Introduction

Quantum effects dominate charge dynamics and radiation production1,2 for charges accelerated by fields with strengths approaching the Schwinger field, Esch = 1.3 × 1018 Vm−13. Such fields exist in extreme astrophysical environments, such as pulsar magnetospheres4, may be accessed by high-power laser systems5,6,7, dense particle beams interacting with plasma8, crystals9 and at the interaction point of next-generation particle colliders10. Radiation reaction affects the energy of inverse Compton scattered (ICS) photons used for various applications11,12,13,14,15,16,17.

Classical radiation reaction theories do not limit the frequency of radiation emitted by accelerating charges and omit stochastic effects inherent in photon emission18, thus demanding a quantum treatment. Two quantum radiation reaction models, the quantum-continuous19 and quantum-stochastic2,20,21 models, correct the former issue, while only the quantum-stochastic model incorporates stochasticity19. Such models are of fundamental importance, providing insight into the effect of the electron self-force on its dynamics in electromagnetic fields.

In astrophysics, radiation reaction is predicted to limit electron-positron cascades, which populate the magnetospheres of pulsars, magnetars and active black holes with plasma4,22 and can strongly affect reconnection in such plasmas23,24. Radiation reaction has been proposed as a dominant factor in gamma-burst generation25 and is expected to influence the dynamics of pair plasmas26, including relativistic current sheets27. Strong-field quantum radiation reaction may substantially affect the interaction point at high luminosity >100 GeV class particle colliders10. Strong electromagnetic fields produced by multi-petawatt laser systems5,6,7,28,29 will enable the exploration of compact particle acceleration30,31 and radiation generation32 (e.g. via inelastic electron-photon scattering, termed Compton scattering33) in higher-power regimes.

In strong-field environments, quantum radiation reaction is expected to dominate laser-solid target interactions34,35, ion-acceleration36 and inverse Compton scattering (ICS), which has recently garnered considerable interest as a means of probing strong-field quantum electrodynamics37,38.

Understanding the effect of radiation reaction on ICS photon spectra is crucial for diverse applications, including industrial, defence, archaeological11,12 and medical13 imaging, nuclear physics14,15,16 and proposed schemes for gamma-gamma colliders17.

The impact of radiation reaction on particle dynamics is characterised by the dimensionless intensity parameter, \({a}_{0}=\frac{{E}_{L}e}{{\omega }_{L}{m}_{e}c}\) and the electron quantum parameter, η = ERF/Esch, where e and me are the magnitude of the electron  charge and electron mass respectively, c is the speed of light in vacuum, EL, ERF are the external electric field (laser) strengths in the laboratory and electron rest frames, respectively, and ωL is the electric field frequency in the laboratory frame. When a0 ≳ 1, both relativistic and multi-(laser) photon effects become important. The regime of strong classical radiation reaction is characterised by αa0η ≃ 1 and η ≪ 1, where α is the fine structure constant2,39. Quantum effects dominate when αa0 ≃ 1 and η ≳ 12,39.

In regimes dominated by classical radiation reaction, an electron emits many photons, which each remove a small fraction of its energy2,21,40; radiation emission is treated as continuous and its impact on electron motion is well-described by the classical Landau-Lifshitz equation18. In the strong-field quantum regime, interactions with the laser field must be treated non-perturbatively and are absorbed into electron basis states by quantising the Dirac field in the presence of the laser field (Furry picture). Photon emission is described perturbatively with respect to these states12 and becomes stochastic, with single emissions removing significant fractions of the electron energy2. In this work, the ‘quantum-stochastic’ model of radiation reaction employs the locally constant field approximation, which assumes emission events are point-like, depending only on local electric and magnetic fields, which are assumed to be constant over the timescale of photon emission3. Between emission events, electron motion is assumed to be classical, which is a good approximation in the ultra-relativistic regime41.

The quantum-continuous model, known as the semi-classical model in the literature, aims to incorporate quantum physics in a classical framework, treating radiation emission as continuous42, but capturing the same rate of change of average electron momentum as the quantum-stochastic model via the inclusion of a correction term, the Gaunt factor19.

To date, six studies9,43,44,45,46 have aimed to measure radiation reaction in strong fields (a0 ≫ 1). Of these studies, only one44 conducted a quantitative comparison of different radiation reaction models, which was limited to 1σ significance.

Using an all-optical setup, we accessed a0 ≈ 10 and η ≤ 0.09, for which strong-field non-perturbative effects dominate and quantum effects are influential, in contrast to early experiments at LINACs47,48,49 and storage rings50, for which a0 < 1. Previous experiments at CORELS45, ELI-NP and APOLLON46, or using crystals9,51,52, did not report an observation of radiation reaction on lepton spectra.

This work exceeds the 5σ significance threshold required for a definitive observation of radiation reaction and presents strong, quantitative evidence favouring quantum models over a classical model. This represents a substantial improvement on previous all-optical experiments, which reported ≤ 3σ evidence of radiation reaction and performed model selection with < 1σ significance due to data scarcity and large uncertainties43,44. The higher significance of our findings is largely due to the greater number of successful collisions reported in this work (> 600) compared to previous all-optical experiments (< 10). This substantial increase in the number of successful collisions was enabled by the implementation of automated timing and pointing stabilisation of the laser.

Notably, our findings clarify outstanding questions raised by previous experiments43,44 regarding the relative validity of different radiation reaction models in the classical-quantum regime.

