Identifying quantum effects in seeded QED cascades via laser-driven residual gas in vacuum

The discrete and stochastic nature of the processes in the strong-field quantum electrodynamics (SF-QED) regime distinguishes them from classical ones. An important approach to identifying the SF-QED features is through the interaction of extremely intense lasers with plasma. Here, we investigate the seeded QED cascades driven by two counter-propagating laser pulses in the background of residual gases in a vacuum chamber via numerical simulations. We focus on the statistical distributions of positron yields from repeated simulations under various conditions. By increasing the gas density, the positron yields become more deterministic. Although the distribution stems from both the quantum stochastic effects and the fluctuations of the environment, the quantum stochastic effects can be identified via the width of the distribution and the exceptional yields, both of which are higher than the quantum-averaged results. The proposed method provides a statistical approach to identifying the quantum stochastic signatures in SFQED processes using high-power lasers and residual gases in the vacuum chamber.


Introduction
High-intensity lasers can provide extreme conditions as a powerful tool for plasma-based accelerators and novel radiation sources [1][2][3][4].It is expected that the focal intensities of 10-100 PW class lasers could approach beyond 10 23 W cm −2 , where strong-field quantum electrodynamics Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Spontaneous electron-positron pair creation out of the vacuum can take place when the field strength is higher than the Sauter-Schwinger critical field E S ≈ 1.3 × 10 16 V cm −1 [39].In strong laser fields, pair creation can be triggered at a much lower field strength [5,40].When the field strength is sufficiently strong, electron-positron pairs which will emit photons capable of decaying into new pairs, leading to QED cascades.It has been shown that the onset of seeded cascade (with one electron at the beginning) can be facilitated at an intensity around 10 24 W cm −2 [18][19][20]41].Electronpositron plasma can be exponentially generated via selfsustained gamma photon radiation and pair-production with even one electron in the propagating or standing wave formed by ultra-intense laser pulses [17,18,21].As the plasma density grows, the laser can eventually be absorbed [42], which determines the upper limit of the strong field in a non-ideal vacuum [17,43].Since seeded QED cascades couple both the stochasticity of photon radiation and pair-production, a strong quantum nature emerges.For instance, the positron yield is stochastic in a non-ideal vacuum setup [20,21].
In this article, we are going to show that the positron yields in seeded QED cascades can be a convenient signal to identify the stochasticity of QED cascades.The statistical distribution of the yields among multiple simulations conforms to a specific distribution, the width of which is larger than that of the photon radiation alone (without coupling) and much larger than the QED-average/semi-classical results.This signature indicates the stochastic nature of QED cascades and the coupling effect of photon radiation and pair-production.The cascade can be triggered by the residual gas in vacuum chambers without the need to fix or inject electrons at the laser focuses, which can potentially test the strong-field theory in the 100 PW-class laser systems [1,3,4].The quantum stochastic effects of QED cascades can be identified by evaluating the positron yield distribution of multiple laser shots hitting the residual gas through a statistical method.

QED Monte-Carlo cascades
The two basic QED processes, nonlinear Compton scattering (NCS) and nonlinear BW process, play the most important roles in QED cascades.Under the assumption of a strong field, the instantaneous photon emission rate is [5,44]: where α f ≈ 1/137 the fine structure constant, γ e the electron relativistic factor, τ c = h mec 2 , h the reduced Planck constant, m e the electron mass, c the speed of light, χ e = F µν Es pν mec and χ γ = F µν Es hkν mec the Lorentz-invariant quantum parameter of the electron and the photon, F µν the electromagnetic field tensor, p ν the electron four-momentum (the photon fourmomentum hk ν ) and E S ≈ 1.32 × 10 18 V m −1 the Sauter-Schwinger critical field [39].K 1 3 (s), K 2 3 (y) the modified Bessel functions and y = χ γ 3χ e(χ e−χ γ ) , respectively.Similarly, the rate of photons decaying into e + e − pairs is [5,44]: (2) where γ e is the relativistic factor of the newborn electron, and γ γ the photon relativistic factor.
For the commonly used codes to simulate interactions with QED processes, the point-like QED events take place on the classical trajectory and are implemented by the Monte-Carlo method [44][45][46] in each time step.Each electron is randomly assigned an optical depth which decreases according to the radiation probability rate and the radiation event is triggered when the optical depth decreases below zero.The photon energy is then determined via inverse sampling of the photon spectrum equation (1).BW pair production follows a similar process where photons decay into pairs of electrons and positrons.

