Selected projects
Improved localconstantfield approximation for strongfield QED codes
The localconstant field approximation (LCFA) is a powerful and
thus widelyused approximation in strongfield 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 strongfield 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 strongfield 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 lowenergy 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 lowenergy 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
lightcone energy in units of the emitting electron lightcone 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 localconstantfield approximation, Phys. Rev. A 98, 012134 (2018).
[3] A. Di Piazza, M. Tamburini, S. Meuren, and C. H. Keitel, Improved localconstantfield approximation for strongfield 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 strongfield 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 strongfield 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 twoelectron wavepacket
in the presence of an electromagnetic plane wave within
strongfield 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 (dashdotted 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 socalled 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 socalled
LorentzAbrahamDirac 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 socalled
LandauLifshitz 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 restframe and
the Schwinger critical electric (magnetic) field strength E_{cr} (B_{cr}),
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 socalled 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
3.8mm thickness (see Fig. 1a). The results in the 10.0mm 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).
Highenergy vacuum birefringence and dichroism in an ultrastrong laser field
In the realm of classical electrodynamics the electromagnetic field experiences no selfinteraction in vacuum. According to QED, however, a finite photonphoton coupling is induced by the presence of virtual charged particles in the vacuum [1]. For lowfrequency electromagnetic fields such vacuum polarization effects are described by the EulerHeisenberg Lagrangian density [2]. The EulerHeisenberg 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 photonphoton coupling.
As the lightbylight scattering cross section attains its maximum at the pairproduction threshold [2], it is natural to consider highenergy 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 highenergy regime, where the EulerHeisenberg 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 strongfield QED prediction for vacuum birefringence is feasible with an average statistical significance of 5σ on the time scale of several days at upcoming 10PW 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 centerofmomentum 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, Highenergy vacuum birefringence and dichroism in an ultrastrong laser field, Phys. Rev. Lett. 119, 250403 (2017).
Nonlinear BreitWheeler pair production in a tightly focused laser beam
The electric field strength where such nonlinear QED effects become sizable
identifies the "strongfield QED" regime and is given by the socalled
Schwinger field: E_{cr}=1.3 × 10^{16} V/cm. Due to the extremely large
value E_{cr}, present and upcoming lasers have to be tightly focused in space
(and in time) in order to aim at values comparable with E_{cr}. Nonetheless, the
value of E_{cr} exceeds by about four orders of magnitudes presently available
laserfield 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 strongfield QED regime can effectively be probed also
nowadays in principle.
Now, all systematic approaches to investigate analytically strongfield 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 strongfield QED regime at present and upcoming laser facilities,
the electrons have to be so highly relativistic that the WentzelKramersBrillouin (WKB)
approximation can be employed (at the nexttotheleading order) to solve analytically
the Dirac equation in the presence of a background laser field practically of arbitrary
spacetime shape. The electron wave functions obtained in this way open the
possibility of investigating analytically and in a systematic way strongfield QED
processes in the presence of a tightly focused laser beam of complex and realistic
spacetime shape by employing the socalled Furry picture. Indeed, we have already
determined analytically the energy spectrum and the angular distribution of the
electronpositron pairs produced in the collision of a photon bunch with an intense
and tightlyfocused laser beam (nonlinear BreitWheeler 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 BreitWheeler 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 BreitWheeler 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 fewcycles laser pulses [2], there are already many envisaged laser facilities that will soon
allow testing QED in the highintensity 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 energymomentum conservation laws and onshell 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 zdirection for electrons colliding headon (or almost headon) with a very intense pulse propagating along the positive zdirection. 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).
Highenergy recollision processes of lasergenerated electronpositron pairs
In quantum field theory in vacuum, Feynman diagrams with particle loops correspond to quantum fluctuations, with the extension, for example, of an electronpositron loop being limited to the Compton spacetime 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 electronpositron pair production the laser field can transfer enough energy to the electron and the positron in the loop to significantly increase the spacetime 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 planewave field reveals that for certain initial conditions the highenergy electron and positron can recollide and annihilate providing energy and momentum for secondary reactions as in an "electronpositron vacuum collider".
The simplest Feynman diagram which contains an electronpositron loop is the leadingorder 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 fluctuationtype processes (yellow curve), corresponding to the creation and the annihilation of the electronpositron pair within the same microscopic formation region, and recollisiontype processes (red curve), corresponding to the creation and the annihilation of the electronpositron 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 highharmonic
generation in atomic physics.