The experimental setup is illustrated in Fig. 1a. Electron beams with mean energy (609 ± 2(stat)(12)(syst)) MeV and a shot-to-shot standard deviation (40 ± 1) MeV (standard errors were calculated using bootstrapping due to non-normal data distributions), generated using laser-driven wakefield acceleration31, collided with a tightly focused, counter-propagating laser pulse with I = (1.0 ± 0.2) × 1021 W cm−2, λL = 2πc/ωL = 0.8 μm, a0 = 21.4 ± 1.8 and 〈η〉 ≤0.13 ± 0.02 at the laser focus, sufficient to probe the strong-field quantum regime. For more details, see ‘experimental methods’.

Fig. 1: Experimental set-up, qualitative comparisons of measured hits and nulls and simulated radiation reaction models.
Fig. 1: Experimental set-up, qualitative comparisons of measured hits and nulls and simulated radiation reaction models.
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a Experimental setup: one laser pulse, focused into a gas jet, drove a wakefield, accelerating electrons. A second, tightly focused, counter-propagating laser pulse collided with the electron beam, which emitted gamma photons. The electron spectrometer consisted of a dipole magnet, which dispersed the electron beam through a wire array onto two LANEX scintillating screens (green). A caesium iodide (CsI) profile screen and stack characterised the transverse profile and spectrum of the emitted gamma radiation, respectively. b Simulated post-collision electron spectra (normalised by integration) and photon spectra illustrating the classical, quantum-stochastic and quantum-continuous model predictions for a0 = 10 (bottom) and a0 = 20 (top). The electron beam and laser pulse collided 40 fs after the laser focus. The transverse and longitudinal laser intensity profiles were Gaussian, with respective full-width half-maxima (FWHM) of 2.47 μm and 30 fs. c Measured electron spectra for hits with high gamma profile yields are shown above those measured for and moderate yields, together with corresponding gamma profile signals. Nulls have been randomly selected.

Figure 1b compares predictions of post-collision electron and photon spectra for the classical, quantum-continuous and quantum-stochastic models. All models predict net electron energy losses (evidenced by lower-energy post-collision electron spectra), which, together with photon yield, scale with increasing a0. These model-independent indicators of radiation reaction are used in the frequentist analysis in section ‘Results and discussions’.

Quantum models prohibit the emission of photons with energies exceeding the electron energy, thus predicting lower energy losses than the classical model, evidenced by the mean post-collision electron energies and photon yields above 100 MeV in Fig. 1b). Unlike the classical and quantum-continuous models, the quantum-stochastic model predicts spectral broadening, arising from probabilistic photon emission42,53,54, indicated by the relative widths of post-collision electron spectra in Fig. 1b.

The model-independent trends in Fig. 1b appear qualitatively in experimental data in Fig. 1c; this is purely illustrative and does not represent the frequentist analysis. In Fig. 1c, electron spectra for hits with moderate and high photon yields exhibit less pronounced high-energy peaks than randomly selected nulls, consistent with fewer high-energy electrons. As photon yield increases, the proportion of charge at high energies decreases.

Results and discussions

In lieu of measured pre-collision electron spectra, electron beams measured for misses and beam-off shots (nulls) are compared to those measured for successful collisions (hits). Hit and null identification is illustrated in Fig. 2a and discussed in the section ‘Methods for frequentist analysis’. The profile screen background subtraction is detailed by Los55.

Fig. 2: Shot selection and summary statistics.
Fig. 2: Shot selection and summary statistics.
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a The shot selection procedure is illustrated. Background-subtracted total counts measured by the gamma profile diagnostic are shown as a function of Q〈γ2〉 for all shots, where Q and γ denote electron beam total charge and Lorentz factor, respectively. A constant, Cγ = 1 × 105 has been added to the normalised total counts for all shots to allow the data to be shown on a logarithmic scale. Nulls (blue, 608 shots) consist of combined misses and beam-off shots. The latter lie within 1σ (cyan, dashed) of the scaling of background gamma yield with Q〈γ2〉 (cyan, continuous), where σ is the standard deviation of the fit to the background scaling. The small fraction of nulls that lie above this threshold are beam-off shots. Hits (red, 687 shots) lie 3σ (orange, dashed) above the background scaling. The grey points cannot be categorised as hits or nulls and thus are excluded from the analysis. The shots analysed using the Bayesian framework are encircled (black). b Simulated post-collision electron spectra, normalised by integration, predicted by different radiation reaction models for a collision between an electron beam (pre-collision spectrum shown) with a Gaussian temporal profile with full-width half-maxima (FWHM) 141 fs and a laser pulse with a0 = 14 and Gaussian transverse and longitudinal intensity profiles with FWHM 2.47 μm and 45 fs, respectively. The collision was offset temporally from the laser focus by 60 fs. The mean energy, 〈E〉, and peak height above the 70th percentile electron energy, P70, which indicates the prominence of the high-energy peak in the spectrum, are shown.