QED-averaged cascades
For the purpose of clarifying the quantum stochastic effects, a QED-averaged estimation of positron yield during cascades [17,21,40,42] is carried out.By using the analytical growth rate of the seeded cascade, the differences between QED-MC and QED-averaged cascades can then be attributed to the stochastic effects of QED and it can decouple the extra randomness of the seeded electrons' initial distribution.The growth rate of pairs in a cascade induced by two linear polarized laser pulses has an empirical fit of [42]: where α ≈ e 2 hc with e the electron charge, K 1 √ a S , χ e ∼ = 1.24µ 3 2 with µ = a αas .Here a = eE meω0c is the normalized laser amplitude with E being the peak electric field, ω 0 the laser angular frequency and a S = eES meω0c .The number of pairs produced by a single electron can then be estimated by N ± ∼ e Γt − 1.In the standing wave formed by two collision laser pulses, the peak field strength experienced by electrons near the collision plane can be approximated by f a (r i ) = a 0 e −r 2 i /w 2 0 with r i the initial radial position.For N seed electrons distributed at different r i , the final electron-positron yield is therefore altered by r i : The parameter t eff is inferred from the QED-MC results for consistency consideration by equating N ± with the mean yield of 1000 QED-MC results.

Test-particle simulation
Obtaining the distribution of the positron yields requires thousands of repetitive simulations of QED cascades.It is computationally expensive to carry out particle-in-cell (PIC) simulations.Therefore, a test-particle simulation is adopted.
The test-particle algorithm solves the Lorentz equation with the Boris pusher and simulates photon emission and pairproduction with Monte-Carlo method, which is adopted by most QED-PIC codes.Unlike PIC, which solves the Maxwell equations in gridded space and interpolates fields onto shaped macro-particles, the test-particle algorithm evaluates the electromagnetic fields at instantaneous particle positions according to predefined laser profiles.We adopt the focused Gaussian pulses [47] in our calculations, which are linearly polarized (LP) along the y-direction and propagate along the x-direction with a wavelength of λ = 800 nm.Since the test-particle algorithm ignores the interaction between particles, simulation parameters such as pulse length and field strength are controlled at the onset of the QED cascade region where the number of generated electron-positron pairs is not too high and a large number of simulations is possible.In such situations, collective plasma effects can be ignored due to the relatively low plasma density.In the following simulations, we choose a field strength of a 0 = 700, spot size of w 0 = 4 µm and pulse length of τ L = 4T 0 , where w 0 and τ L corresponds to the 1/e of the Gaussian profile.The simulation time step is dt = T0 100 , which is proved to be sufficient to model NCS and BW processes in the parameter region [45].We chose a relatively small spot size and short pulse length to suppress further exponential growth of produced pairs [19,22] for computational consideration.According to our findings to be presented, longer or larger pulses will both result in stronger stochastic signatures.

