Efficient ion re-acceleration in laboratory-produced interpenetrating collisionless shocks

TL;DR

This study demonstrates that colliding collisionless shocks in a laboratory setting can re-accelerate protons efficiently, providing insights into cosmic ray acceleration mechanisms through combined experimental and simulation approaches.

Contribution

The paper introduces a novel laboratory platform for studying proton re-acceleration in colliding collisionless shocks, advancing understanding of cosmic ray acceleration.

Findings

Interpenetrating shocks boost proton energy significantly.

Reacceleration occurs via bouncing in convective electric fields.

Laboratory results support shock-shock collision theories in astrophysics.

Abstract

Although the origin of cosmic rays (CRs) remains an open question, collisionless magnetized shock waves are widely regarded as key sites for particle acceleration. Recent theories further suggest that shock-shock collisions in stellar clusters could provide the additional acceleration needed to explain the observed high-energy CR spectrum. Here, we investigate this hypothesis through a laser-based experiment that creates magnetized plasma conditions similar to astrophysical environments. Our results demonstrate that interpenetrating collisionless shocks can significantly boost the energy of ambient protons previously energized by the individual shocks, while also improving the overall acceleration efficiency. Numerical kinetic simulations corroborate these findings, revealing that protons are reaccelerated via their bouncing motion in the convective electric fields of the colliding magnetized flows. By allowing to highly energize ambient protons, our novel colliding-shock platform opens the prospect to test the long-discussed mechanism of diffusive shock acceleration in a controlled laboratory setting.

Tables

Table 1. Table 1 : Key dimensionless parameters of typical astrophysical shocks and those produced in the reported experiment. The ratio of the proton mean-free-path ( λ mfp \lambda_{\rm mfp} ) over the shock width, given by the ion inertial length d i d_{i} , designates the collisionality of the system. The proton mean-free-path is given by v s ​ h ​ τ i v_{sh}\tau_{i} where v s ​ h v_{sh} is the shock velocity and τ i \tau_{i} is the ion collisional time, calculated using the ion temperature in the shock. ρ i \rho_{i} is the Larmor radius of upstream (US) protons having, in the reference frame of the shock, an incoming velocity equal to that of the shock in the laboratory frame. The Alfvénic Mach number M A M_{A} is calculated with respect to the US values, i.e. it is given by the ratio between v s ​ h v_{sh} and the Alfvén velocity v A ≡ B / μ 0 ​ n p ​ m p v_{A}\equiv B/\sqrt{\mu_{0}n_{p}m_{p}} , where B is the magnetic field strength, m p m_{p} the proton mass and n p n_{p} the proton plasma density. For the laboratory shock, T i T_{i} ≈ \approx 200 eV and n p n_{p} ≈ 10 18 ​ cm − 3 \approx 10^{18}\ \rm cm^{-3} . For interplanetary shocks [ 18 , 19 ] , n p n_{p} ≈ 20 ​ cm − 3 \approx 20\ \rm cm^{-3} , T i T_{i} ≈ 10 ​ eV \approx 10\ \rm eV , B ≈ 10 ​ nT \approx 10\ \rm nT and M A M_{A} is in the range of 3 − 20 3-20 . For shocks induced by SN remnants [ 20 ] in the interstellar medium (ISM), we use T i T_{i} ≈ 300 ​ eV \approx 300\ \rm eV , B ≈ 1 − 5 ​ μ B\approx 1-5\ \mu G for the ISM magnetic field [ 21 ] and 0.01 − 10 0.01-10 cm -3 for its density. Since SN remnants have velocities ranging from a few thousands km/s, for young ones [ 22 ] , to the 1000 − 5000 1000-5000 km/s range, for older ones [ 23 ] , we thus have M A M_{A} in the several tens-hundreds range. In galaxy clusters [ 24 , 25 ] , T i T_{i} ≈ \approx 2 keV, n p n_{p} ≈ 10 − 2 ​ cm − 3 \approx 10^{-2}\ \rm cm^{-3} , and v s ​ h v_{sh} is in the range 1000 − 2000 1000-2000 km/s with B ≈ 0.1 − 1 ​ μ B\approx 0.1-1\ \mu G, so we also have M A ∼ 100 M_{A}\sim 100 . Finally, for lobes and winds of active galactic nuclei [ 26 ] , their velocity can be up to 10000 10000 km/s or more, and in the circumgalactic medium we have 0.01 − 0.1 ​ μ 0.01-0.1\ \mu G at a density of 10 − 2 − 10 − 3 10^{-2}-10^{-3} cm -3 , with T i T_{i} ≈ \approx 1 keV, which results in M A ∼ 100 − 1000 M_{A}\sim 100-1000 . As for the overshoot [ 27 ] of the downstream (DS) magnetic field with respect to the US one, it is calculated as B D ​ S / B U ​ S B_{DS}/B_{US} .

