion of radiation-reaction effects in the full quantum regime amounts in completely determining the evolution of the quantum state representing a single electron initially free, which then enters the background field. This is, of course, a formidable task, as it also involves, e.g., electron-positron pair production originating from the photons emitted by the electrons. Thus, in [2] we have limited ourselves to the so-called moderately quantum regime, which is experimentally relevant and where essentially pair-production remains negligible. Thus, the problem is still single-particle and a clear comparison between classical and quantum results is feasible. In Fig. 1 classical and quantum spectra with and without radiation-reaction effects included are shown. Inclusion of radiation-reaction effects in the quantum regime has mainly three effects: (i) increase of the spectral yield at low energies, (ii) shift to lower energies of the maximum of the spectral yield, and (iii) decrease of the spectral yield at high energies. The physical reason is that, due to radiation reaction, the electron loses its energy by emitting several relatively low-energy photons when the laser field is not at the maximum amplitude yet, and the probability of emitting one photon in the high-energy region is less than if radiation reaction is neglected. Figure 1 also shows that the classical treatment of radiation reaction (short dashed, blue curve) artificially enhances the above three effects of radiation reaction, which is due essentially to the classical overestimation of the average energy emitted by the electron.

Fig. 1.Multiphoton Compton spectra as a function of the photon energy in units of the initial electron energy calculated quantum mechanically with (solid, black line with error bars, see Ref. [2] for details) and without (long dashed, red line) radiation reaction. For the sake of comparison, the corresponding classical spectra with (short dashed, blue line) and without (dotted, magenta line) radiation reaction are also shown (see [2] for additional numerical details). Figure from Ref. [2], copyright of the American Physical Society.


[1] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Elsevier, Oxford (1975).
[2] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Rev. Lett. 105, 220403 (2010).

Determining the carrier-envelope phase of intense few-cycle laser pulses

An important ingredient of the generation of ultrastrong laser pulses is the compression of the laser energy to shorter and shorter time scales. Temporal compression down to two laser cycles or even to one single laser cycle has already been achieved experimentally in different frequency ranges. In this few-cycle regime the response of atoms and molecules to a laser pulse becomes dependent on the carrier envelope phase (CEP) of the driving field, i.e., the phase difference between the carrier wave and the envelope function. Experimental determination of the CEP, however, has been possible so far only for laser intensities up to 1014-1015 W/cm2. Presently available peak laser intensities are of the order of 1022 W/cm2 and ultrashort laser pulses of such high intensities are envisaged (see for example the PFS project under development in Garching, Germany). Therefore, it is highly desirable to have a procedure to determine the CEP of short laser pulses also when their intensity largely outruns the realm of applicability of the traditional determination schemes. We have theoretically described a method of determining in principle the CEP of a strong (intensity larger than 1020 W/cm2) short laser pulse by employing multiphoton Compton scattering [1]. The method exploits the fact that an ultrarelativistic electron emits radiation almost exclusively in a narrow cone along its instantaneous velocity. Thus, determining the electron's angular emission pattern from a scattering event by an ultra-short laser pulse provides knowledge about the electron's trajectory and in turn about the CEP of the driving field (see Fig. 1).

Fig. 1. Typical multiphoton Compton scattering spectra for two different physical situations: in parts a) and b) recoil effects in the photon spectra are negligible, while in parts b) and c) they are important and taken into account. In both cases the angular width of the emission region is compared with analytical predictions for two different values of the CEP (white horizontal lines): -&pi/10 and -&pi/5 in parts a) and b), respectively, and 0 and &pi/4 in parts c) and d), respectively. Figure from Ref. [1], copyright of the American Physical Society.


[1] F. Mackenroth, A. Di Piazza, and C. H. Keitel, Phys. Rev. Lett. 105, 063903 (2010).

A matterless double-slit

When light passes through a double-slit under certain conditions, it creates a series of bright and dark fringes on a screen some distance away. This is due to interference between the signals at each slit and demonstrates the wave-like nature of light. If one pictures light as discrete photons and attempts to measure which slit the light chose, the pattern will disappear, demonstrating the particle-like nature of light. The double-slit experiment has been central to the development and understanding of quantum mechanics and the wave-particle duality of nature. So far, all the experimental double-slit schemes proposed and realized have involved matter. However, quantum electrodynamics predicts that electromagnetic fields can also interact in vacuum through charged virtual particles (vacuum fluctuations). By exploiting this pure quantum interaction we have envisaged a matterless double-slit scenario consisting only of light [2]. In the proposed scenario two separated, parallel Gaussian laser beams form the " slits" that are probed by a third Gaussian laser beam which is diffracted to generate an interference pattern entirely from light (see Fig. 1, where the strong beams come from the right and the probe beam is counterpropagating to them). In Fig. 1 a typical interference pattern is shown, where typical alternating maxima and minima are visible. In [2] the possibility of observing this effect experimentally is also envisaged by employing upcoming laser systems like the Extreme Light Infrastructure (ELI) and the Exawatt Center for Extreme Light Studies (XCELS).

Fig. 1. A matterless double-slit: a wider probe laser beam counter-propagates antiparallel to two, tightly-focused, separated, ultra-intense laser beams, generating a diffraction pattern due to vacuum polarization. Electric and magnetic field for a fixed probe polarization are also shown. The interference pattern has a structure with alternating maxima and minima typical of double-slit experiments.


[1] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Rev. Lett. 97, 083603 (2006).
[2] B. King, A. Di Piazza, and C. H. Keitel, Nature Photon. 4, 92 (2010).