Fixed seeded electrons
We start from the simplest situation where two colliding LP laser pulses with electrons are fixed at the origin to demonstrate the quantum stochastic effects and statistic scaling law in QED cascades.More realistic considerations will be discussed later.The evolution of the positron yields for a different number of seeded electrons N seed =1, 10, 100, 1000 is shown in figure 1.One can see that although each set of simulations has the same initial condition, the growth and final yields may significantly differ from each other when the number of seeds is low, indicating the quantum stochastic feature of the QED cascades [21].It should be noted that for N seed = 1 cascades are not triggered in most simulations for the considered parameters, and the positron yields distribution gathers near zero.By increasing the number of seeded electrons, the positron yield coverages near the mean value N marked by the dashedlines, indicating a more deterministic behavior of the QED cascades.One can find that N increases almost linearly with N seed in statistics, which could be modeled by the analytical calculation [21,42].
Another noticeable feature is the exceptional positron yields that significantly exceed the mean yield and other results, as indicated in figure 1.For N seed = 1 some simulations generate more than 50 positrons and one simulation generates about ∼100 positrons, much larger than the mean value at about N ≈ 7.As N seed increases, such a deviation is significantly depressed, and the statistics become more deterministic, as shown by figures 1(g) and (h).It can be predicted that the QED cascades triggered by thin foil [42,48,49] or gases of moderate density [19,22,50] are much more predictable and the exceptional yields will be absent.
The positron yield distribution and the exceptional shots reflect the quantum/stochastic nature of photon radiation, radiation-reaction, pair-production, and their coupling.It requires a certain probability for electrons/positrons to emit high-energy photons and for those photons to decay into pairs at specific phases, resulting in abundant positron yields that stem from a sequence of incessant improbable QED events.It should be emphasized that individual photon radiation or pair-production does not lead to similar effects, which will be discussed later.To quantitatively describe the quantum effects, the normalized positron yield distributions are illustrated in figure 2(a).The relative width of the distribution can be modeled by the standard deviation σ which is shown in figure 2(b) along with the mean yield N. For N seed = 1, the distribution displays the strongest quantum stochasticity, wherein cascading is not triggered in most simulations, but it produces a maximum yield that is ten times greater than the mean yield.As N seed increases, both the relative widths and exceptional yields shrink and the statistic distributions become more concentrated and deterministic, as shown by the black-dotted line in figure 2(b).This can be interpreted as the transition of the statistics from quantum to classical, where each particle exhibits quantum behaviour but the statistics of the particle system behaves more classically as the particle number increases.It should be noted that photon radiation and pair-production are quantum processes with no analogy in classical physics, but the statistics can be deterministic and classical.At the same time, N increases linearly to N seed as expected, following equation ( 4), which models the quantum-averaged positron yields.
The above results reflect the quantum stochastic nature of the coupling between the photon radiation and pair-production as mentioned before, and individual photon radiation or pairproduction processes will not induce stronger stochasticity.The normalized photon yields of seed electrons without BW pair production are shown in figure 3 that presents the individual stochasticity of photon radiation.The photon yield distributions are more concentrated than the results in figure 2, indicating lower uncertainty of the individual processes.In cascades, the number and energy of photons radiated by electrons are distributed in a wide range, which will induce a wider distribution of the pairs produced by these photons.The uncertainty of photon radiation and pair-production are then coupled and exhibit higher stochasticity.
On the other hand, the quantum stochastic effects can be further coupled with the classical stochastic accelerations [51], in which the particle's stochastic acceleration is triggered by the random-walk-like motion (wandering path) in the standing wave.These classical effects are naturally included in the calculation by solving the Lorentz equation, which are then imprinted to the radiated photons.As the number of seeding particles increases, the positron yield converges, denoting the limit of classical stochasticity.It is also interesting that the stochastic photon emission could reduce the classical stochastic heating, which could contribute to the existence of attractors in the phase space and even show a transition from chaotic to regular dynamics as the field strength increases [8,50,52,53].

