Abstract
Ongoing progress in laser and accelerator technology opens new possibilities in highfield science, notably to investigate the largely unexplored strongfield quantum electrodynamics (SFQED) regime where electronpositron pairs can be created directly from lightmatter or even lightvacuum interactions. Laserless strategies such as beambeam collisions have also been proposed to access the nonperturbative limit of SFQED. Here we report on a concept to probe SFQED by harnessing the interaction between a highcharge, ultrarelativistic electron beam and a solid conducting target. When impinging onto the target surface, the beam self fields are reflected, partly or fully, depending on the beam shape; in the rest frame of the beam electrons, these fields can exceed the Schwinger field, thus triggering SFQED effects such as quantum nonlinear inverse Compton scattering and nonlinear BreitWheeler electronpositron pair creation. Through reduced modeling and kinetic numerical simulations, we show that this singlebeam setup can achieve interaction conditions similar to those envisioned in beambeam collisions, but in a simpler and more controllable way owing to the automatic overlap of the beam and driving fields. This scheme thus eases the way to precision studies of SFQED and is also a promising milestone towards laserless studies of nonperturbative SFQED.
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Introduction
Advances in multipetawatt laser systems^{1,2,3} should soon allow strongfield quantum electrodynamics (SFQED) effects^{4,5,6} to be induced and probed in extremeintensity lasermatter interactions^{7,8,9,10,11,12,13,14,15,16,17,18}. The main experimental challenge will be to access, in a controlled manner, a regime characterized by a quantum parameter χ = E^{*}/E_{cr} larger than unity, that is, where charged particles experience in their rest frame an electric field \({E}^{* }=\gamma \left\vert {{{{{{{{\bf{E}}}}}}}}}_{{{{{\parallel }}}}}/\gamma +{{{{{{{{\bf{E}}}}}}}}}_{\perp }+{{{{{{{\bf{v}}}}}}}}\times {{{{{{{\bf{B}}}}}}}}\right\vert \simeq \gamma \left\vert {{{{{{{{\bf{E}}}}}}}}}_{\perp }+{{{{{{{\bf{v}}}}}}}}\times {{{{{{{\bf{B}}}}}}}}\right\vert\) stronger than the Schwinger critical field^{19}\({E}_{{{{{{{{\rm{cr}}}}}}}}}={m}_{e}^{2}{c}^{3}/e\hslash \simeq 1.3\times 1{0}^{18}\,{{{{{{{\rm{V}}}}}}}}\,{{{{{{{{\rm{m}}}}}}}}}^{1}\) (m_{e} is the electron mass, e the elementary charge, c the speed of light, ℏ the reduced Planck constant, v the particle velocity, γ its Lorenz factor ≫ 1, E_{∥} and E_{⊥} the electric field components respectively parallel and perpendicular to v, and B the magnetic field). Strongfield QED processes such as quantum nonlinear inverse Compton scattering (NICS), equivalent to strongfield quantum synchrotron radiation, and nonlinear BreitWheeler (NBW) electronpositron pair creation become prominent when χ ≳ 1^{6,20}, up to the point of profoundly modifying the dynamics of physical systems subject to such conditions^{21,22,23,24,25}. In recent years, using laser pulses of ~10^{21} W cm^{−2} intensity, it has already become possible to explore some features of NICS in the marginally quantum regime^{14,26,27}.
Different schemes have been proposed to achieve, and diagnose^{28}, wellcontrolled conditions under which the quantum parameter approaches or exceeds unity. The most common strategy relies on the headon collision of an intense laser pulse with a highenergy electron beam. In the rest frame of the beam, the laser field strength is boosted by a factor of ~2γ, possibly leading to high χ values when combining an ultrarelativistic beam and an ultraintense laser pulse. The first experimental observation of strongfield QED effects in such a configuration was made at SLAC^{29,30}, using a ~47 GeV electron beam and a laser pulse focused to a mildly relativistic intensity of ~0.5 × 10^{18} W cm^{−2}. This seminal experiment attained χ ≈ 0.3 and yielded a total of about one hundred BreitWheeler positrons^{29}. Since then, alloptical laserbeam schemes involving laserwakefieldaccelerated electron beams^{31,32,33,34} have been considered as well^{9,10,35}, the first experimental tests of this configuration achieving χ ≈ 0.25^{26,27}. These scenarios, though, are characterized by oscillatory driving fields, with tens of femtosecond timescales, and probably even longer given common intensity contrast issues. Such features should contribute to blurring the strongfield QED observables, and so seem nonoptimal towards precision studies.
