Reaching supercritical field strengths with intense lasers

All perturbative approaches to quantum electrodynamics (QED) break down in the collision of a high-energy electron beam with an intense laser, when the laser fields are boosted to `supercritical' strengths greater than the critical field of QED. This leads to cascades of photon emission and electron-positron pair creation, as well as the onset of substantial radiative corrections. We identify the important role played by the collision angle in mitigating energy losses to photon emission that would otherwise prevent the electrons reaching the supercritical regime. We show that a collision between an electron beam with energy in the tens of GeV and a laser pulse of intensity $10^{24}~\text{W}\text{cm}^{-2}$ at, crucially, normal incidence is a viable platform for studying the breakdown of perturbative strong-field QED. Our results have implications for the design of near-term experiments as they predict that certain quantum effects are enhanced at oblique incidence.

Experimental exploration of nonperturbative quantum electrodynamics (QED) is challenging as large electromagnetic fields comparable to the critical field of QED E cr = 1.3 × 10 18 Vm −1 [1,2] are required. Nevertheless, ever-increasing laser intensities [3][4][5] make it possible to probe fields that are effectively supercritical, i.e., have magnitude greater than E cr . This is achieved using the Lorentz boost when ultrarelativistic electrons collide with an intense laser pulse [6,7], as the parameter χ e that controls the importance of nonlinear QED is the rest-frame electric field normalized to the critical field strength. χ e is covariantly expressed as χ e = |F µν u ν |/E cr [8], where F is the electromagnetic field tensor, u the electron fourvelocity, and E cr = m 2 /e is the critical field strength. We use natural units in which = c = 1 (e is the elementary charge, m the electron mass) throughout.
In the supercritical regime χ e 1, particle dynamics is dominated by cascades of photon emission and electron-positron pair creation [8][9][10][11]. The importance of studying these phenomena is motivated by their relevance to high-field astrophysical environments, such as magnetars [12][13][14], and to laser-matter interactions beyond the current intensity frontier [15,16]. It is also of considerable theoretical interest, as when αχ 2/3 e approaches unity (α is the fine-structure constant), radiative corrections to quantum processes become so large that all current, perturbative, predictions fail [17,18] and strong-field QED becomes fully nonperturbative.
In this Letter we show how the collision of an intense laser pulse with an ultrarelativistic electron beam may be used to probe the supercritical regime. A significant obstacle to this is posed by radiation reaction, an accelerating charge's loss of energy to photon emission, which strongly reduces u at χ e 1, thereby suppressing χ e itself [19][20][21][22]. We show that this can be mitigated by appropriate choice of the angle at which the beams collide. We present a theoretical expression for the maximum χ e , which predicts, contrary to the expectation that the ideal geometry is counterpropagation, that oblique incidence is favoured for laser pulses of high intensity or long duration. This enhances certain quantum effects on radiation reaction, namely straggling [23,24] and stochastic broadening [25]. As a result, not only will laser-electron collision experiments that are practically constrained to oblique incidence [26] still detect signatures of quantum effects, but these signatures can be stronger than they would be in a head-on collision. Furthermore, we show that at the intensity and electron energy necessary to probe radiative corrections, normal incidence is strongly favoured to reduce radiative losses that would otherwise prevent reaching such high χ e .
High-power lasers compress energy into ultrashort pulses that can be focussed almost to the diffraction limit. The theoretical upper bound on χ e is obtained by treating the laser as a pulsed plane electromagnetic wave with peak dimensionless amplitude a 0 = eE 0 /(mω 0 ), peak electric field strength E 0 and central frequency ω 0 , and neglecting radiative losses. Then where θ is the collision angle (defined to be zero for counterpropagation) and γ 0 1 is the initial Lorentz factor of the electron. The largest possible quantum parameter is χ 0 = χ e (θ = 0).