Figure 2b illustrates two model-independent signatures of radiation reaction, identified by extensive simulation work provided in Supplementary Fig. 11 in Supplementary Information A.3 and as shown by Los et al.56. These are used to quantitatively examine whether differences between hits and nulls exceeded shot-to-shot variation in electron spectra and whether any such differences are consistent with radiation reaction. Compared to the pre-collision spectrum, simulated post-collision spectra have lower mean energies, 〈E〉, and less pronounced peaks above the 70th percentile energy, P70, calculated using electron spectra normalised by integration. The 70th percentile energy is the energy under which 70% of the electron beam charge lies. Changes in P70 reflect the redistribution of charge due to spectral broadening. Spectral broadening may reflect stochasticity or spatial and temporal misalignments between the laser pulse and electrons, which cause equally energetic electrons to experience different laser intensities and hence energy losses.

In Fig. 3a, b, there are fewer high-energy electron spectra and fewer strongly-peaked spectra for hits compared to nulls, meaning fewer high-energy and more low-energy electron beams in the former population compared to the latter. Two-sample Kolmogorov-Smirnov (KS) tests (selected for applicability to arbitrary distributions) confirm this, indicating the null hypothesis that hits and nulls originated from the same distribution can be rejected at the 5σ (p = 5.3 × 10−9) and 4σ (p = 2.7 × 10−5) levels for P70 and 〈E〉, respectively.

Fig. 3: Model-independent analysis of electron energy loss and photon yield.
Fig. 3: Model-independent analysis of electron energy loss and photon yield.
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Distributions of a 〈E〉 and b P70 for measured hits (red) and nulls (blue). Hit and null distributions have been normalised to the total number of shots in each. Hit and null distributions of (c) mean 〈E〉, denoted \(\langle \widetilde{E}\rangle\), and d mean P70, denoted \({\widetilde{P}}_{70}\), obtained by bootstrapping hit and null distributions in a, b respectively. Bottom: e Mean 〈E〉 for the 687 hits and 607 nulls analysed, binned logarithmically by gamma profile yield normalised to Q〈γ2〉, error bars reflect the standard error of 〈E〉 per bin; different bins contain different numbers of shots. f Similar to e, for P70.

The distribution of mean values, \(\langle \widetilde{E}\rangle\), calculated for the distribution of 〈E〉 in Fig. 3a using bootstrapping (see section ‘Bootstrapping’) is shown in Fig. 3c. The distribution of mean values for P70, denoted \({\widetilde{P}}_{70}\), is shown in Fig. 3b. The hit distribution means for \(\langle \widetilde{E}\rangle\) and \({\widetilde{P}}_{70}\) lie 3σ (p = 2.0 × 10−4) and 5σ (p = 3.3 × 10−7) below the corresponding means for nulls, respectively. The close agreement between the significance values obtained using KS tests and by bootstrapping attests to their reliability. The highly significant observation of lower mean electron energies and lower peak heights for hits compared to nulls is consistent with electron energy loss and spectral broadening and constitutes strong evidence of radiation reaction.

As illustrated in Fig. 2a, photon yields for all background sources (betatron radiation, bremsstrahlung) and thus for nulls, scale positively with Q〈γ2〉, where Q and 〈γ2〉 denote the total charge and expected value of γ2 for the electron beam, respectively. The photon counts in Fig. 3e, f are normalised by Q〈γ2〉 to remove this background scaling to first-order. After this correction is applied, a small residual positive correlation remains for the nulls.

For hits in Fig. 3e, f, P70 and 〈E〉 decrease with increasing normalised photon yield. For the brightest hits, P70 and 〈E〉 lie 15σ and 7σ below the null means, respectively. Thus, the electron beam has lower energy for successful collisions compared to unsuccessful collisions, consistent with radiation reaction.

By contrast to the positive scaling observed for nulls, hits exhibit a negative scaling with gamma yield, consistent with energy loss and hence radiation reaction. The correlation coefficients for hits and nulls differ significantly, by 40σ and 34σ for P70 and 〈E〉, respectively (see Supplementary Fig. 10, Supplementary Information A.3). The negative correlations observed for hits demonstrate that different physical mechanisms dominate radiation production compared to nulls and confirm that energy losses ‘switch-on’ for successful collisions.

The highly significant observation of electron energy loss during successful collisions, together with the strong correlation between increasing photon yield and greater electron energy loss, constitutes an observation of radiation reaction on particles.

All-optical experiments offer advantages such as natural timing and alignment of the electron beam and the colliding laser, but also present challenges. Parameters such as laser intensity and the relative size, duration and spatio-temporal overlap of the electron beam and the colliding laser pulse strongly affect post-collision electron and photon spectra, but are not measured on-shot and vary substantially between shots. This precludes the straightforward model comparison illustrated in Fig. 1b and necessitates an analysis that accounts for uncertainties due to unknown pre-collision electron spectra and unknown collision parameters when comparing radiation reaction models. This was achieved using Bayesian inference.

If a model, M, depends on parameters with unknown values, Bayesian inference estimates these parameters by constructing the probability that the model is accurate given the data observed, D, called the posterior probability, P(M ∣ D) (this notation integrates over all parameter values). The most likely parameter values are those that optimise the posterior probability.

The posterior probability is calculated using Bayes’ theorem, given in Eq. (1)57,

$$P(M| D)=\frac{P(D| M)P(M)}{P(D)}$$
(1)

where the likelihood, P(D ∣ M), is the probability of observing the data given the model, the prior probability, P(M), represents prior beliefs about the model before observing data and P(D) is the (constant) probability of observing the data. In the absence of an analytic expression for the posterior, the posterior distribution is computed algorithmically by iteratively sampling model parameters, calculating the corresponding likelihood and hence updating the posterior until convergence is reached. For efficiency, high-probability regions of the parameter space are preferentially sampled.