Experimental considerations
Increasing the number of fixed seeded electrons in the focus is only an analog of the transition from quantum statistics to classical statistics, which is not experimentally possible.Previous studies have focused on injecting high-energy seed particles into the strong laser fields to initiate the cascades [41,54,55], but this is also challenging to achieve in practice.Here we propose a different approach, where we use the low-density residual gas in the vacuum chamber as the seed particles.In our modeling, the gas molecules (100% N 2 for simplicity) are pre-ionized by the strong laser fields and randomly distributed around the laser focal area.The electrons of N 2 are randomly distributed in the simulation box of 48 µm × 60 µm × 60 µm and the number of electrons is determined by the gas number density n gas (or vacuity) and the volume of the interaction region.The counter-propagating laser pulses enter the interaction region from opposite sides of the simulation box along +x and −x direction and trigger cascading.
We simulate seeded cascades with four different gas densities n gas = 10 15 , 10 16 , 10 17 , 10 18 m −3 , in which n gas = 10 15 m −3 is close to the lowest attainable vacuity of the vacuum chamber for PW laser facilities.Figure 4(a) shows the normalized positron yield distribution for 1000 simulations.For low gas densities (⩽10 17 m −3 ) the exponential distribution shows a strong stochastic signature since cascades in most simulations are not initiated due to the low densities.The highest yield is as high as Nmean > 100 at n gas = 10 15 m −3 , much higher than the fixed seeded electrons, since the mean yield of low-density gas is lower than that of the fixed electrons.At higher density (∼ 10 18 m −3 ) the statistic distribution shows a classically convergent trend, similar to the results in figure 2(a).
However, the high yield at low densities originates from the additional stochasticity induced by the randomness of the initial distribution of the electrons.For the sole estimation of this aspect, the semi-classical (QED-averaged cascade) results, as introduced in the methods section, are shown in figure 4(b) where the stochastic effects from QED are averaged and the yield distribution can be solely attributed to the randomness of the initial electron locations.By comparing the widths of the distributions in figures 4(a) and (b), we find that the location-induced randomness is significantly lower than the QED-induced randomness.The widths σ and mean yields N are compared in figure 4(c), where the mean yields of the QED and QED-averaged results coincide but the widths of the yield distribution of the latter are one-order-of-magnitude lower than the former, indicating that the strength of QED stochasticity exceeds the randomness of initial distribution of the locations and that the gap inbetween represents the stochasticity of QED.
Therefore, two experimental signatures can provide proof of quantum stochastic effects: (1) the statistical distribution and the transition of statistics from quantum stochastic to classical, derived from the interaction between the coupling stochastic effects of quantum nature and the statistically convergent effect induced by multiple seeds.The QED-induced distribution is distinctly different and broader than the semiclassical/QED-averaged calculation; (2) the exceptional prolific shots with much higher yield than the mean expectation for thin gas, where few pairs should be produced in semiclassical/QED-averaged calculations.The yield may be two orders higher than the mean value, thus it can be easily identified by statistical methods.
Considering that 1000 shots is a heavy task for high-power (>100 PW) laser systems due to their low repetition rates, we estimate the minimum required shots to reproduce the distribution of the deviation σ.As shown in figure 5, the black lines represent the widths of yields of QED (solid line) and QED-averaged (dashed line) results for N shots = 1000, and the same in figure 4(c), and the shaded areas represent the 5% and 95% percentiles of possible σ that can be observed for shot numbers of N shots = 10 (gray), 50 (red) and 100 (green).For very few shots like N shots = 10, it is more probable to produce zero-results at low gas densities and the possible σ can be lower than expected.It should take more than 100 shots to resolve the statistical deviation at low density.As the gas density increases, the percentile area shrinks rapidly for both the QED and QED-averaged results, i.e. the transition to classical.The QED results can be well distinguished from the QEDaveraged results for N shots ⩾ 50.Therefore, more than 50 shots are required to capture the accurate deviation of positron yields and the transition from quantum to classical as the gas density increases.

The optimal laser parameter
Here, we estimate the stability of the statistical law for three densities n gas = 10 16 , 10 17 , 10 18 m −3 with different laser parameters, in order to find the optimal experimental conditions  for detection.We consider realistic parameters for near-future experiments: (a) a 0 varies from 650 to 800 and τ 0 = 4T 0 , w 0 = 4 µm, (b) τ 0 varies from 2T 0 to 5T 0 and a 0 = 700, w 0 = 4 µm, (c) a 0 = 700, τ 0 = 4T 0 with w 0 varies from 2 µm to 5 µm.The results are shown in figure 6 for three gas densities.
In general, increasing a 0 , τ 0 or w 0 all lead to higher mean yield of cascades (red lines), which means more QED events, and more significant stochastic features can be observed.This can be verified by the gray lines.It is worth noting that increasing w 0 increases the number of electrons in the colliding volume, which is equivalent to increasing the gas density.On the other hand, it increases the interaction time of cascades in laser fields, which is similar to increasing the pulse length.As already shown in figures 5 and 6(b), the dependence of σ on gas density and pulse length are opposite.As a result, σ shows relatively weak dependence on w 0 as shown by the gray lines in figure 6(c).Therefore, tightly focused lasers are preferable for the observation of the stochastic effects of QED cascades.This is different from [19,22] that for optimal cascade development a larger laser spot size is needed.The pulse lengths and field strengths are constrained in the selected region for computational consideration.Predictions can be extrapolated from the results for longer pulse lengths and lower field strengths accessible in future 100 PW-class laser systems [1][2][3][4].

