The light that takes back an electron's energy

The light that moves an electron and pays it nothing followed an electron through a pulse of intense light and found it left at rest, displaced but unpaid: a plane wave cannot give a free electron net energy, because two quantities — its sideways canonical momentum and the combination $\gamma - p_z/mc$ — are conserved throughout. The argument left one thing out deliberately. An electron shaken by light radiates, and the radiation carries away energy and momentum that the conservation laws did not account for.

For an electron at rest that omission is harmless, and the force a charge exerts on itself put a number on why: the radiation-reaction force on an electron in the most intense laser light built is under a millionth of the force the light exerts. This essay sends the electron the other way — head-on into the light, at nearly the speed of light — and finds the same force grown until it takes most of the electron’s energy. Nothing about the force has changed except the frame in which it is measured.

The field an electron sees when it runs into the light #

A laser’s strength is measured by the dimensionless field $a_0 = eE/m\omega c$, the momentum the field can give an electron in one cycle in units of $mc$. It is about one at an intensity of $10^{18}$ watts per square centimetre for near-infrared light, and the most powerful lasers reach $a_0$ of a few hundred. For an electron at rest, $a_0$ of a hundred means it is thrown about at nearly the speed of light, but the force of its own radiation is still tiny compared with the field’s push, because the radiated power depends on the square of the acceleration and the acceleration, at the laser’s frequency, is modest by the standards of the charge’s own time scale.

An electron running head-on into the light at Lorentz factor $\gamma$ sees something else. Transformed into its frame, the light’s frequency is raised by the Doppler factor $\gamma(1 + \beta) \approx 2\gamma$, and so is the electric field. A 500 MeV electron, with $\gamma = 1000$, meets light whose field in its own frame is two thousand times what the laboratory measures and whose frequency is two thousand times higher. The power a charge radiates grows as the square of the field it is accelerated by, the Larmor law that a charge that turns must glow began from, and the radiation, emitted forward along the electron’s motion, carries away momentum that the electron must lose.

The numbers make the turnover concrete. A field of $a_0 = 100$ at a wavelength of 0.8 micrometres is an intensity of about $2\times10^{22}$ watts per square centimetre, the scale of the strongest lasers now operating. Radiated power grows as the square of the field in the electron’s frame and, for head-on motion, as the square of the Doppler factor once more because the frequency rises too; between them, running into the light at $\gamma = 1000$ multiplies the radiated power, measured in the electron’s own frame, by something like $(2\gamma)^2 = 4\times10^6$. A force that was a millionth of the applied one for an electron at rest becomes comparable to it. The physics of the self-force has not changed: the same Larmor power, the same classical electron radius, the same small coupling. What has changed is which frame the electron is in when the light arrives, and radiation reaction is the one place in classical electrodynamics where that choice decides whether an effect exists.

A light-front momentum that is no longer conserved #

The cleanest description uses the quantity that the light alone conserves. For an electron moving against a plane wave, $u_- = \gamma(1 + \beta)$, measured along the direction of the light, does not change however hard the electron is shaken: it is the conserved quantity that left the electron unpaid in the pulse of the previous argument. Its conservation has a reason that survives into the new problem. A plane wave depends on position and time only through the combination $t - z/c$, so it is unchanged by a shift along the light’s direction made together with an equal shift in time, and the momentum belonging to that symmetry is conserved. A radiating electron breaks the symmetry not in the light but in itself: the photons it emits, travelling forward along its own motion, carry away light-front momentum, and the electron has to give it up. With radiation reaction included, in the form Lev Landau and Evgeny Lifshitz gave it — the reduced equation that avoids the runaway solutions of the original — it changes, at a rate set by the square of the field:

$$\frac{du_-}{d\varphi} = -\frac{2}{3}\,r_e k_0\,u_-^2\,a(\varphi)^2,$$

where $\varphi$ is the laser phase, $k_0$ the laser’s wavenumber and $r_e$ the classical electron radius. The equation can be solved exactly, as Antonino Di Piazza showed in 2008:

$$u_-(\varphi) = \frac{u_{-0}}{1 + \tfrac23 r_e k_0 u_{-0}\int a^2\,d\varphi'}.$$

The loss is governed by the product of three numbers: $r_ek_0$, the electron’s classical size in laser wavelengths, a few parts in a hundred million; $u_{-0} \approx 2\gamma_0$, the Doppler boost; and the integral of $a^2$ over the pulse, which for a ten-cycle pulse of peak $a_0$ is about twelve times $a_0^2$. For $\gamma_0 = 1000$ and $a_0 = 100$ their product is about 3.5, and the electron’s light-front momentum, and with it its energy, falls by a factor of 4.5.

The loss happens in steps. Twice in every laser cycle the field passes through a maximum, the electron is bent most sharply, and it radiates a burst; between maxima it coasts. The steps are largest at the peak of the pulse, where $a^2$ is largest, and they shrink as the electron loses energy, because a slower electron sees a weaker field in its own frame. The integral form shows the same saturation: the energy kept falls as one over one plus the integrated field, not exponentially, so doubling the pulse’s energy does not take twice as much from the electron.

How the loss grows with the electron’s energy #

The striking feature of the solution is that it depends on the electron’s energy as well as the laser’s. Faster electrons lose a larger fraction.