As pre-collision electron spectra were not measured for successful collisions, distributions of pre-collision electron spectra were predicted for these shots using a neural network58, given the measured laser energy, plasma density and longitudinal profile of plasma re-combination light. The distribution variance reflects the prediction uncertainty, which, together with experimental uncertainties, is accounted for in the Bayesian analysis.

Collision parameters that could not be measured were inferred using Bayesian inference56. To avoid over-fitting and excessive computational run times56, only a subset of collision parameters was inferred and all other parameters were fixed. We chose to infer τe, a0 and the longitudinal displacement of the collision from the laser focus, Zd, as they have the highest expected impact on the post-collision electron and photon spectra and exhibit degeneracies with parameters that were not inferred. Degeneracies allow different combinations of collision parameters to produce the same collision distributions of η and a0, denoted by \(\widetilde{\eta }\) and \({\widetilde{a}}_{0}\) and hence the same observables, as discussed in section ‘Free and fixed parameter selection, Bayesian inference test cases’ and by Los et al.56. Hence, the inference procedure returns ‘effective’ values for τe, a0 and Zd; so called as they reflect the distributions of \(\widetilde{\eta }\) and \({\widetilde{a}}_{0}\) that reproduce the observables, rather than accurately representing electron beam and laser properties.

Transverse misalignments between the electron beam and the laser pulse, rd, were assumed to be 0. To maximise the probability that this condition was met, only the ten shots with the highest gamma yields normalised by Q〈γ2〉 were analysed. Constraints in computational resources limited the number of shots analysed; each inference required ≈19200 CPU h, 60 GB per CPU.

Three inferences were performed per shot; one for each model of radiation reaction. During each inference, different sets of collision parameters were combined with the predicted pre-collision electron spectrum to reconstruct different collisions and predict corresponding post-collision electron and photon spectra for the relevant radiation reaction model.

Although the ‘true’ parameters are unknown, the relative validity of two models can be compared using the ratio of model evidence or Bayes factor (see Eq. (6), section ‘Analytical Methods’). The model evidence is obtained by integrating the likelihood weighted by the parameter priors over all parameter space. Bayes factors provide a more robust metric of model performance compared to a frequentist approach (e.g. a reduced chi-squared for a least-squares fit), which only compares the validity of a model to a null hypothesis for the ‘best-fit’ parameters.

The Bayesian analysis was tested on realistic electron spectra for various simulated collisions with differing collision parameters. For each test, the Bayesian analysis consistently favoured the correct model and inferred the first moments of the collision distributions of \(\widetilde{\eta }\) and \({\widetilde{a}}_{0}\), respectively denoted by \(\langle \widetilde{\eta }\rangle\) and \(\langle {\widetilde{a}}_{0}\rangle\), within 1σ of their simulation values. Model differentiation was only accurate for rd ≤ 1.6w0, where w0 = 2.5 μm was the laser waist at focus56.

Measured and inferred post-collision electron and photon spectra are shown in Fig. 4 for the highest normalised gamma yield shot. Additional results are provided in the Supplementary Information A.2. In Fig. 4a, the 〈E〉 and P70 measured post-collision lie 3σ and 11σ below the neural net pre-collisionpredictions, respectively; note systematic uncertainties cancel as the correlation coefficient is 1. Only the quantum-stochastic model infers both 〈E〉 and P70 within 1σ of the measured spectrum, indicating marginally higher performance. Both quantum models predict more accurate photon spectra with lower uncertainties than the classical model, as indicated by the mean photon energies in Fig. 4b. Typically, the inferred photon energy deposition lies within 1σ of the measured energy deposition, confirming the goodness of fit for this diagnostic. The requirement to reproduce both electron and photon spectra using only three fitting parameters places a stronger constraint than an analysis considering a single diagnostic, which risks overfitting. This accounts for the discrepancy between measured and inferred mean photon energies and highlights the importance of considering multiple diagnostics simultaneously.

Fig. 4: Bayesian inference results for the highest gamma-yield shot normalised to Q〈γ2〉 (shot 2).
Fig. 4: Bayesian inference results for the highest gamma-yield shot normalised to Q〈γ2〉 (shot 2).
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Measured data (red) and predictions for the classical (green), quantum-continuous (blue), and quantum-stochastic (magenta) models, which inferred \(\langle {\widetilde{a}}_{0}\rangle=6.2\pm 1.0\) and \({\sigma }_{{a}_{0}}=1.2\pm 0.3\), \(\langle {\widetilde{a}}_{0}\rangle=6.8\pm 0.9\) and \({\sigma }_{{a}_{0}}=0.4\pm 0.1\) and \(\langle {\widetilde{a}}_{0}\rangle=6.7\pm 0.9\) and \({\sigma }_{{a}_{0}}=0.4\pm 0.1\), respectively. a Measured and inferred post-collision electron spectra. The shaded region on the measured spectrum reflects the systematic error due to uncertainties in the positions of the magnet and scintillating screens. For the former, 〈E〉 = (564.1 ± 0.0(10.3)) MeV, P70 = (0.83 ± 0.00(0.01)) GeV−1. The distribution of pre-collision electron spectra predicted by the neural network (orange), for which 〈E〉 = (574.1 ± 3.9(10.7)) MeV, P70 = (1.29 ± 0.05(0.02)) GeV−1 and its median (black). b Measured and inferred photon energy deposition in each scintillation crystal as a function of propagation distance in the CsI photon spectrometer. Error bars indicate the standard deviation of scintillation light in each crystal combined with the standard error from the photon number calibration. The mean photon energy measured was (63.3 ± 5.8) MeV.