Progress in focusing and compressing highenergy accelerator beams^{36}, notably using wakefieldinduced plasma lenses^{37,38,39} and energychirped beams^{40}, can also be leveraged to reach strongfield QED conditions^{41,42}. Thus, Yakimenko et al.^{43} recently proposed to collide two ultrarelativistic (~100 GeV), highly focused (~10 nm), highcharge (~1 nC) electron beams to achieve χ ≫ 1, and ultimately enter the nonperturbative SFQED regime characterized by α_{f}χ^{2/3} ≳ 1 (α_{f} = 1/137 is the finestructure constant) in which the conventional theoretical framework of strongfield QED breaks down^{44,45,46,47,48,49}. In this scenario, the self fields of one beam act for the other beam as the laser field in a laserbeam collision, yet with the significant advantage that those fields are halfcyclelike and subfemtosecond, thus more adapted, in principle, to clean strongfield QED measurements. This scheme, however, necessitates extreme beam parameters and is highly sensitive to the very precise beam alignment required to ensure proper overlap and stable headon collisions. In the likely event of shottoshot jittering, one should rely on the detected gammaray and positron spectra to infer, via a modeldependent deconvolution, the actual interaction conditions, a hurdle in the way of developing a reliable testbed of strongfield QED models.
Here, as a simpler alternative, we propose to employ a single ultrarelativistic, highly focused and compressed beam and make it interact with its own self fields that are reflected (in a broad sense) off the surface of a soliddensity plasma^{50}. Under certain conditions, to be detailed later, this scenario can be viewed as a beambeam collision^{41,42} in which the colliding beam is replaced with the incoming beam’s image charge on the target (see Fig. 1). In this simple picture, the total radial electric field vanishes at the plasma surface, whereas the azimuthal magnetic fields of the incoming and image beams add up. The microscopic sources of the image beam fields in the z ≤ 0 space are the induced surface plasma currents that act to screen the incident beam self fields inside the soliddensity plasma (z > 0). Still, compared to beambeam collisions, the beamplasma scheme is much easier to implement as it does not involve a second, perfectly aligned electron beam. Even more importantly, it guarantees the overlap of the beam with the plasmainduced fields. Therefore, knowing the initial beam parameters, one can predict with high accuracy the time evolution of the fields, and hence of the χ parameter experienced by each beam particle, a key advantage for wellcontrolled strongfield QED measurements.
The objective of this work is to demonstrate, through a detailed analysis based on theoretical modeling and advanced particleincell (PIC) simulations, the potential of ultrarelativistic beamplasma collisions for precision strongfield QED studies. Besides the strong control over the interaction conditions inherent to this scheme, the requirement is to reach χ values well above unity, and over time scales short enough that each beam electron emits on average less than one gammaray photon. This ensures that the time history of the χ parameter is only determined by the initial beam parameters and not by the beam evolution during the interaction, which would complicate the benchmarking of strongfield QED models.
Our paper is structured as follows. First, we present a proofofconcept PIC simulation that reproduces the results of the beambeam setup. We then analyze the constraints posed to the beam shape and electron target density in order to achieve reflection of the beam self fields. Next, considering the case of an electron beam of fixed energy (10 GeV) and charge (1 nC) colliding with a gold foil target, we identify the optimal beam density and aspect ratio to reach high χ values. Finally, we demonstrate that the gammaray photon and positron spectra generated in this configuration can provide unambiguous experimental signatures of strongfield QED, that is, of the nonlinear inverse Compton scattering and nonlinear BreitWheeler processes that are found to largely prevail over the competing, undesirable Bremsstrahlung and BetheHeitler/Coulomb Trident processes.
Results
Proofofconcept PIC simulation
To assess the validity of the simple “image beam” picture, we have simulated the beamplasma interaction using the threedimensional (3D) PIC CALDER code^{51}. Because the beam and the plasma have rotational symmetry about the propagation axis z, cylindrical coordinates will be used throughout the paper with (r, θ, z) denoting the radial, azimuthal and longitudinal components respectively. Furthermore, since the simulation uses a moving window, the longitudinal comoving coordinates (ξ = z − ct, τ = t) will be adopted. The beam center position is set at ξ = 0, and τ = 0 corresponds to the time at which the central slice of the beam crosses the plasma surface. In this first simulation, the 10 GeV beam has a total charge of Q = 1 nC and a Gaussian density profile of rms length σ_{z} = 54 nm and width σ_{r} = 271 nm. The peak beam density is then n_{b,peak} ≃ 10^{23} cm^{−3}. The beam is initialized together with its self fields in vacuum. The plasma target has a sharp boundary and a uniform electron density n_{p} = 10^{24} cm^{−3} ≃ 10n_{b,peak}, a value typical of fully ionized solid foils. The simulation parameters are further detailed in the Methods section.