Fig. 1. Probability for the absorption of n laser photons by an electronpositron loop (in arbitrary units). Yellow curve: vacuumfluctuationtype contribution, red curve: recollisiontype 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 highdensity electronpositron pair plasmas in the laboratory
Electronpositron plasmas are emitted as ultrarelativistic jets
in different astrophysical scenarios under extreme conditions,
like during gammaray 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 ultrarelativistic electronpositron 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
skindepth 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 electronpositron 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) electronpositron
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
MonteCarlo simulation code FLUKA
(see in particular the red crosses in Fig. 1),
which among others also includes electronelectron
and electronpositron interactions, atomic scattering
and other breaking mechanisms, together with highenergy
processes like muonantimuon 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 ultrarelativistic 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 highdensity electronpositron pair plasmas in the laboratory, Nature Commun. 6, 6747 (2015).
Plasmabased generation and control of a single fewcycle highenergy ultrahighintensity 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: fewcycle
duration, highenergy and ultrahigh intensity. Already in
nonrelativistic atomic physics, it has been demonstrated that quantum
processes can be controlled by manipulating the pulse shape of
fewcycle laser pulses [1]. In order to achieve the same goal also in
the ultrarelativistic regime and in the realm of strongfield QED,
fewcycle laser pulses with tunable carrierenvelope phase (CEP) are
required with peak intensities largely exceeding 10^{20}
W/cm^{2} [2]. In [3] we put forward the concept of a laserboosted soliddensity parabolic relativistic
"mirror", interacting with a superintense
counterpropagating laser pulse, to generate a CEP tunable fewcycle
pulse with multijoule energy and peak intensity exceeding 10^{23}
W/cm^{2}. 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 multidimensional
particleincell simulations show that a single fewcycle, multipetawatt laser pulse with
several joule of energy and with peak intensity exceeding 10^{23}
W/cm^{2} can be generated already employing nextgeneration
highpower laser systems. In addition, the carrierenvelope phase of
the generated fewcycle pulse can be tuned provided that the
carrierenvelope 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 fewcycle reflected pulse with 5.8
fs duration, 6.8 J energy and 2.3 × 10^{23} W/cm^{2}
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 socalled LandauLifshitz (LL) equation. Furthermore,
in the ultrarelativistic, "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 headon 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 10cycle 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).
Tabletop laserbased source of femtosecond, collimated, ultrarelativistic positron beams
Ultrarelativistic, highlycollimated positron beams have been generated so far by employing largescale electron
accelerators. The electron beams produced by the accelerator collides with a solid target (typically gold or lead),
generating on the rear side electronpositron bunches, whose oppositelycharged 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 electronpositron 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 wakefield 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
ultrarelativistic energies (>100 MeV), and with a narrow angular distribution (~ 3 mrad) in a tabletop setup.
The possibility of generating such highenergy lepton beams is of central importance for astrophysics due
to their similarity to jets of long gammaray bursts.


Fig. 1. Tabletop experimental setup for the production of short, narrow, and ultrarelativistic
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 ultraintense 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 lowestorder being the emission of two photons. This process is called
nonlinear double Compton scattering. The possible detection of
twophoton 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 singlephoton
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 singlephoton background
is suppressed and thus the detection of the twophoton signal becomes possible. We have performed this task
by demonstrating an angular separation between the single and twophoton 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
singlephoton emission cone.


Fig. 1. Twophoton emission spectra for observation of both photons inside the singlephoton 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 10^{20} W/cm^{2}, 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 radiationreaction
effects [2]. Such effects are included via the LandauLifshitz 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
10^{23} W/cm^{2}, when quantum radiationreaction effects
start affecting substantially the electron dynamics [1].