In Fig. 5, individual shots yield weak (Bayes factor < 3.259) but consistent evidence favouring the quantum models over the classical model, increasing the credibility of the former. The Bayes factors for the quantum-continuous and quantum-stochastic models lie consistently around 1, signifying their comparable performance. As the inferred parameters for each shot are independent, model evidence can be combined by multiplication. The combined Bayes factors over 10 shots indicate strong evidence favouring the quantum-stochastic and quantum-continuous models over the classical model, but insufficient evidence to distinguish between the quantum models.

Fig. 5: Bayesian comparison of radiation reaction models.
Fig. 5: Bayesian comparison of radiation reaction models.
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Bayes factors for individual shots (circles) and combined over ten shots (triangles) are shown. Weak (white), substantial (light shading) and strong (dark shading) evidence favouring model 1 (red) or model 2 (blue) are categorised according to the half-log scale convention outlined by Kass and Raftery59. The dashed grey line indicates equal performance of the compared models.

The quantum models better replicate both photon and electron spectra simultaneously compared to classical predictions, as they predict lower energy losses. The electron spectra largely determine the posterior location, while photon spectra predominantly affect the posterior shape as their likelihood distributions are far narrower; inferred photon spectra have fewer degrees of freedom and thus are less accurate than inferred electron spectra, but their fractional uncertainties are not substantially greater. Thus, both diagnostics play a critical role in constraining the posterior. As electron spectra contribute similarly to the posterior probability across all models, as indicated by their inferred 〈E〉 and P70, relative model performance is determined by the photon spectrum. Quantum models yield higher posterior probabilities than the classical model as they infer photon spectra more accurately and with lower uncertainty. The novel ability to analyse photon and electron spectra within one self-consistent framework is critical for model comparison and a key strength of this Bayesian approach.

The quantum-stochastic model inferred \(0.05\le \langle \widetilde{\eta }\rangle \le 0.1\) and \(7\le \langle {\widetilde{a}}_{0}\rangle \le 13\) across ten shots. For the inferred \(\langle {\widetilde{a}}_{0}\rangle \ge 7\), the transverse offset was rd ≤ 0.64 w0, well within the range of rd for which model selection is accurate.

Analytic classical and quantum-stochastic models60,61 and a numerical quantum-continuous model provide an independent corroboration of the Bayesian analysis. A Gaussian is fitted to the high-energy peak in the pre-collision electron spectrum in Fig. 4a. Using the \(\langle {\widetilde{a}}_{0}\rangle\) inferred by the Bayesian analysis, the corresponding post-collision spectrum is predicted analytically for each model. The mean energy, 〈Ep〉, and standard deviation, σp, of the pre-collision peak are 〈Ep〉 = 864 MeV and σp = 32 MeV (4%). The post-collision peak predicted by the classical model has 〈Ep〉 = 807 MeV and σp = 19 MeV (2%), while quantum-continuous and quantum-stochastic models predict peaks with 〈Ep〉 = 803 MeV and σp = 34 MeV (4%) and 〈Ep〉 = 814 MeV and σp = 58 MeV (16%), respectively, consistent with Arran et al.62 and Yoffe et al.63. The predicted post-collision spectra are consistent with the measured data, validating the Bayesian analysis. The post-collision spectral widths predicted analytically appear to indicate the feasibility of model selection between all three models for these collision conditions. However, as large variations in laser intensity during the collision also cause spectral broadening, more precise knowledge of collision conditions is needed to differentiate between quantum models.

The onset of quantum effects and reduced accuracy of the Landau-Lifshitz model observed for \(0.05\le \langle \widetilde{\eta }\rangle \le 0.1\) and \(7\le \langle {\widetilde{a}}_{0}\rangle \le 13\) motivate the use of quantum-corrected models in this regime. Quantum-corrected radiation reaction models predict lower energy losses for charged particles in strong fields compared to classical models, with wide-ranging consequences. Quantum-corrected ICS photon spectra have fewer high-energy photons; this has implications for ICS-based photon sources used for imaging and nuclear physics. In astrophysical environments, a quantum-corrected synchrotron emission reduces emission and cooling rates of pulsars and galactic nuclei jets and affects synchrotron emission-based estimates of magnetic field strengths. Quantum-corrected radiation models indicate higher centre-of-mass energies in particle accelerators, while for laser-solid target ion acceleration, such models predict stronger sheath fields and hence higher ion energies.

In conclusion, we report a highly significant observation of radiation reaction on electron and photon spectra and present quantitative, strong evidence to favour two quantum models over a classical model, for \(0.05\le \langle \widetilde{\eta }\rangle \le 0.1\) and \(7\le \langle {\widetilde{a}}_{0}\rangle \le 13\), due to the lower energy losses the former predict. While spectral broadening was observed, insufficient knowledge of collision parameters and large uncertainties on predicted pre-collision electron spectra inhibited our ability to determine whether this arose due to stochasticity. Model differentiability would improve with stable, mono-energetic electron beams and strongly constraining priors motivated by collision parameter measurements. Lower variation in transverse alignment or in collisions with an expanded, higher-power laser would boost statistics at higher a0, facilitating investigations of quantum phenomena over a greater range of η and providing new insight into the validity of the quantum-continuous and quantum-stochastic models in these regimes.