Experiments at a 0 0.4, χ e 0.3 have demonstrated nonlinear QED effects including pair creation [6,7], and recently evidence of radiation reaction has been reported at a 0 10, χ e 0.1 [27,28]. At present, the highest field strengths are equivalent to a 0 50 [29,30], or χ 0 1 at γ 0 m 1 GeV; a 0 > 100 is expected in the next generation of high-intensity laser facilities [31,32]. The stronger the electromagnetic field, the lower the electron energy that is needed to reach high χ e . In experiments with aligned crystals where the field strength ∼ 10 13 Vm −1 [33], χ e 1 and evidence of quantum arXiv:1807.03730v1 [physics.plasm-ph] 10 Jul 2018 radiation reaction require 100-GeV electron beams [34]. χ e > 1 will also be probed in beam-beam interactions in the next generation of linear colliders [35,36]. Despite the strong spatial and temporal compression of laser pulses, it is inevitable that the electron will have to traverse a finite region of space over which the field strength ramps up before it reaches the point of peak a 0 . If significant energy loss takes place during this interval, the electron's χ e will be much smaller than that predicted by eq. (1). We now derive an expression for the maximum χ e reached by an electron that takes radiative losses and the laser pulse's spatial structure into account, following the analysis given in [37].
Consider an electron colliding at angle θ with a linearly polarized laser pulse that has Gaussian temporal and radial intensity profiles of size τ and r 0 respectively. Here τ is the full width at half maximum (FWHM) of the temporal intensity profile and r 0 is the waist of the focussed pulse (the radius at which the intensity falls to 1/e 2 of its peak). The point at which χ e is maximized is defined by (γa) = 0, where a is the envelope of the electric field and primes denote differentiation with respect to phase. We substitute into this extremization condition the cycle-averaged radiated power γ = αmχ 2 e g(χ e )/[3(1+cos θ)ω 0 ], where the Gaunt factor g(χ e ) ≤ 1 accounts for quantum corrections to the photon spectrum [9], and approximate γ γ 0 , following [37]. Then the maximum quantum parameter χ max satisfies Here χ 0 is the largest possible quantum parameter [eq. (1) with θ = 0] and the classical radiation reaction parameter R c = αa 0 χ 0 [20,38]. The dependence of χ max on the laser pulse structure is captured by an effective duration (per wavelength) n eff = ω 0 τ /[2π 1 + τ 2 tan 2 (θ/2)/(r 2 0 ln 4)]. For simplicity, we approximate that the temporal and radial intensity profiles have constant size.
In the limit χ max 1, eq. (2) has a solution in terms of the Lambert function W : W (δ 2 ) is strictly increasing for δ ≥ 0 and therefore χ e decreases with increasing δ. Unlike eq. (1), eq. (3) does not depend symmetrically upon a 0 and γ 0 , as δ ∝ a 2 0 γ 0 . Hence it is more beneficial to increase γ 0 than a 0 when aiming for very large χ e . Physically this is because the photon emission rate has a stronger dependence on a 0 than on γ 0 ; by minimizing the number of emitted photons we also mitigate the radiative losses that would reduce χ e . To show that eq. (2) can be used as a quantitative prediction of the largest χ e that is reached in a laserelectron beam collision, we compare its predictions to the results of single-particle Monte Carlo simulations. These model a QED cascade of photon emission and pair creation by factorizing it into a product of first-order processes [39,40], which occur along the particles' classical trajectories at positions determined by integration of QED probability rates that are calculated in the locally constant field approximation [8]. This 'semi-classical' approach is valid when a 3 0 /χ e 1 because the formation lengths of the photons and electron-positron pairs are much smaller than the laser wavelength and interference effects are suppressed [41].