Quantity	Astrophysical systems	Laboratory
${\lambda }_{\mathrm{mfp}}/d_i$	${10}^7-{10}^{13}$	65
${\rho }_i/d_i$	$10-{10}^3$	14
Alfvénic Mach number $M_A=v_{sh}/v_A$	$\sim 3-100$s	$\sim 10$
Magnetic field overshoot $B_{DS}/B_{US}$	$\sim 2-20$	$\sim 3-6$

Full text

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1]LULI - CNRS, CEA, UPMC Univ Paris 06 : Sorbonne Université, Ecole Polytechnique, Institut Polytechnique de Paris - F-91128 Palaiseau cedex, France 2]Sorbonne Université, Observatoire de Paris, Université PSL, Laboratoire d’étude de l’Univers et des phénomènes eXtrêmes, LUX, CNRS, F-75005 Meudon, France 3]Central Laser Facility, STFC Rutherford Appleton Laboratory, Oxfordshire OX11 0QX, United Kingdom 4]Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom 5]ELI-NP, "Horia Hulubei" National Institute for Physics and Nuclear Engineering, 30 Reactorului Street, RO-077125, Bucharest-Magurele, Romania 6]Laboratoire de micro-irradiation, de métrologie et de dosimétrie des neutrons, PSE-Santé/SDOS, IRSN, 13115 Saint-Paul-Lez-Durance, France 7]Center for Plasma Physics, School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom 8]University of Bordeaux, Centre Lasers Intenses et Applications, CNRS, CEA, UMR 5107, F-33405 Talence, France 9]INRS-EMT, 1650 boul, Lionel-Boulet, Varennes, QC, J3X 1S2, Canada 10]LNCMI, UPR 3228, CNRS-UGA-UPS-INSA, Toulouse 31400, France 11]Department of Astronomy & Astrophysics, University of Chicago, Chicago, Illinois 60637, USA 12]Enrico Fermi Institute, The University of Chicago, Chicago, Illinois 60637, USA 13]CEA, DAM, DIF, F-91297 Arpajon, France 14]Université Paris-Saclay, CEA, LMCE, F-91680 Bruyères-le-Châtel, France 15]Laboratoire Univers et Particules de Montpellier CNRS/Université de Montpellier, Place E. Bataillon, 34095 Montpellier, France 16]University of Palermo, Department of Physics and Chemistry, Palermo, Italy 17]Laboratoire de Physique des Plasmas (LPP), CNRS, Observatoire de Paris, Sorbonne Université, Université Paris-Saclay, École polytechnique, Institut Polytechnique de Paris, F-91120 Palaiseau, France 18]INAF–Osservatorio Astronomico di Palermo, Palermo, Italy

Efficient ion re-acceleration in laboratory-produced interpenetrating collisionless shocks

W. Yao

  

I. Cohen

  

P. Suarez Gerona

  

H. Ahmed

  

A.F.A. Bott

  

S. N. Chen

  

M. Cook

  

R. Lelièvre

  

P. Martin

  

T. Waltenspiel

  

P. Antici

  

J. Béard

  

M. Borghesi

  

D. Caprioli

  

A. Ciardi

  

E. d’Humières

  

M. François

  

L. Gremillet

  

A. Marcowith

  

M. Miceli

  

T. Seebaruth

  

S. Orlando

  

J. Fuchs

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(August 27, 2025)