Figure 2a, b show 2D (ξ − r) density slices of the beam in vacuum (a) and of plasma electrons when the beam has just entered the plasma (b). The corresponding maps of the radial electric field E_{r} and azimuthal magnetic field B_{θ} are displayed in Fig. 2c, d. In Fig. 2e is plotted the temporal evolution of the fields as seen by a beam electron located at (r, ξ) = (1.6σ_{r}, 0). This position, indicated by a cross in Fig. 2a–d, is where the beam self fields are the highest. As expected, the beam self fields are fully screened inside the dense plasma: the E_{r} field nearly vanishes at the surface while the B_{θ} field is approximately doubled.
Figure 2e also shows the evolution of E^{*}/γ ≃ E_{r} − cB_{θ}, which represents the net radial focusing force experienced by a beam electron. This quantity provides a figure of merit for strongfield QED effects that is independent of γ and allows χ = E^{*}/E_{cr} to be directly evaluated when a particular γ is specified. Note that the self fields of a relativistic beam (and thus the reflected fields) are independent of γ, so that χ simply scales as γ.
Our simulation confirms that, as in the beambeam scenario, but in a much easier to realize and more robust setup, the beamplasma configuration allows beam electrons to probe transient electromagnetic fields of amplitude similar to the beam self fields, which can trigger interactions with χ ≫ 1 at high Lorentz factors. Initially, when propagating in vacuum, the beam electrons are subjected to a negligible transverse force (E^{*} ≃ 0) as the transverse electric and magnetic self fields closely compensate each other in the laboratory frame. Just prior to entering the plasma, the beam electrons experience the reflected fields which impart a transverse force of magnitude comparable with that of the self fields. For a beam energy of 10 GeV, this would translate into an electron quantum parameter χ ≃ 5. However, as soon as the beam penetrates the plasma, E^{*}/γ rapidly drops due to plasma shielding. Thus, akin to the beambeam scheme, the beam electrons undergo a high χ value over a very brief duration only, of the order of σ_{z}/c, which is beneficial for precision studies of strongfield QED.
Propagating or evanescent fields
The properties of the reflection process are determined by the beamtoplasmadensity ratio α = n_{b,peak}/n_{p} and the beam widthtolength ratio β = σ_{r}/σ_{z}. A necessary condition for the plasma to efficiently screen the self fields (yielding a nearly vanishing electric field at the surface) is that the electron plasma density is much larger than the beam density, that is, α ≪ 1. Yet, even if the condition of vanishing electric field at the surface is satisfied, the image beam model only applies in the socalled radiating regime^{50} associated with a beam larger in width than in length, that is, with β ≫ 1 (pancakeshaped beams). Then, the reflection process generates fields that propagate (in the direction of negative z) and diffract over an approximate Rayleigh length \({\sigma }_{r}^{2}/{\sigma }_{z}\gg {\sigma }_{z}\). Diffraction is negligible during the collision so that these propagating fields can be approximated to those of the image charge. In the nonradiating regime β ≪ 1 (cigarshaped beams), the plasmainduced fields are evanescent, and so confined to the vicinity of the vacuumplasma boundary. This configuration, not captured in the image beam approach, was already considered for focusing purposes several decades ago^{52,53,54}. A unified treatment of those two limits is provided by the pseudophoton model developed by Sampath et al.^{50}, which we will exploit in the remainder of this Section.