Fig. 1. Angularresolved 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. HarShemesh 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 selfinteraction in a strong laser field
Electrons which interact with planewave laser fields are normally described by Volkov states in strongfield QED calculations.
These are exact solutions of the interacting Dirac equation with a classical fourpotential and thus the background electromagnetic
planewave 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 strongfield QED also affect the properties of onshell particles. In [1] we have calculated the
leadingorder modifications of the Volkovstates in the finestructure constant by solving the SchwingerDirac equation.
We have shown that the quasimomentum describing an electron inside a (quasi)monochromatic laser field undergoes
a pure quantum contribution due to radiative corrections. Beside this, the spindynamics of the electron is significantly
altered due to the electron selfinteraction. For a planewave laser field the Dirac equation predicts that the quasienergy of
the electron inside the laserfield is degenerated with respect to the spin quantum number.
Similar to the Lambshift for electrons bound to a nucleus, nonlinear 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 laserpulse (see Fig. 1). According to the
Dirac equation, the calculated asymmetry should be zero. A measurement of a nonzero spin asymmetry would therefore be a clear signature
for nonlinear 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 carrierenvelope phase (CEP) for
an optical laser pulse with a duration of 8 fs. The laser pulse is linearly polarized along the xdirection and
collides headon with electrons having an energy of 500 MeV and a spinpolarization along the ydirection.
After the interaction, the spin of the electron is measured along the zdirection. 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 energymomentum loss alters the electron's trajectory [1].
The radiationreaction problem is the determination of the equation of motion of an electron by including selfconsistently the effects
of the emitted radiation on the electronâ€™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 selfconsistent inclusion of radiationreaction 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., electronpositron pair production originating from the photons
emitted by the electrons. Thus, in [2] we have limited ourselves to the socalled moderately quantum regime, which is experimentally relevant and
where essentially pairproduction remains negligible. Thus, the problem is still singleparticle
and a clear comparison between classical and quantum results is feasible. In Fig. 1 classical and quantum spectra
with and without radiationreaction effects included are shown. Inclusion of radiationreaction 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 lowenergy photons when the laser field is not at the maximum
amplitude yet, and the probability of emitting one photon in the highenergy 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 carrierenvelope phase of intense fewcycle 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 fewcycle 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 10^{14}10^{15} W/cm^{2}. Presently available peak laser intensities are of the order of 10^{22} W/cm^{2} 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 10^{20} W/cm^{2}) 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 ultrashort 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 doubleslit
When light passes through a doubleslit 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 wavelike 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 particlelike nature of light. The
doubleslit experiment has been central to the development and understanding of quantum mechanics and the waveparticle duality of nature. So far,
all the experimental doubleslit 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 doubleslit 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 doubleslit: a wider probe laser beam counterpropagates antiparallel to two, tightlyfocused, separated, ultraintense 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 doubleslit 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 socalled LandauLifshitz equation [1]. In this equation radiation reaction is included as an additional force acting on
the electron (selfforce). 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 E_{cr} 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 socalled 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. &omega_{0}
is the laser angular frequency. The black lines indicate the theoretical predictions of the spectra cutoff. 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).
Laserphoton merging in strong laser fields
In the collision of a highenergy 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 E_{cr}, it has to be taken
into account exactly in the calculations. This is achieved by ab initio quantizing the electronpositron field
in the presence of the laser field (Furry picture). The final observables show a complex, nonperturbative dependence on the laser field parameters.
This renders these results very appealing because nonperturbative vacuum polarization effects can in principle be measured with this setup. In [2,3] we have
also explored alternative setups where nonperturbative vacuum polarization effects can be in principle measured.


Fig. 1. Angular distribution of photons resulting from twophoton Thomson scattering alone (dotted line) and from twophoton Thomson scattering plus twolaser
photon merging (continuous line). The proton beam collides headon with the laser beam that propagates along the polar axis and &theta indicates the polar angle.
The parameter &chi_{2} 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).
Lightbylight 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
electronpositron pairs that are present in the vacuum (see Fig. 1). In [3] we have shown that a strong optical standing wave "diffracts" an Xray 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 y_{d} with respect to the interaction point (see Fig. 1).


Fig. 1. Ellipticity and polarization rotation angle acquired by a linearly polarized Xray 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).
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 HarShemesh, Ben King, Felix Mackenroth, Sebastian Meuren