Methods

Experimental methods

The experiment was conducted using the dual-beam Gemini laser at the Central Laser Facility, Rutherford Appleton Laboratory, UK. An f/40 off-axis parabola was used to focus one arm of the linearly polarised, two-beam system to a transverse full-width half-maximum (FWHM) of the focal spot intensity of (35 ± 3) μm × (40.6 ± 1.2) μm. An off-shot Grenouille measurement of the FWHM duration of the laser intensity yielded (59.5 ± 2.5) fs. The laser delivered (6.8 ± 0.6) J to the target, corresponding to an a0 = 1.0 ± 0.15 (standard deviation given).

The laser-wakefield drive beam was focused into a 15 mm supersonic gas jet with a trapezoidal density profile and 5 mm ramps. An average electron density of ≈(1.1 ± 0.2) × 1018 cm−18 was attained at the peak of the trapezoidal profile with He gas, doped with 1% N2 to induce ionisation injection64.

The electron beam and photons produced by ICS propagated through the hole in the f/2 parabola, which had an acceptance angle of 42 mrad. A ∫ B(x) dx = 0.4 T m dipole magnet was used to disperse the electron beam through a wire array onto two sequential LANEX screens, which were imaged by two cooled 16-bit cameras. The electron spectrum was subsequently retrieved with the aid of a tracking algorithm, which computed the trajectories of electrons through the magnetic field. The wire array and two screens allowed the degeneracy between the electron beam energy and pointing into the magnet to be de-convolved65,66,67. The systematic uncertainty in the retrieved electron energy due to the uncertainties in the relative positions of the magnet, lanex screens and gas jet was

$${\zeta }_{e}\,[{\rm{MeV}}]={C}_{e}E\,{[{\rm{MeV}}]}^{2},$$
(2)

where E denotes electron energy and Ce = 32.45 × 10−6 MeV−1.

The wakefield accelerator produced (140 ± 1) pC electron beams with mean and standard deviation energy (609 ± 2(stat)(12)(syst)) MeV and shot-to-shot standard deviation (40 ± 1) MeV, respectively (standard error given). Electrons with energies < 300 MeV could not be measured. A radial source size of < (0.7 ± 0.1) μm was assumed, in line with previous measurements68,69. The FWHM energy-dependent electron beam divergence, θD, measured along the axis transverse to the dispersion plane by the LANEX screens, which measured the electron spectrum, was \(({b}_{1}-{b}_{2}\sqrt{\gamma {m}_{e}\,[{\rm{GeV}}]})\), where \({b}_{1}=1.3{0}_{-0.19}^{+0.26}\,\rm{mrad}\), \({b}_{2}=0.2{6}_{-0.28}^{+0.24}\,{\rm{mrad}}\,{{\rm{GeV}}}^{-0.5}\). The axial symmetry of the electron beam divergence was confirmed using linear Thomson scattering70.

The colliding laser pulse was focused at the rear of the gas jet by an f/2 parabola with a 25.4 mm on-axis hole. The laser intensity profile had transverse FWHM (2.5 ± 0.2) μm × (2.1 ± 0.1) μm and FWHM duration (45.0 ± 2.5) fs. Due to energy losses in the laser system, including the on-axis hole in the f/2 parabola, the energy on-target was (6.13 ± 0.02) J, yielding a peak a0 = 21.4 ± 1.8.

Spatial and temporal overlap of laser pulses

Accessing collision a0 ≳ 10 required ≲ 1 μm and ≲ 10 fs precision spatio-temporal overlap of the electron beam and laser focus. This required a careful alignment procedure, detailed below.

A micron knife-edge 90∘ prism, imaged using a × 10 microscope objective, was used to overlap the two laser pulses spatially and temporally, where the latter was achieved using spatial interferometry, as demonstrated previously44,71. By optimising the contrast of the interference pattern, the two laser pulses were synchronised to within ± 10 fs. To time the colliding pulse, two additional effects needed to be corrected for, namely the reduced non-linear group velocity of the wakefield-driver laser pulse in the plasma and the longitudinal displacement of the electron beam from the wakefield driver by N plasma wavelengths, where \(N=\frac{1}{2}\) for an electron beam travelling at the dephasing limit with velocity close to c. Thus, the longitudinal collision position was shifted closer to the gas jet by δz,

$${\delta }_{z}=\frac{3d}{4}\frac{{n}_{e}}{{n}_{c}}+N\frac{{\lambda }_{L}}{2}\sqrt{\frac{{n}_{c}}{{n}_{e}}}$$
(3)

where ne and \({n}_{c}=\frac{{\epsilon }_{0}{m}_{e}{\omega }_{L}^{2}}{{e}^{2}}\) are the plasma and critical densities and d is the distance from the upstream edge of the plasma to the injection point.

Fluctuations in ambient temperature altered the temporal and spatial alignment between the electron beam and the colliding laser over the course of shooting. Thus, implementing timing and pointing stabilisation was key to obtaining a high number of successful collisions.

Once timed and aligned, references for optimal timing and alignment were taken using a spectral interferometer in the laser area and a diagnostic of the f/40 beam pointing, respectively. Long-term drifts in spatial and temporal alignment could be corrected for by adjusting the tip and tilt of a mirror in the f/40 beamline and by altering the path length of one of the laser arms, respectively.