In Fig. 3a, the evolution of E^{*}/γ as experienced by a beam electron located at (r, ξ) = (1.6σ_{r}, 0) and as computed by the pseudophoton method, is represented for different sizetolength ratios β. While the agreement with the image beam model is excellent for β = 10, one can see that both the maximum value of E^{*}/γ and its duration (normalized to the beam length σ_{z}) are reduced at lower β, as a consequence of the induced fields turning evanescent. These two effects are quantified in Fig. 3b and c. First, in the nonradiating regime β ≪ 1, the maximum field amplitude experienced by a beam electron is equal to the maximum of the beam self electric field \({E}_{r,\max }^{b}\), in contrast with the doubled amplitude predicted by the image beam model, valid in the radiating regime β ≫ 1. The transition from \(\max ({E}^{* }/\gamma )=2{E}_{r,\max }^{b}\) to \(\max ({E}^{* }/\gamma )={E}_{r,\max }^{b}\) occurs around β ≃ 1 [see Fig. 3b], with β = 1 (spherical beam) providing a reasonably high peak field strength, \(\max ({E}^{* }/\gamma )=1.65{E}_{r,\max }^{b}\). Second, we observe that the duration τ_{rms} over which a beam electron is subjected to a high value of E^{*}/γ (see Methods for the definition of τ_{rms}) diminishes when the plasmainduced fields become increasingly evanescent. The cτ_{rms}/σ_{z} ratio evolves from a constant value ≃ 0.3 at β > 1 [with E^{*}/γ having a halfGaussian shape in Fig. 3a, corresponding to a propagating field] to smaller values at β ≪ 1 [with E^{*}/γ showing an exponential decay in Fig. 3a, corresponding to an evanescent field], with cτ_{rms}/σ_{z} ≃ 0.22 at β ≃ 0.1.
The pseudophoton model^{50} therefore predicts that the highest field E^{*}/γ and highest duration τ_{rms} are reached in the radiating regime, β ≫ 1, but also that comparable performance can be obtained for β ~ 1. However, using cigarshaped beams with β < 0.1 should considerably reduce the time (in units of σ_{z}/c) over which the fields are experienced.
Plasma screening and transparency
Given the scaling of the beam self field at fixed beam charge Q, \({E}_{r,\max }^{b}\propto {\alpha }^{2/3}{\beta }^{1/3}\), it seems desirable to increase α to boost the reflected field. However, when the beam density approaches the plasma density (α ~ 1), the plasma is no longer able to perfectly screen the beam self field, which in turn may hinder the expected enhancement of the reflected field.
To evaluate the impact of imperfect plasma screening, we have carried out PIC simulations using the quasi3D CALDERCIRC code^{55} (see “Methods”), varying α from 0.005 to 0.7 for β = 0.2, 1 and 5, and keeping the charge constant at Q = 1 nC and the plasma density at n_{p} = 1 × 10^{24} cm^{−3}. Figure 4 shows that, for α = 0.005, the lowest simulated value, the PIC results reasonably agree with the pseudophoton model [see Fig. 3b], with the maximum of E^{*}/γ going from \(1.9{E}_{r,\max }^{b}\) at β = 5 (propagating fields for pancakeshaped beams) to \(1.2{E}_{r,\max }^{b}\) at β = 0.2 (evanescent fields for cigarshaped beams).
However, when α is increased, the PIC results deviate substantially from the pseudophoton model, which assumes perfect screening and is therefore independent of α. At α ≃ 0.1, the β = 1 and β = 5 curves cross each other, and at α ≃ 0.7, the hierarchy of E^{*}/γ with β is fully reversed. When going to α ≳ 1, the reflection is strongly degraded and the interaction enters the blowout regime^{56,57} where the plasma electrons are fully expelled from the beam path, causing an ion cavity to form in the wake of the beam (see Supplementary Note 1). This regime, which can continue in the bulk of the plasma and is accounted for in our parametric simulation study below, starkly departs from the concept proposed here, based on the ultrashort surface interaction of the beam with its reflected (in a broad sense) self field. In comparison, the blowout mechanism generally occurs over longer timescales (set by the foil’s thickness), during which both the fields acting upon the beam and its resultant phase space may vary significantly, and not in a wellcontrolled way. Under our conditions, the nonlinear beam dynamics is made even more complex due to the abundant creation of electronpositron pairs that is enabled by the longer interaction time. These charged particles can strongly alter the fields experienced by the beam electrons, thus further hampering precision investigations of strongfield QED.
Figure 5 shows that working around α ≃ 0.5 can provide a beamplasma collision optimized for high χ, with a reasonably high maximum value of \({E}^{* }/\gamma \simeq {E}_{r,\max }^{r}\) and a weak dependence on β. Beamplasma collisions are therefore constrained by the achievable plasma density of the target and the need of α to stay below 1, which together limit E^{*}/γ.