**Although the origin of cosmic rays (CRs) remains an open question, collisionless magnetized shock waves are widely regarded as key sites for particle acceleration [1]. Recent theories further suggest that shock-shock collisions in stellar clusters [2, 3] could provide the additional acceleration needed to explain the observed high-energy CR spectrum [4, 5, 6, 7]. Here, we investigate this hypothesis through a laser-based experiment that creates magnetized plasma conditions similar to astrophysical environments. Our results demonstrate that interpenetrating collisionless shocks can significantly boost the energy of ambient protons previously energized by the individual shocks, while also improving the overall acceleration efficiency. Numerical kinetic simulations corroborate these findings, revealing that protons are reaccelerated via their bouncing motion in the convective electric fields of the colliding magnetized flows. By allowing to highly energize ambient protons, our novel colliding-shock platform opens the prospect to test the long-discussed mechanism of diffusive shock acceleration in a controlled laboratory setting. **

One interesting locale where shock-shock collisions can be found is in clusters of hot giant stars, where frequent and correlated supernova (SN) explosions take place [8]. The interplay between stellar winds and SN events in these regions gives rise to large-scale structures known as “superbubbles” [9], within which various super-critical***I.e., shocks with a magnetosonic Mach number $M_{ms}=v_{sh}/\sqrt{v_{A}^{2}+c_{s}^{2}}$ exceeding, in a MHD framework and for perpendicular shocks, the critical value of 2.7[10] over which ion are reflected by the shock - here, $v_{sh}$, $v_{A}$ and $c_{s}$ are the shock, ambient Alfvénic and sound velocities, respectively. shock-shock collisions occur, including wind-wind, SN-wind, and SN-SN shock interactions [4, 11].

Given that astrophysical shocks generally propagate through turbulent media [12], the magnetic field obliquity at the shock front is inherently random. Here, we focus on perpendicular shocks (i.e., where the background magnetic field is perpendicular to the shock normal), which are known to be efficient particle accelerators [13, 6].

Several theoretical models have attempted to understand the acceleration of cosmic rays (CRs) in colliding-shock configurations. For instance, Vieu et al. [5] proposed that, although the fundamental acceleration process remains unchanged when two shocks collide, spectral hardening can occur for the highest-energy particles that are scattered between the two shock fronts. Complementarily, Malkov and Lemoine [7] suggested that converging perpendicular shocks could efficiently re-accelerate preexisting CRs. Hence, the goal of the presently reported experiment: to test the re-acceleration of ions in such situation of colliding super-critical shocks.

Our experimental setup (see Methods) investigates ion acceleration from two converging, perpendicular, super-critical shocks. These shocks, as their astrophysical counterparts, are “collisionless” because the width of their transition layer, given by the ion inertial length $d_{i}$ [10], is much smaller than the collisional mean-free-path ($\lambda_{mfp}$) of the ions (see Table 1). Instead of Coulomb collisions, shock formation is mediated by wave-particle interactions in self-induced or pre-existing electromagnetic fields. Also note that in our conditions, as in astrophysical systems, the ion Larmor radius of upstream protons encountering the shock is much larger than the shock width. Such structures are ubiquitous phenomena and have been observed across a huge range of scales [10], from astrophysical ($10^{16}\sim 10^{25}$ m) [14], to space ($10^{8}\sim 10^{12}$ m) [15] plasmas, and down to laboratory environments ($10^{-6}\sim 10^{-3}$ m) [16, 17].

We begin by characterizing the individual shocks produced in the experiment (see Methods). As illustrated in Fig. 1a, using a high-power laser, we create a high-velocity plasma piston from an irradiated target. As this piston expands into a magnetized, tenuous hydrogen ambient gas, a shock is driven. To diagnose the shock dynamics, we employ ultra-fast proton radiography (see Methods). For this, an auxiliary laser generates a short burst of non-perturbative, high-energy (MeV) protons, which probe the electromagnetic fields within the expanding piston and shock [28]. The broad energy spectrum of the probing protons allows the temporal evolution of the plasma to be resolved, as protons of different energies traverse the plasma at different times. To capture the shock evolution over several nanoseconds, we introduce a controlled delay between the generation of the piston and the probing protons. This allows us to scan the probing window across the full shock evolution within the field-of-view (limited by the opening of the magnetic coil).

Fig. 1b presents a snapshot of a single piston and the resulting shock (see Supplementary Material for a comprehensive characterization of the shock dynamics, velocity, and energy). During the initial phase of shock formation and propagation – up to the peak of the laser pulse – the average shock propagation velocity is measured to be $\sim 3000\,\rm km/s$. Given the plasma density and magnetization, this corresponds to an Alfvénic Mach number of $M_{A}>10$, similar to shocks observed in diverse astrophysical systems (Table 1).