Starting with head-on collisions, we show how the distribution of χ max , the largest quantum parameter attained by the electron on its passage through the laser pulse, is affected by the pulse duration. The electron initial Lorentz factor is set to be one of 5 × 10 3 , 2 × 10 4 , and 10 5 . The laser a 0 is chosen such that χ 0 is 1, 10 and 100 respectively. The laser frequency is fixed at ω 0 = 1.55 eV and the pulse duration τ is varied from 2 to 200 wavelengths. The distributions of χ max shown in fig. 1 demonstrate that increasing the pulse duration strongly reduces the number of electrons that reach large quantum parameter. This behaviour is captured by eq. (2), which we find to be a good quantitative prediction of the 90th percentile of the distribution. The connection to radiation reaction becomes clear when we solve eq. (2) to find the largest laser pulse duration τ for which χ max > χ 0 /2. We find that τ 8E 0 /P(χ 0 /2), where E 0 is the initial energy of the electron and P the average radiated power (including quantum corrections) of an electron with given χ. The larger P is, the shorter the pulse must be to ensure that at least 10% of the electrons reach a quantum parameter of at least χ 0 /2. Equation (1) leads us to expect that quantum effects are strongest in the head-on collision geometry, where the geometric factor 1 + cos θ is largest. However, unless the pulse duration is as little as a few cycles in length (at which point radiation 'quenching' is possible [42]), radiation reaction strongly reduces the number of electrons that get close to the maximum possible χ e . This can be mitigated by moving to collisions at oblique incidence, because the spot to which a laser pulse is focussed (∼ 2 µm) is typically smaller than the length of its temporal profile (20 fs [32], 30 fs [29,30] or 150 fs [31]). Even though the maximum possible χ e at θ > 0 is smaller than that at θ = 0, many more electrons get close to the maximum because the interaction length is shorter and radiative losses are reduced. This is illustrated in fig. 2, where the collision angle θ for which χ max is largest is plotted for two exemplary pulse durations τ = 10λ and 50λ (27 and 130 fs respectively at a wavelength of 0.8 µm). The shorter the pulse duration, the larger a 0 can be before the head-on collision ceases to be optimal. As the laser amplitude is increased, radiation reaction becomes stronger and the optimal angle increases away from zero. The increased χ max at oblique incidence enhances two quantum effects: the emission of photons with energy comparable to that of the electron, and the stochastic broadening of the electron energy spectrum.
In fig. 3 we show how these two signatures are affected by the collision angle θ in a QED cascade when χ 0 = 10 and the laser pulse duration is one of τ = 10λ and 50λ. The dependence of the distribution of χ max on the angle is different in the two cases: whereas it is approximately constant at χ max 5 for the shorter pulse, the maximum quantum parameter is strongly suppressed for θ < 15 • for the longer pulse. We find that not only is χ max max- imized at θ 45 • rather than at 0 • , as shown in fig. 2, but that the value at 90 • is twice that at 0 • . The reduced apparent pulse duration at normal incidence more than compensates for the reduction in the geometric factor in χ e because it reduces the electron's total loss of energy to radiation. Our theoretical scaling eq. (2) captures both these effects, in close agreement with the simulation results.
The number of high-energy photons is especially sensitive to the highest χ e reached by the electron [24,43]. Accordingly, consider the number of photons N γ with energy ω > γ 0 m/2 in the absence of electron-positron pair creation [the dashed lines in fig. 3(b)]. For the shorter pulse, N γ is almost independent of the collision angle, whereas for the longer pulse, it is maximized at θ 45 • and suppressed for θ < 15 • [44]. In both cases the dependence of N γ on θ mimics that of χ max . When depletion of the photon spectrum due to electron-positron pair creation is included, the optimal angle is increased to 90 • for both pulse durations. This is because the reduced interaction length at normal incidence suppresses the pair creation probability [37], as shown in fig. 3(d).