The effect of laser pointing instability
Studies have shown that the time delay between the pulses has a minor impact on the cascades, but the transverse mismatch between pulses can prevent cascading [22].Taking into consideration the pointing instability, we assume that each pulse has an independent offset ∆ in each transverse direction (y and z) that follows ∆ = r • δw, where r is a random number that follows a normal distribution.We simulate cascades for δw = 0, 1, 2, 3, 4 µm and w 0 = 4 µm.The results in figure 7 show that the average yields are suppressed when the misalignment increases since the number of electrons in the interaction region decrease when the two colliding pulses are misaligned.Although misalignment of laser spots adds extra randomness to the process, the trend of increased σ for lower gas densities remains.Moreover, as there is a trend of insufficient shots for statistics in figure 5 at lower density, the pointing instability requires more results to guarantee more reliable statistics.However, to better observe the effects, larger yields are preferred for a higher signal-noise ratio, and the laser misalignment should be well controlled.

The pre-pulse influences
For a PW laser system one must consider the influence of prepulses.For the investigated field strengths, the pre-pulses can fully ionize low-Z atoms like hydrogen, but relatively high-Z atoms (like oxygen and nitrogen) may only get partially ionized before the main pulse arrives.This effect has been discussed in several works such as [19,22], in which the hydrogen/oxygen gas target is used for QED cascade.Since the pre-pulses are unable to fully ionize the inner-shell electrons, it is hard for the pre-pulses to sweep out the electrons in the interaction region.In a recent study on vacuum cleaning experiments [56] using pre-pulse-like lasers to ionize and eliminate the residual gas near the laser focus, it is found that while these ionized particles are pushed outward, a significant amount of residual particles also drift into the laser propagation path, indicating negligible effects of pre-pulses.The prepulses overall act as a modification of the gas densities near the laser focus, and increasing the gas density could compensate for the decrease in electrons.

Conclusion
In summary, we studied the seeded QED cascades driven by two counter-propagating laser pulses, and found that the positron yields follow specific distributions among multiple simulations, which reflects the stochastic nature of QED cascades.The quantum stochastic effect is greatly enhanced by the coupling between photon radiation and pair-production in the seeded cascade.The enhanced effect can be observed in the collision between ultra-intense laser pulses and thin gases, which are usually residual molecules in the vacuum chamber.The quantum stochastic effect can be quantified by the width of the distribution and exceptional events of the positron yields.Quantum effects are significant when the gas density is moderately low and the results become more deterministic at higher densities.For the experimental observation of quantum stochastic effects, a tightly focused laser is preferred and the pointing stability of the laser pulses should be well controlled for optimal yields.The proposed scheme can be validated with the 100 PW laser systems.

Figure 2 .
Figure 2. (a) The positrons' yield distribution of 4 sets of simulations (N seed =1, 10, 100, 1000), each of which is normalized by the mean yield of 1000 independent simulations.(b) The normalized distributions' standard deviation σ (black dots) and mean yield N (red dots) versus the number of seed N seed .

Figure 3 .
Figure 3.The normalized photons' yield distribution without BW pair production.

Figure 4 .
Figure 4. (a) The positrons' yield distributions of 4 sets of simulations (ngas = 10 15 , 10 16 , 10 17 , 10 18 m −3 ), each of which is normalized by the mean yield N. (b) The positrons' yield distributions of the semi-classical calculations that only consider the randomness of the initial electron locations.(c) The normalized standard deviation σ (black) and the mean yield N (red) versus the gas density ngas, where the solid-dotted lines are from simulations and the dashed-triangle lines are from semi-classical (QED-averaged) calculations.

Figure 5 .
Figure 5.The standard deviation σ of normalized positron yields at different densities ngas for limited number of laser shots of N shots = 10 (gray), 50 (red) and 100 (green).The lines are the results of N shots = 1000, same in figure 4(c).The shaded areas represent the 5% and 95% percentiles of possible σ of N shots shots.