Proton radiography not only captures the global shock dynamics but also the magnetic structures in the shocked plasma. A detailed analysis of the proton deflections in this region (see the Supplementary Material and Figs. S2 and S3) reveals that they are primarily caused by magnetic fields [29]. Using the PROBLEM algorithm [30] (see Methods), we can reconstruct a path-integrated magnetic field map (along the $z$-axis, the direction of the applied field) from the proton deflectometry data.

Fig. 1c displays the magnetic field map derived from the raw proton radiography image shown in Fig. 1b. As anticipated from both astrophysical observations and simulations [10, 20], we observe a rim of compressed magnetic field around the shock front, which is preceded by long filamentary structures, likely caused by the Weibel instability [31] in the upstream (US) region. The compressed magnetic field has a field strength 3 to 6 times higher than in the US. This enhancement aligns with expectations for supercritical astrophysical shocks [32, 27, 19], as summarized in Table 1. Consistent with satellite measurements [32, 33] and astrophysical simulations [34], the compressed magnetic region exhibits strong corrugation, likely due to instabilities [35]. Additionally, we detect a chaotic magnetic field structure behind the shock front, potentially of kinetic origin [36], with a field strength comparable with that within the shock itself.

By generating two face-to-face supercritical shocks, we can investigate their impact on ambient particles. The setup is illustrated in Fig. 2a. We use the same diagnostic technique as in the single-shock case to sample the shock dynamics. The two targets are positioned 4 mm apart (see Methods), ensuring the shocks interact early, while their individual velocities remain high. As evidenced in Fig. 2b, the two shocks, once formed, encounter each other at around $t\approx 1\,\rm ns$, at which point they are both supercritical with $M_{ms}\approx 14$. Fig. 2b1-b3 show that proton radiography accurately captures the collision dynamics – i.e., before, during, and after the collision – as well as the subsequent merging of the two structures.

The local plasma conditions are further characterized by Thomson scattering (see Methods). For a single shock, the inferred upstream electron density and temperature are approximately $n_{e}\approx 10^{18}\ \rm cm^{-3}$ and $T_{e}\approx 100\pm 20\,\rm eV$, respectively. The temperature rises to around $\approx 190\pm 20$ eV in the downstream (DS). Similar values are measured in the upstream in the double-shock case. However, this configuration yields a higher downstream temperature ($\approx 250\pm 20\ \rm eV$), as the shock collision introduces an additional heating source to the plasma [37]. These measured values are used to calculate the ion collisional time used in Table 1.

We now analyze the energy spectra of nonthermal protons, collected from the plasma along the external magnetic field (see Fig. 1a and Fig. 2a for the arrangement of the spectrometer in the experiment). The proton spectra are shown in Fig. 3. For a single shock, ambient protons (blue) are energized – likely via shock surfing acceleration [38] – up to a cutoff energy of $\sim 80\,\rm keV$, consistent with our previous results [17]. Integrating the spectrum and assuming isotropic emission over $2\pi\,\rm sr$ leads to a total of $\sim 3\times 10^{10}$ ($\sim 5\,\rm nC$) accelerated protons (above an energy of $20\,\rm keV$), with a total kinetic energy of 0.18 mJ. From energy conservation (see Supplementary Material), the energy contained within one shock is estimated to be $\sim 17\,\rm J$. Given the 4 mm target separation, upstream protons (having, in the reference frame of the shock, an incoming velocity equal to that of the shock in the laboratory frame) undergo $\gtrsim 2$ Larmor gyrations over this distance. From this, we can infer a shock-to-proton energy conversion efficiency of $\sim 0.005\,\%$ per Larmor period. This value aligns well with simulation predictions [39], further validating the astrophysical relevance of our laboratory shocks.

Remarkably, when two shocks collide (red points, Fig. 3), the proton cutoff energy increases fivefold to $\sim 400\,\rm keV$, while the total energy in accelerated protons rises to $\sim 2.7\,\rm mJ$, corresponding to $\sim 2\times 10^{11}$ protons ($\sim 30\,\rm nC$). This represents an eightfold increase in shock-to-proton energy conversion efficiency compared to the single-shock case.