The efficiency of the plasma shielding is not only determined by the beamtoplasma density ratio but also to the response time of the plasma electrons \(\sim {\omega }_{p}^{1}\) (ω_{p} is the electron plasma frequency) compared to the beam duration σ_{z}/c. When ω_{p}σ_{z}/c ≲ 1, i.e., when the beam duration is shorter than, or comparable with the inverse plasma frequency (or, equivalently, when the beam length is shorter than the plasma skin depth k_{p} ≡ ω_{p}/c), the plasma electrons have insufficient time to react to the beam self fields, which hence cannot be properly screened. As the plasma turns transparent to the self fields, the maximum value of E^{*}/γ is strongly reduced, with, for example, \(\max ({E}^{* }/\gamma )\simeq 0.1{E}_{r,\max }^{b}\) at k_{p}σ_{z} = 0.5 (see Supplementary Note 2). This limit, however, is very hard to reach in practice given the extremely thin skin depth of soliddensity plasmas (\({k}_{p}^{1}\simeq 5\,{{{{{{{\rm{nm}}}}}}}}\) at n_{p} = 1 × 10^{24} cm^{−3}). Therefore, all results reported here use much longer bunches such that k_{p}σ_{z} > 1.
Target ionization and parametric study
To probe strongfield QED at high χ values, the beamplasma scheme requires an electron beam with large self fields and thus of high density, yet lower than that of the plasma electrons to ensure efficient shielding and reflection of the beam self fields. The plasma should thus be as dense as possible. A conductor with free electrons can be a natural choice for the solid target. However, by leveraging the possible ionization of bound electrons in a highZ material by the beam self fields, higher electron plasma densities can be envisioned.
To maximize the effective plasma electron density, we have considered a gold target (Z = 79), whose atoms have one free electron (giving an electron plasma density \({n}_{p,{1}^{+}}=5.9\times 1{0}^{22}\,{{{{{{{{\rm{cm}}}}}}}}}^{3}\)) and can provide, via field ionization, up to 76 additional electrons (giving a plasma density \({n}_{p,7{7}^{+}}=4.5\times 1{0}^{24}\,{{{{{{{{\rm{cm}}}}}}}}}^{3}\)) for the beam parameters considered here. We expect that the denser the beam, the stronger its self fields, and the higher the target ionization will be. A drawback of field ionization, however, is to make the relative beam density (α) vary dynamically as a function of the instantaneous plasma electron density, which may increase the complexity of the problem.
To evaluate the induced fields and χ values that can be achieved in a gold target for different beam parameters, we have carried out a set of PIC simulations using the quasi3D CALDERCIRC code^{55} in which field ionization^{58} is enabled (see “Methods” for numerical details). This parametric study was performed for beam densities ranging from 1.77 × 10^{23} cm^{−3} to 1.77 × 10^{25} cm^{−3} and β ∈ (0.2, 1, 5), keeping the charge constant at Q = 1 nC. The target was initialized as a semiinfinite, neutral plasma composed of Au^{1+} ions and free electrons of uniform density 5.9 × 10^{22} cm^{−3}. With these parameters, our simulations show that the beam is able to ionize the target surface atoms to Au^{57+} for n_{b} = 1.77 × 10^{23} cm^{−3} and β = 1, and to Au^{77+} for n_{b} ≥ 1.06 × 10^{25} cm^{−3} and β ∈ (0.2, 1, 5) before the central slice of the beam hits the surface (see Supplementary Note 3).
For simplicity, Fig. 5 shows the results of this parametric study using the definition \(\alpha ={n}_{b,{{{{{{{\rm{peak}}}}}}}}}/{n}_{p,6{0}^{+}}={n}_{b,{{{{{{{\rm{peak}}}}}}}}}/60{n}_{p,{1}^{+}}\), (i.e. with respect to the electron density of a Au^{60+} plasma), a reasonable approximation for gold ionization between Au^{57+} and Au^{77+}. The time evolution of E^{*}/γ (and hence χ) is observed to be sharply peaked for α ≤ 0.7, a behavior characteristic of a beamplasma collision where the dominant fields are those reflected by the surface or the evanescent fields in its vicinity. Curves with latetime plateaus, particularly visible for β = 0.2 and α ≥ 2, illustrate the case where the physics is essentially dominated by the blowout mechanism, whereby plasma electrons are fully expelled from the beam path, around the surface as well as in the bulk of the plasma, allowing intense plasmainduced fields to be maintained at large times. For β = 5, the peak values at α ≥ 2 are not substantially increased with respect to α = 0.7, thus highlighting the degraded reflection process. Furthermore, in these cases (α ≥ 2, β = 5), E^{*}/γ rapidly drops after its maximum despite the formation of an ion cavity. The reason for this behavior is that the highest fields in the cavity created by a pancakeshaped, short beam are located behind it, and so hardly act upon it. In addition, the peak of E^{*}/γ is substantially shifted from cτ = 0, due to the strong forward push undergone by surface plasma electrons when the beam enters the target^{59}. This accounts for the compressed plasma surface that is seen in Fig. 2b.