Classical radiation reaction effects below the radiation dominating regime

An electron in a laser field emits electromagnetic radiation and the emission modifies the electron trajectory and then the emission spectrum itself. In the realm of classical electrodynamics the modified equation of motion of the electron which accounts for the reaction of the electromagnetic emission onto the electron motion is the so-called Landau-Lifshitz equation [1]. In this equation radiation reaction is included as an additional force acting on the electron (self-force). In [2] we have solved exactly and in a closed analytic form this equation when the external field is represented by a plane wave with arbitrary polarization and spectral content. We have shown that the typical parameter which determines the magnitude of the radiative effects is given by R=αχξ. In this expression &alpha=1/137 is the fine structure constant, &chi is the laser field amplitude in the rest frame of the electron in units of Ecr and &xi is amplitude of the electron oscillating relativistic momentum in the laser field in units of its rest mass times the speed of light. When the parameter R is close to unity, one enters the so-called radiation dominated regime where the effect of the radiation reaction force is comparable with that of the Lorentz force. In [3] we have found a different regime that lies well below the radiation dominated regime (R << 1) but where the effects on the electron spectra of the radiation reaction are manifest and measurable (see Fig. 1). In this regime the change of the electron momentum along the laser propagation direction (longitudinal momentum) due to radiation reaction in one laser period is of the order of the longitudinal momentum itself in the laser field and the electron, initially counterpropagating with the laser beam, undergoes a reflection only due to radiation reaction. In this way the effects of radiation reaction become measurable at laser intensities feasible in the near future.

Fig. 1. Comparison of the energy spectra in rad-1 emitted by an electron colliding head on with a strong laser beam without (part a)) and with (part b)) inclusion of the self force. The angle &theta is the polar angle with the laser field propagating along the positive polar axis. &omega0 is the laser angular frequency. The black lines indicate the theoretical predictions of the spectra cut-off. Figure from Ref. [3], copyright of the American Physical Society.


[1] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Elsevier, Oxford (1975).
[2] A. Di Piazza, Lett. Math. Phys. 83, 305 (2008).
[3] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Rev. Lett. 102, 254802 (2009).

Laser-photon merging in strong laser fields

In the collision of a high-energy proton and a laser field, the laser photons can merge into one single photon as a result of the quantum interaction between the laser field and the proton Coulomb field [1]. The proton has unique features which allow for the detection of this effect. In fact, on the one hand, the proton is light enough to be accelerated to very high energies. As a result, the laser field in the rest frame of the proton is strongly enhanced with respect to its value in the laboratory frame and it can be close to the critical field. On the other hand, the proton is heavy enough that the multiphoton Thomson scattering of the laser photons by the proton is reduced (see Fig. 1). In fact, multiphoton Thomson scattering represents a background of our process. Since the laser electric field in the rest frame of the proton can be of the same order of Ecr, it has to be taken into account exactly in the calculations. This is achieved by ab initio quantizing the electron-positron field in the presence of the laser field (Furry picture). The final observables show a complex, non-perturbative dependence on the laser field parameters. This renders these results very appealing because non-perturbative vacuum polarization effects can in principle be measured with this setup. In [2,3] we have also explored alternative setups where non-perturbative vacuum polarization effects can be in principle measured.

Fig. 1. Angular distribution of photons resulting from two-photon Thomson scattering alone (dotted line) and from two-photon Thomson scattering plus two-laser photon merging (continuous line). The proton beam collides head-on with the laser beam that propagates along the polar axis and &theta indicates the polar angle. The parameter &chi2 is proportional to the laser field amplitude in the rest frame of the electron and the figure shows that in the important part of the spectrum it is of the order of unity. Figure from Ref. [1], copyright of the American Physical Society.


[1] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Rev. Lett. 100, 010403 (2008).
[2] A. Di Piazza, A. I. Milstein, and C. H. Keitel, Phys. Rev. A 76, 032103 (2006).
[3] A. Di Piazza and A. I. Milstein, Phys. Rev. A 77, 042102 (2007).


Light-by-light diffraction

QED predicts that electromagnetic fields interact in vacuum giving rise to a number of interesting effects [1,2]. This interaction is mediated by the virtual electron-positron pairs that are present in the vacuum (see Fig. 1). In [3] we have shown that a strong optical standing wave "diffracts" an X-ray probe in a similar way as if it was an aperture in a wall. The presence of the standing wave modifies the polarization state of the probe. If the probe is initially linearly polarized, then after the interaction it will result elliptically polarized with the main axis of the ellipse rotated with respect to the initial polarization direction. The values of the ellipticity ε and of the polarization rotation angle ψ depend, among other parameters, on the observation distance yd with respect to the interaction point (see Fig. 1).

Fig. 1. Ellipticity and polarization rotation angle acquired by a linearly polarized X-ray probe field after interacting with a strong optical standing wave. Figure from Ref. [3], copyright of the American Physical Society.


[1] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Rev. D 72, 085005 (2005).
[2] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Plasmas 24, 032102 (2007).
[3] A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, Phys. Rev. Lett. 97, 083603 (2006).



For further information and for related projects, please contact Antonino Di Piazza.





Old group pictures

July 2017, from left to right: Tobias Wistisen, Matteo Tamburini, Antonino Di Piazza, Fabien Niel, Sergey Bragin, Alessandro Angioi, Maitreyi Sangal, Archana Sampath.

July 2015, from left to right: Sergey Bragin, Antonino Di Piazza, Matteo Tamburini, Alessandro Angioi, Sebastian Meuren, Dmitry Karlovets, Rashid Shaisultanov.

August 2012, from left to right: Norman Neitz, Sebastian Meuren, Antonino Di Piazza, Matteo Tamburini, Felix Mackenroth

May 2011, from left to right: front row: Ashutosh Sharma, Antonino Di Piazza, Norman Neitz; back row: Omri Har-Shemesh, Ben King, Felix Mackenroth, Sebastian Meuren


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