It is worth noting that, when the two colliding shocks are subcritical – as achieved by increasing the standoff distance between the piston-generating targets to 7 mm (see Methods) – no further boost in the energy of ambient protons is observed. This result aligns with our earlier experiment, in which two shocks collided $\sim 10\,\rm ns$ after piston initiation, with a magnetosonic Mach number of only 1.1 at the time of collision [40], and no proton reenergization was detected.

To elucidate our experimental results, we have performed kinetic particle-in-cell (PIC) simulations using the SMILEI code [41]. The simulations were conducted in 1D3V geometry (1D in space and 3D in momentum space) and initialized with the experimental plasma parameters (density, temperature, velocity, and magnetic field).

Fig. 4a illustrates the spatiotemporal evolution of the convective electric field $E_{y}$ in the colliding-shock case. The two shocks form within the first 1 ns of the simulation, and propagate at an initial velocity of $v_{sh}\approx 2750\pm 250\,\rm km/s$, close to the experimental observation and corresponding to an initial kinetic energy of $E_{sh}\equiv 0.5m_{p}v_{sh}^{2}\approx 40\,\rm keV$.

The shocks collide at $1.25\pm 0.25\,\rm ns$ at the center of the simulation box ($x=0$), where the local plasma density increases tenfold compared to its unperturbed value (not shown), with their velocities decreasing to $1000\pm 200\,\rm km/s$.

Particle tracking allows us to identify the energization mechanism at work during the shock collision. Fig. 4a shows four representative particle trajectories overlaid on the field evolution map. Their transverse velocity, $v_{y}$, is indicated by the color scale at the top. After reflecting off either shock front, these particles gain most of their energy by traveling back and forth in the unperturbed regions of the two flows. There, as they propagate against either flow, they are energized by the local convective electric field, i.e., when $E_{y}$ is positive (or negative), the particles have their $v_{y}$ also being positive (or negative) [40, 37].

This is confirmed in Fig. 4b, where we plot the work performed on a single particle by the convective ($E_{y}$) and electrostatic ($E_{x}$) fields. While the latter accounts for the initial modest energy gain ($\sim 10\,\rm keV$), the convective field largely dominates after $t\approx 1.5\,\rm ns$, with its work eventually exceeding $300\,\rm keV$ by $t\approx 3\,\rm ns$. In stark contrast, for a single shock – where the second colliding flow and its associated motional field are absent – the energization is significantly weaker. Here, the energy gain from $E_{x}$ remains comparable, but the contribution from $E_{y}$ oscillates between gain and loss, likely due to shock surfing (Fig. 4c).

Fig. 3 shows that the simulated overall proton spectra for both configurations are in excellent agreement with the observations, with a cutoff energy approaching $10\,E_{sh}$ in the double-shock case.

Given this enhanced acceleration efficiency in colliding shocks and by combining them with plasma turbulence, as recently enabled in laser-driven experiments [30, 42], our platform is a promising tool for the experimental study of diffusive shock acceleration (DSA) [43]. DSA, a leading theory for the origin of cosmic rays, relies on particles scattering off magnetic turbulence and bouncing many times across the shock front [44], but it requires an initial energy boost to get started. Our findings show that shock collisions could provide that crucial “kick-off” energy, enabling direct investigation of the fundamental processes that shape the universe’s most energetic particles.

Methods

Experimental setup

The experiment was performed in the target area west (TAW) of the Rutherford Appleton laboratory VULCAN laser facility (U.K.). The experimental setup combines one or two high-power laser produced plasmas, a large-scale background gas, and an externally applied, large-scale, homogeneous and steady (at the spatial and temporal scales of the experiment) magnetic field [45].

Each laser pulse, of 1053 nm wavelength, has 200 J in energy over a 1.4 ns duration in full-width-at-half-maximum (FWHM). The laser temporal shape is Gaussian. Each laser is focused on its respective target with a 200 micron focal spot in diameter with a super Gaussian spatial shape, inducing an on-target peak intensity of about $10^{14}$ W/cm2.