From this parametric study, we conclude that in order to achieve ultrashort surface interaction of the beam with the plasmainduced fields, comparable in strength with the beam self fields, optimal performance is obtained for α ∈ [0.2, 0.7] and β ∈ [0.2, 1]. The quantum parameter experienced by the beam particles then shows a sharply peaked temporal variation (of typical duration ~ σ_{z}/c) with maximum values χ ≃ 5−15 for a beam energy of 10 GeV. Pushing to higher α or β (a very challenging experimental objective given the difficulty of compressing beams to such levels), only provides small improvement or leads to a transition to the more complex, nonlinear blowout regime, detrimental to wellcontrolled investigations of strongfield QED.
Strongfield QED observables
From the previous results, one can predict that the interaction of a 10 GeV electron beam characterized by Q = 1 nC and σ_{r} = σ_{z} = 55 nm (β = 1) with a solid gold target (α = 0.7) will entail massive generation of γray photons via NICS (e^{−} + nω → e^{−} + γ) in the strongfield QED (χ ≫ 1) regime. In this process, a beam electron absorbs several lowenergy photons (nω) from the electromagnetic field and emits a single γ ray of high energy. The associated Feynman diagram is sketched in the inset of Fig. 6a, the double line representing the dressed state of the electron Dirac field in a strong electromagnetic field.
In turn, when subjected to the same driving fields as the beam electrons, the NICS γ rays of energy ε_{γ} > 2m_{e}c^{2} can rapidly convert into electronpositron pairs via the NBW process (γ + nω → e^{−} + e^{+}) if their quantum parameter, \({\chi }_{\gamma }=({\varepsilon }_{\gamma }/{m}_{e}{c}^{2}) {{{{{{{{\bf{E}}}}}}}}}_{\perp }+c{\hat{{{{{{{{\bf{k}}}}}}}}}}_{\gamma }\times {{{{{{{\bf{B}}}}}}}} /{E}_{{{{{{{{\rm{cr}}}}}}}}}\), exceeds unity (\({\hat{{{{{{{{\bf{k}}}}}}}}}}_{\gamma }\) is the photon propagation direction)^{6}. The inset of Fig. 6b shows the corresponding Feynman diagram.
To assess accurately the efficiency of these two strongfield QED processes, and confirm that they fully control the highenergy photon and positron emissions, we present in Fig. 6 the results of a 3D PICQED simulation of the above scenario in the case of a 100 nmthick gold foil. In addition to NICS and NBW^{60}, the simulation describes the competing processes mediated by the Coulomb field of the target nuclei, namely, Bremsstrahlung together with BetheHeitler and Trident pair generation^{61}.
Upon crossing the foil surface, the beam electrons experience χ values as high as ~25 [see Fig. 6b and e], causing prolific NICS photon production, as expected. Figure 6a shows that γ rays are created with a doughnutshaped distribution, which originates from the spatial distributions of the beam density and χ parameter. Figure 6d further reveals that the γray spectrum (photon number per 0.1% energy bandwidth) is relatively flat and extends all the way up to the electron beam energy. It is to be emphasized that, despite the extreme fields at play, the interaction time ~σ_{z}/c is short enough that the average number of γ rays emitted per beam electron remains lower than unity (this number is measured to be ~0.46 for photon energies >0.5 GeV, i.e., 5% of the energy of the individual beam electrons). The effective absence of successive highenergy photon emissions ensures that NICS occurs under wellcontrolled conditions (only determined by the initial beam parameters), and justifies a posteriori its neglect in our previous design study. Moreover, the total beam energy is found to drop only by ~15% during the interaction, essentially from NICS (the energy loss induced by the plasmainduced fields remains negligible). Such a weak variation in beam energy can further simplify the modeling of the problem.