Another important signature of quantum effects is (1) (dashed lines), which neglects radiative losses, with eq. (2), which includes them, for an electron colliding with a laser pulse (wavelength λ = 0.8 µm, duration τ = 50λ and focal spot size r0 = 2 µm). We see that collisions at normal incidence (orange) are very strongly favoured over head-on (blue) when radiative losses are accounted for.
broadening of the electron energy spectrum [25], caused by the stochasticity of the underlying photon emission process [23]. The variance of the energy distribution σ 2 γ is studied in detail in [45][46][47][48], where it is shown that the temporal evolution of the variance is governed by two competing terms: one that arises from the fact that the radiated power is larger for higher energy electrons, which favours decreasing σ γ , and a stochastic term, which favours increasing σ γ . The broadening term dominates if χ e is large and the pulse duration is short. Both of these cause oblique incidence to be favoured for the scenario explored in fig. 3: χ max is larger at θ > 0 (or at least not significantly reduced) and the interaction length is shorter as well. Figure 3(c) shows that the variance of the post-collision energy is larger for larger θ [49], and that this is not changed appreciably by electron-positron pair creation. That high-energy photon emission and energy broadening can be enhanced at oblique incidence is also true for lower χ e , as shown in the Supplemental Material for χ 0 = 1.
We now consider the collision parameters necessary to reach αχ 2/3 e 1, where strong-field QED becomes fully nonperturbative. By this we mean that perturbation theory with respect to the dynamical electromagnetic field breaks down [17]: for example, the lowest order correction to the strong-field QED vertex V [50]. (Recall that the theory is already nonperturbative in the sense that amplitudes must be calculated to all orders in coupling to the background electromagnetic field a 0 if a 0 > 1 [8].) Reaching such large χ e is therefore of fundamental interest, but experimentally challenging. Strictly the calculation cannot be done for αχ 2/3 e ∼ 1, because we would need all the radiative corrections; however, we can estimate when they become significant by using our results to find the collision parameters necessary to reach, say, χ e = 100, at which the vertex correction is of order 10% and radiative corrections are non-negligible. We emphasize that while our analysis of the radiative losses neglects these corrections, the crucial physical insight remains accurate, as the energy loss which reduces χ max from χ 0 occurs within the intensity ramp where radiative corrections are less important. The dashed lines in fig. 4 show the minimum γ 0 and a 0 if χ e were given by eq. (1): it is evident that the ideal collision angle θ = 0. This is no longer the case when dynamical effects are taken into account.
Using eq. (2) to estimate the minimum energy and laser intensity instead, we find that collisions at θ = π/2 are strongly favoured for a pulse with duration τ = 50λ. (The validity of the scaling at this χ e is checked against simulations in the Supplemental Material.) The additional electron energy or laser intensity necessary to compensate radiative losses, which is indicated by the vertical (horizontal) gaps between the solid and dashed lines, can be substantial. At a 0 = 1000 and θ = 0, for example, the minimum energy must be increased by more than an order of magnitude, from the naive estimate of 8.4 GeV, to 180 GeV. The gradient of the lines indicates that the necessary increase in γ 0 is always smaller than the equivalent increase in a 0 . As discussed earlier, this is because of the stronger dependence of the photon emission rate on a 0 .
We find that the ideal scenario for reaching χ e = 100 and the onset of radiative corrections is not a head-on collision, but rather a collision at 90 • . At a 0 = 1000, which is equivalent to an intensity of 2×10 24 Wcm −2 at a wavelength of 0.8 µm, the required energy is ∼40 GeV. This is readily achievable with conventional accelerators [6,7], and the necessary laser system is of the kind being commissioned at the ELI facilities [31]. Advantages of this geometry are that χ max is independent of the pulse duration and that the high-energy products of the collision are directed away from the focussing optic of the laser pulse.
In summary, we have studied how to reach large quantum parameter in the collision of an electron beam with an intense laser pulse. Our scaling for the maximum χ e , which is verified by Monte Carlo simulations, predicts that the optimal collision geometry is not head-on for long or high-intensity laser pulses. The shorter interaction length at oblique incidence compensates for the geometric reduction in χ e , causing signatures of quantum effects to be enhanced. Our results show that a collision at normal incidence is a viable platform for studying the onset of the breakdown of perturbative strong-field QED at αχ 2/3 e > 0.1.