Each target is a Teflon disk (CF2), with a thickness of 2 mm, thicker than the ablation induced by the lasers. The distance between the laser-irradiated surface of the two targets is either 4 or 7 mm. The setup is immersed in a large-volume gas, produced by pulsing a gas jet from a nozzle, so that the density is homogeneous in between the targets. The ambient gas is hydrogen, at a low density of $10^{18}$ cm*-3* for a 7 bars backing pressure. The externally applied magnetic field is 10 T, and the direction of the magnetic field is along the z-axis (i.e., perpendicular to the shock propagation direction, which is along the x-axis).

The diagnostics include proton radiography (PR) [28], Thomson scattering (TS) [46], and a magnetic spectrometer for measuring the spectra of the protons.

The probing protons are accelerated from a secondary 15 $\mu m$ Au target using an auxiliary ultra-short 15 ps duration, 250 J energy laser pulse, which is focused to $10^{19}$ W/cm2 to generate a broadband, divergent proton beam with a maximum energy 15 MeV, through the target normal sheath-acceleration (TNSA) mechanism [28]. The proton beam is accelerated along the z-axis, i.e. along the same axis as the external magnetic field. The protons then propagate through the plasma, after which they are collected by a stack of radio-chromic films (RCF) [28] with thin filters between them. The different layers in our RCF stack configuration that are appropriate for field reconstruction are sensitive to proton energies of $E=9.7,12,14$ MeV. The dose maps generated by lower energy protons displayed caustics and could not be used for field reconstruction [30] (see details in the Supplementary material). The distance between the proton source and the plasma was $5.05$ cm, the distance between the plasma and the RCF stack was $7.24$ cm. The resulting magnification was thus $1.69$, and the length of the probed plasma across the magnetic field is assumed to be $3$ mm. The temporal evolution of single shocks was followed within the temporal interval from -0.5 ns to 3.7 ns with respect to the peak of the laser pulse. The magnetic field is reconstructed from the proton radiography images using the PROBLEM code (PROton-imaged B-field nonlinear extraction module) [30]. The magnetic fields are retrieved by solving the logarithmic parabolic Monge-Ampère equation for the steady-state solution of the deflection field potential. The algorithm is based on using an adaptive mesh and a standard centered second order finite difference scheme for the spatial discretizations and a forward Euler scheme for the temporal discretization [47].

For TS, we also used another auxiliary laser, this time a long-pulse (also 1.4 ns FWHM in duration), frequency-converted to 527 nm, and which was propagated into the plasma along the z-axis, alternatively to the protons used for PR. The Thomson scattered light was collected in the vertical direction, to ensure measurements in the collective regime, and was analyzed by optical spectrometers coupled to streak cameras, as in our previous experiment on single shocks [17].

To measure the proton spectra, we collected the energized ambient protons along the z-axis (i.e. along the magnetic field axis), alternating with the PR and TS diagnostics. As in our previous experiment on single shocks [17], we used a calibrated magnetic spectrometer to resolve the energy of the protons. The protons were detected, in a time-integrated manner, using absolutely calibrated BAS-TR imaging plates [48] equipped with filters to ensure that ions other than protons did not reach the detector.

Simulation setup

The shock collision is modeled with the fully kinetic Particle-In-Cell (PIC) code SMILEI [41] in one-dimensional (1D) geometry, as in our previous work [17]. The simulation box is initialized to be $L_{x}=6144d_{e}=33$ mm, with the spatial resolution $d_{x}=0.2d_{e}=1.1\ \mu$m, with the electron inertial length $d_{e}=c/\omega_{pe}=5.3\ \mu$m, and the electron plasma frequency $\omega_{pe}=(n_{e0}q_{e}^{2}/m_{e}\epsilon_{0})^{1/2}=5.6\times 10^{13}$ s*-1*. Here, the initial ambient electron number density is $n_{e0}=1.0\times 10^{18}$ cm*-3*; $c$, $m_{e}$, $q_{e}$ and $\epsilon_{0}$ are the speed of light, the electron mass, elementary charge, and the permittivity of free space, respectively.