Figure 6e shows that the quantum parameter χ_{γ} of the NICS photons can reach values above ~10. Figure 6b and c detail the spatial distribution of the resultant NBW pairs during and after the passage of the electron beam through the target. The interaction results in the generation of a positron beam of ≳10^{21} cm^{−3} density and ≳10 pC charge (above 1 GeV) that copropagates with the (approximately 10 × denser) γray beam [see Fig. 6c and e]. As for the photons, the energy spectrum of the positrons extends up to the initial beam energy, yet drops faster beyond its maximum at ~2 GeV [Fig. 6d].
Importantly, our advanced PICQED simulation further reveals that the competing (Bremsstrahlung, BetheHeitler and Coulomb Trident) processes that may generate undesirable γray and positron background signals are all negligible for the thin (100 nm) gold target considered. In detail, the number of NICS γ rays (NBW pairs) is about 10^{3} (resp. 1.5 × 10^{3}) times higher than due to Bremsstrahlung (resp. BetheHeitler and Coulomb Trident processes). The outgoing γray and positron spectra can therefore serve as unambiguous observables of the ultrashorttimescale, strongfield QED processes triggered during the beamplasma collision, a result of prime significance for future experiments.
Conclusions
Laboratory investigations of strongfield QED remain difficult nowadays due to the extreme electromagnetic fields and particle energies required to reach χ values well in excess of unity. We have demonstrated that the self fields of a very dense, ultrarelativistic charged particle beam can be harnessed, when coupled with an even denser plasma target, to achieve strongfield QED conditions. In particular, we have shown that a SLACclass, 10 GeV, 1 nC electron beam with dimensions σ_{r} = σ_{z} = 55 nm can probe strongfield QED processes when colliding with a thin gold foil. The electron (photon) quantum parameter can reach values as high as χ ~ 25 (resp. χ_{γ} ~ 15), so that up to ~10 pC of ultrarelativistic (≥1 GeV) positrons can be produced through NICS followed by NBW. The latter processes turn out to largely dominate other sources of γ rays (Bremsstrahlung) and pairs (BetheHeitler, Coulomb Trident), and hence the beamplasma scheme can provide unequivocal strongfield QED signatures, free of undesirable background.
As the χ parameter is proportional to the beam’s Lorentz factor, increasing the latter would allow one to access even more extreme conditions (e.g. χ ~ 100 at 40 GeV), or to relax the constraints on the beam parameters for a fixed χ value. For instance, given the α^{2/3}β^{1/3} scaling of the self field, the same χ value could be reached by going from 10 GeV to 40 GeV and lowering α by a factor of (40/10)^{3/2} = 8, keeping β and Q constant.
While beambeam collisions at future 100GeVclass particle colliders could, in principle, enter the nonperturbative SFQED regime (\(\chi\; \gtrsim\; {\alpha }_{{{{{{{{\rm{f}}}}}}}}}^{3/2}\simeq 1600\))^{43}, their experimental realization appears highly challenging due to alignment issues. At this stage, the beamplasma setup, which involves a single beam only and guarantees perfectly controlled interactions, seems to offer a much easier avenue to acceleratorbased, fundamental strongfield QED studies in the χ ~ 10 − 100 regime, and ultimately beyond by boosting the beam energy.
Methods
Particleincell simulations
The 3D PIC code CALDER^{51} (for Figs. 2 and 6) and the quasi3D PIC code CALDERCIRC^{55} (for Figs. 4 and 5) were used to simulate the interaction between the electron beam and the plasma or gold target. In CALDERCIRC runs, because of the axisymmetry of the problem, only the first mode (m = 0) of the azimuthal decomposition was included, making it equivalent to a 2D (r, z) simulation. The boundary conditions along the transverse direction were set to be reflective. The reason for this choice is that standard absorbing conditions were found unable to treat properly the beam self fields that propagate tangentially to the transverse boundary of the domain, and would extend, in actual conditions, well beyond it. Reflective conditions proved satisfactory enough to reduce detrimental boundary errors. All simulations reported in the paper used mobile ions but discarded Coulomb (elastic) collisions, which previous tests found of negligible influence. The electron beam was initialized with zero temperature as emittance effects are negligible over the relatively short propagation distances considered. As the beam preserves its shape throughout the simulation, we only display it before entering the target in Fig. 2a. For the simulations shown in Figs. 2, 4 and 5, in which the beam electrons propagate ballistically, each cell initially contained 2 plasma ions, 10 plasma electrons and 5 beam electrons. For the simulation depicted in Fig. 6, each cell was filled with one plasma ion, 5 plasma electrons and 3 beam electrons, and the beam evolved selfconsistently.