Long term timing and pointing drifts were corrected using an automated feedback loop. Thus, the remaining misalignment between the electron beam and the colliding laser resulted from shot-to-shot variation in the beam paths and the laser pointing due to vibrations. The temporal jitter between the two laser arms was assumed to be normally distributed with a standard deviation ± 30 fs. An additional source of uncertainty in the timing between the electron beam and colliding laser stems from the unknown value of d, which we assume to be uniformly distributed with lower and upper bounds of 0 mm and 10 mm, respectively. To correct for δz, the path between the two beams was reduced by 20 fs. Thus, the offset in timing between the two beams, \(\Delta t\approx \frac{{\delta }_{z}}{c}\), 2.7 fs ≤ Δt ≤ 45.8 fs. The standard deviations of the radial positions of the colliding laser and the electron beam due to pointing variations were measured to be (0.53 ± 0.26) μm and (17.5 ± 0.5) μm, respectively, which correspond to a standard deviation in transverse alignment of (17.5 ± 0.6) μm.

Gamma radiation diagnostics

The angular distribution of gamma radiation was measured using a 50 × 50 × 10 mm profile screen consisting of 1 × 1 × 10 mm CsI(Tl) crystals separated by 0.2 mm titanium oxide spacers. Dimensions have the format (horizontal × vertical × depth). The front of the profile screen was coated with 0.5 mm titanium oxide. The profile screen was placed outside the vacuum chamber, (2244 ± 4) mm from the interaction and was imaged using a cooled 16-bit CCD camera.

The energy deposition of gamma photons was measured using a 50 × 50 × 150 mm dual-axis CsI(Tl) scintillator, comprised of alternating layers of horizontally and vertically oriented 5 × 5 × 50 mm CsI(Tl) crystals, which were held in place by a 3D printed 1 mm nylon frame and separated by 1 mm rubber spacers to prevent light leakage between crystals. Two cooled 16-bit CCD cameras imaged the scintillation light from above and laterally. The calorimeter was placed outside the vacuum chamber, (3570 ± 3) mm from the interaction.

Geant472,73 simulations were used to obtain the energy deposition in the CsI photon diagnostics70 as a function of incident photon energy, as demonstrated in Behm et al.74. The maximum and minimum photon energies used to calculate energy deposition were 0.01 MeV and 1 GeV, respectively. These simulations included the chamber geometry, large objects inside the chamber such as the dipole magnets and all materials placed in the beam path, including a 1 mm alumina laser block, a 25 μm Kapton window with a 375 μm Kevlar backing sheet and a 25 μm aluminium foil. Variations in the scintillation efficiency of the crystals and in the efficiency of the imaging system were characterised and subsequently compensated for by comparing the measured and simulated response of the calorimeter to bremsstrahlung generated by an electron beam propagating through a (1.5 ± 0.1) mm PTFE target with radiation length much less than the radiation length of a 1 GeV electron beam.

The ICS spectrum, SICS, has the characteristic shape

$${S}_{ICS}=A{\left(\frac{{E}_{\gamma }}{{E}_{\gamma c}}\right)}^{-\frac{2}{3}}{e}^{-\left(\frac{{E}_{\gamma }}{{E}_{\gamma c}}\right)}$$
(4)

where A and Eγc represent photon number and the critical energy of the spectrum, respectively. Bayesian inference was used to obtain the values of A and Eγc for which the energy deposition calculated using Eq. (4) fitted the measured energy deposition.

Methods for frequentist analysis

Following the approach employed by Cole et al.44, the photon yield measured by the gamma profile diagnostic was used to identify hits and nulls. The total yield measured by the profile screen, Yγ, is expected to scale with the electron beam total charge, Q and expected value of γ2, 〈γ2〉:

$${Y}_{\gamma }=({C}_{{\rm{ICS}}}{a}_{0}^{2}+{C}_{{\rm{BKG}}})Q\langle {\gamma }^{2}\rangle$$
(5)

where the first and second terms describe the contributions of ICS and background (e.g. due to bremsstrahlung) to the total yield, respectively and CICS, CBKG, are scaling constants. The scaling for ICS with photon yield holds for \(\gamma {a}_{0}^{2} < 5.5\times 1{0}^{5}\)75. Multiple sets of shots in which the counter-propagating laser was not fired were taken to obtain the characteristic background scaling with Q〈γ2〉. Misses and hits, classified as shots which produced yields within 1σ and above 3σ of the background scaling, respectively, are shown in Fig. 2.

In this work, we present a controlled experiment in which the only difference between hits and nulls is the presence or absence of a collision between a laser and an electron beam, respectively. Systematic changes in electron beam properties and the effect of background radiation mechanisms were mitigated and accounted for, if necessary.

Long-term drifts in the laser pointing and timing, which could produce systematic changes in the electron spectra and overlap of the electron beam and laser at the collision, were minimised using automated correction of the spatial and temporal overlap of the wakefield drive laser and the colliding laser. Hits and nulls were interleaved to ensure null electron beams were representative of hits.

Background radiation mechanisms, namely betatron and bremsstrahlung, were characterised using laser-off shots. Distributions of the mean electron beam energy, 〈E〉, and height of its spectral peak above the 70th percentile energy, P70, were obtained for hits and nulls. Both background radiation mechanisms produce a positive scaling between photon yield and Q〈γ2〉.