The simulation box consists of 30720 cells. For each cell, we put 2048 particles for each species. The simulation lasts for $2.5\times 10^{5}\omega_{pe}^{-1}\sim 4.0$ ns. For the boundary conditions, the electromagnetic fields are absorbed, while particles are removed. Note that enough room is left between the boundary and the plasmas, so that the boundary conditions do not affect the dynamics of the interpenetrating shocks. The ambient plasma lies in the middle of the simulation box, having a size of 4 or 7 mm, like those in the experiments. We set two shocked plasmas from the left/right sides, each having a size of about 13 mm, a density of $2n_{e0}$, and a drifting velocity of $v_{0}=2500$ km/s towards the middle. Note that in our simulation, the ion species is proton with its real mass ($m_{p}/m_{e}=1836$), and the shock width is initialized as one ion inertial length $d_{i}\approx 200\ \mu$m in between the ambient plasma and the shocked plasma.

The initial temperature of the ambient plasma is $T_{e0}=T_{i0}=100$ eV, as in the experiment. The initial temperature of the shocked plasma for the double shock case is $T_{e,DS}=T_{i,DS}=250$ eV, and that for the single shock case is $T_{e,SS}=T_{i,SS}=190$ eV. The shocked plasma temperature in single shock case is a bit lower than that of the double shock one, as verified by our experimental measurements (see main text).

The externally applied magnetic field is homogeneously distributed along the simulation box. The magnetic field direction is along the $z$-axis, with a strength of 10 T, i.e., $B_{z}=10$ T (which corresponds to $\omega_{ce}/\omega_{pe}=0.03$, where $\omega_{ce}=q_{e}B/m_{e}$). Note that during the shock collision, the magnetic field in the downstream around the shock has a compression ratio over 10. Thus, for the protons drifting along the shock, with an initial velocity of $v_{x0}=2500$ km/s, their Larmor radius is $r_{Li}=0.26$ mm and gyro-period is $\tau_{Li}=0.6$ ns.

To reproduce the macroscopic expansion of the plasma plume, simulations of the experiment using the hydrodynamic radiative multi-temperature code FLASH [49] have also been performed. In these two-dimensional (2D)-axisymmetric simulations, using the target, gas jet and laser parameters of the experiment but with no external magnetic field, the shock front reaches to a maximum velocity of 500 km/s, about six times slower than in the experiment. In the simulation, the laser energy deposition is calculated by inverse Bremsstrahlung [50], and in this experiment we should be in a case where inverse Bremsstrahlung is dominant. The impossibility to reproduce with FLASH the plasma plume dynamics for the first ns after launching the shock lies elsewhere, and the front of the plume must be driven by other kinetic effects at later times. Thus, the analysis of the acceleration mechanism is based on the PIC simulations.

Acknowledgments

The authors would like to thank the laser, engineering and target fabrication teams of the Central Laser facility (U.K.) for their expert support. This work was supported by funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (Grant Agreement No. 787539, JF). We acknowledge support by the IMPULSE project by the European Union Framework Program for Research and Innovation Horizon 2020 (grant agreement no. 871161) , and EPRSC (grant EP/P010059/1). This research was also supported in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP). The computational resources of this work were supported by the HPC resources of MesoPSL financed by the Region Ile de France and the project EquipMeso (reference ANR-10-EQPX29-01) of the programme Investissements d’Avenir supervised by the Agence Nationale pour la Recherche. It was also supported by the National Sciences and Engineering Research Council of Canada (NSERC) and Compute Canada (Job: pve-323-ac, PA), and by NASA (grant 80NSSC24K0173) and NSF (grant AST-2510951).

Author Contributions Statement

J.F. conceived the project. W.Y., H.A., M.C., S.N.C., R.L., P.M., J.B., T.W. and J.F. performed the experiments, with support from P.A., and M.B. W.Y., I.C., A.F.A.B., S.N.C., and J.F. analyzed the data. W.Y. and P.S-G. performed and analyzed the SMILEI simulations, with discussions with A.C., M.F., T.S., L.G., E.d.H. and J.F. The bulk of the paper was written by W.Y., S.N.C., I.C. and J.F., with inputs from L.G., S.O., M.M., A.M. and D.C. All authors commented and revised the paper.

Materials and correspondence

Correspondence and material requests should be addressed to julien.fuchs@polytechnique.edu.

Competing Interests Statement

The authors declare no competing interests.

Data availability All data needed to evaluate the conclusions in the paper are present in the paper. Experimental data and simulations are respectively archived on servers at LULI and LERMA laboratories and are available from the corresponding author upon reasonable request.

Code availability The code used to generate Fig. 3 and Fig. 4 is SMILEI, which is detailed in the Methods section.