The 3D (x, y, ξ ≡ z − ct) simulation domain associated with Fig. 2 extends over 8σ_{r} × 8σ_{r} × 8σ_{z}, and contains 1624 × 1624 × 650 cells. The time step is of 1.8 as and the plasma has a uniform electron density n_{p} = 1 × 10^{24} cm^{−3}. The 3D (x, y, ξ) simulation domain corresponding to Fig. 6 has dimensions of 9σ_{r} × 9σ_{r} × 9σ_{z}, discretized into 553 × 553 × 650 cells. The time step is of 1.2 as and the plasma is initialized with Au^{1+} ions and a uniform density of 5.9 × 10^{22} cm^{−3}. Synchrotron and Bremsstrahlung radiation, pair creation through nonlinear BreitWheeler, BetheHeitler and Coulomb Trident processes, and field ionization are described simultaneously^{60,61}.
The quasi3D simulations of Figs. 4 and 5 consider a plasma initialized with Au^{1+} ions and a uniform density of 5.9 × 10^{22} cm^{−3}. Field ionization is activated, each ionization event leading to a new electron macroparticle. Therefore, ionization from Au^{1+} to Au^{77+} produces 76 new electrons per gold ion. These simulations cover a (r, ξ) domain of size 8σ_{r} × 10σ_{z} with:

for β = 1, a mesh made of 200 × 500 cells and time steps of 7.7 as, 4.7 as, 3.1 as, 2.2 as, 1.6 as for, respectively, α = 0.05, 0.2, 0.7, 2, 5.

for β = 0.2, a mesh made of 200 × 2500 cells and time steps of 4.3 as, 2.8 as, 1.8 as, 1.3 as, 0.9 as for, respectively, α = 0.05, 0.2, 0.7, 2, 5.

for β = 5, a mesh made of 1000 × 500 cells and time steps of 2.6 as, 1.6 as, 1.1 as, 0.7 as, 0.5 as for, respectively, α = 0.05, 0.2, 0.7, 2, 5.
Definition of the interaction time plotted in Fig. 3c
The typical time τ_{rms} over which a beam electron experiences the fields is defined as the rms duration:
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Code availability
The CALDER code has been used to generate the results presented in this manuscript. This code has already been characterized with a full description reported in the references provided in the manuscript^{51,55,58,60,61}.
Change history
10 July 2023
A Correction to this paper has been published: https://doi.org/10.1038/s4200502301294x
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Acknowledgements
This work was performed in the framework of the E332 Collaboration. E332 is a SLAC experiment which aims at the study of nearfield coherent transition radiation in beammultifoil collisions and the resulting selffocusing, gammaray emission and strongfield QED interactions. The work at CEA and LOA was supported by the ANR (UnRIP project, Grant No. ANR20CE300030). The work at LOA was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (MPAC project, Grant Agreement No. 715807). P.S. was supported by the ANR under the programme “Investissements d’Avenir” (Grant No. ANR18EURE0014). We acknowledge GENCITGCC for granting us access to the supercomputer IRENE under Grants No. 2020A0080510786, 2021A0100510786, 2021A0110510062 and 2022A0120510786 to run PIC simulations. J.R.P. was supported by DOE NNSA LRGF fellowship under grant DENA0003960. The work at SLAC was supported by U.S. DOE FES Grant No. FWP100331 and DOE Contract DEAC0276SF00515. U.C.L.A. was supported by U.S. Department of Energy Grant No. DESC001006 and NSF Grant No. 1734315.
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Simulations, analytical work and data analysis were carried out by A.M. and P.S., with the support and supervision of X.D., L.G., M.T. and S.C.A.M., P.S., R.A., H.E., F.F., S.G., M.G., M.J.H., C.H.K., A.K., M.L., Y.M., S.M., Z.N., B.O., J.R.P., D.S., Y.W., X.X., V.Z., X.D., L.G., M.T., S.C. discussed the results and their interpretation. A.M., P.S., L.G. and S.C. wrote the manuscript with inputs from all authors. S.C. supervised the project.
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Matheron, A., San Miguel Claveria, P., Ariniello, R. et al. Probing strongfield QED in beamplasma collisions. Commun Phys 6, 141 (2023). https://doi.org/10.1038/s42005023012634
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DOI: https://doi.org/10.1038/s42005023012634
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