All radiation reaction models predict a reduction in 〈E〉 and P70 (the latter is predicted for all models if the electron beam interacts with a range of laser intensities), for hits compared to nulls and a photon yield which increases with decreasing P70 and decreasing 〈E〉. These model-independent metrics of radiation reaction were compared for hits and nulls.

No known systematic background effect can produce the reduction in 〈E〉 and P70 for successful collisions and the negative scaling of 〈E〉 and P70 with photon yield simultaneously. Background radiation sources produce a positive scaling of photon yield with Q〈γ2〉, opposite to the trend predicted for radiation reaction. As hits and nulls were interleaved, both systematic and random changes in laser and electron properties would affect both hits and nulls equally, in contrast to radiation reaction models' predictions.

Given that the only difference between hits and nulls is the presence or absence of a collision between the electron beam and laser, the simultaneous observation of all of the above signatures of radiation reaction signatures for hits and their absence for nulls constitutes a definitive observation of radiation reaction.

Analytical methods

Free and fixed parameter selection, Bayesian inference test cases

A full, detailed account of the implementation and testing of the Bayesian inference framework and the forward models used therein is provided in ref. 56.

Several parameters (including electron beam source size and chirp, laser duration, transverse offset, etc) have been assigned fixed values in the forward model (i.e. are not inferred). Free and fixed parameters were chosen in accordance with the following criteria:

  • The expected effect of variation. This incorporates both the probability of parameter variation by a given amount and the impact of this variation on the post-collision electron and photon spectra. Parameters fixed due to their small expected effect include laser duration and focal spot size and electron beam chirp.

  • Shot selection. The ten shots which produced the highest CsI profile screen yields, normalised to Q〈γ2〉, were analysed using the Bayesian framework. As the laser intensity decreases most steeply with transverse (rather than longitudinal) misalignment, by analysing the highest yield shots the probability of a large transverse offset is reduced.

  • Degeneracy. If changes in two (or more) collision parameters engender similar alterations in post-collision observables, it is possible to fix one of these parameters and vary the second to reproduce the effect of the fixed parameter. For example, if the electron beam has finite divergence, varying the longitudinal position of the collision alters the size of the electron beam at the collision. This produces post-collision observables similar to those obtained by varying the electron beam source size. Degeneracy allows changes in the electron beam source size, divergence and transverse offset from the laser focus to be compensated by free parameters (laser energy, longitudinal offset of the collision from focus, electron beam duration). The laser energy was chosen as a free parameter to enable the Bayesian inference procedure to tackle shot-to-shot variations therein.

The laser, electron and collision parameters which were measured, estimated or inferred based on previous measurements, are summarised alongside their assigned values in the forward models in Supplementary Table 1, Supplementary Table 2 and Supplementary Table 3, respectively, in the Supplementary Information A.1.

Bayesian inference implementation and testing

The Bayesian inference procedures used the Markov chain Monte Carlo76 from the Python package emcee. Lack of on-shot parameter measurements necessitated broad priors56.

Extensive testing of the Bayesian analysis56, which included inferences on synthetic data where the fixed collision parameters (e.g. Zd) were assigned different values from the forward models, revealed that for all test cases the inference procedure yielded weak evidence favouring the correct model; none of the test cases yielded false positives. This is a clear demonstration of the accuracy of the model selection capabilities of the Bayesian framework we have developed. In all test cases, the highest performing model(s) inferred \(\langle {\widetilde{a}}_{0}\rangle\) and 〈η〉 within 1σ of the input (correct) value. This shows the analysis infers the physical parameters governing the collision accurately to first order, in spite of the simplifying assumptions made.

Bayes factors

The Bayes factor, PX/PY, for models X and Y, used to perform model comparison, is defined as the integral over the marginalised posterior/likelihood

$$\frac{{P}_{X}}{{P}_{Y}}=\frac{\int \,p({\phi }_{X}| {M}_{X})p(D| {\phi }_{X},{M}_{X})d{\phi }_{X}}{\int \,p({\phi }_{Y}| {M}_{Y})p(D| {\phi }_{Y},{M}_{Y})d{\phi }_{Y}}$$
(6)

where, ϕX, ϕY are the parameter vectors which characterise models MX and MY, respectively. The integrals in Eq. (6) do not have analytic solutions and are challenging to compute numerically due to the complex shape of the posterior distribution. Therefore, the Bayes factor was approximated using leave-one-out cross-validation with Pareto-smoothed importance sampling (LOO-PSIS)77 available from the Python package arviz78. As the inferred parameters and hence the posterior probabilities for each shot are independent, their product yields the total Bayes factor.

Bootstrapping

Bootstrapping, chosen for its applicability to non-normal distributions, was used to compute the errors on population means of the hit and null distributions for 〈E〉 and P70, shown in Fig. 3. Bootstrapping was also used to compute the mean and standard error for the null scaling of photon counts measured by the profile screen with Q〈γ2〉, illustrated in Fig. 2. In each instance where bootstrapping was employed in the analysis, a sample size equal to that of the dataset was used and the data was re-sampled 10,000 times. The robustness of the bootstrapping analysis was verified by increasing the number of re-samples by a factor of 100, which did not significantly affect the results. The random randint package from the numpy library in Python was used to perform sampling.