The fraction lost is controlled by $\gamma_0 a_0^2$. Doubling the electron’s energy has the same effect as raising the laser’s field by $\sqrt2$, which is twice the intensity. A two-gigaelectronvolt electron, of the kind a laser-plasma accelerator now produces in a few centimetres — the wave an electron rides to a gigaelectronvolt followed how — loses half its energy to a pulse of $a_0 \approx 27$, an intensity reached by many laboratories. The radiation goes into a beam of gamma rays along the electron’s original direction, with energies up to a large fraction of the electron’s own: the same inverse Compton scattering that the photon a fast electron throws back followed for a weak laser, now strong enough that each electron scatters thousands of photons and feels the recoil of every one.

Where the energy goes #

The energy the electron loses does not disappear into heat; it leaves as a beam of gamma rays, narrowly collimated along the electron’s original direction, within an angle of about $a_0/\gamma$ — the angle through which the field swings the electron. The photons have a broad spectrum, like the synchrotron radiation of an electron bent in a magnet, because in its own frame the electron is being bent through a sharp arc twice every cycle of the light. The characteristic photon energy of that spectrum is about $\tfrac32\gamma^2a_0\hbar\omega_0$: for a 511 MeV electron in a field of $a_0 = 30$, about seventy megaelectronvolts, a seventh of the electron’s energy in a single photon.

That number is already a warning. A smooth, classical outflow of energy is a good description when each photon carries a negligible share of the radiating particle’s energy, so that the losses add up to a steady drag. When a typical photon takes a seventh, the electron’s energy changes in jumps, and the size of the jumps is set by Planck’s constant rather than by the laser field. The ratio of the characteristic photon energy to the electron’s energy is $\tfrac34\chi$, in terms of a parameter that measures how quantum the emission is, and it is the subject of the next section. The gamma rays themselves are useful: a beam of photons of tens of megaelectronvolts from a centimetre-scale source, emitted in a few femtoseconds, is a probe of nuclei and a source for nuclear physics that does not need a kilometre-scale accelerator.

The energy bookkeeping of the collision is lopsided in an instructive way. A bunch of a nanocoulomb of electrons at a gigaelectronvolt carries a joule; a laser pulse of a few hundred terawatts lasting thirty femtoseconds carries about ten. If the electrons lose half their energy, half a joule leaves as gamma rays — and almost none of it comes from the laser. Each scattering absorbs laser photons of an electronvolt or two and emits a gamma ray of tens of megaelectronvolts, and the difference is paid by the electron. The light acts as a wiggler, an undulator whose period is the laser’s wavelength and whose strength is $a_0$, and the electron radiates its own energy away in it. That is why the effect needs the electron to be fast rather than the laser to be energetic: the reservoir drained is the electron’s kinetic energy, and the light’s role is to bend it sharply enough, often enough, to make it radiate.

Where the classical theory gives out #

The classical formula treats the radiation as a smooth outflow of energy. That stops being right when the energy of a single emitted photon becomes a sizeable fraction of the electron’s, because the electron cannot give up more than it has and does not radiate continuously but in quanta. The measure of that is the quantum parameter

$$\chi = \frac{E_{\text{rest frame}}}{E_{\text{cr}}} \approx 2\gamma\,a_0\,\frac{\hbar\omega_0}{mc^2},$$

the field in the electron’s frame divided by the critical field $m^2c^3/e\hbar$, about $1.3\times10^{18}$ volts per metre — the field at which the vacuum itself becomes unstable to producing electron–positron pairs. The field nobody can transform away described it as the Schwinger field and noted that no laboratory field has come within four orders of magnitude of it. That is true in the laboratory frame. An electron running into the light at a few gigaelectronvolts gains three or four orders of magnitude by the Doppler boost alone, and in its own frame it meets the critical field with lasers that exist. When $\chi$ is small the classical description is accurate; when it approaches one, a typical emitted photon carries a good part of the electron’s energy.

Quantum electrodynamics gives the radiated power as the classical power multiplied by a factor $g(\chi)$ that is one for small $\chi$ and falls steadily as $\chi$ grows. The suppression has a simple origin: the classical spectrum extends to photon energies that the electron does not have, and quantum mechanics cuts the spectrum off at the electron’s energy. At $\chi$ of a tenth, the classical formula is already fifty per cent too large; at one, more than five times.

For a two-gigaelectronvolt electron the quantum correction matters at every field strength at which radiation reaction is large. The corrected electron keeps more of its energy, and the difference grows with the field: at $a_0 = 100$ the classical theory has it keep seven per cent, the corrected one twenty-three. And the averaged correction drawn here still hides the larger change. Quantum emission is a sequence of discrete photons emitted at random, so two electrons entering identical pulses leave with different energies; some are lucky and emit little, a phenomenon called straggling, and the beam that leaves has a spread of energies that the classical theory cannot produce from a beam with none.

Where the regimes lie #

The two conditions — classical loss of order one, and $\chi$ of order one — depend differently on the electron’s energy. The first needs $\gamma a_0^2$ large, the second $\gamma a_0$.

Below about five gigaelectronvolts, with near-infrared light, a laser can make radiation reaction dominate the electron’s motion while the emission is still nearly classical; above that energy, the emission turns quantum before the classical loss becomes large. The first experiment in this territory was SLAC’s E-144 in the mid-1990s, which sent 46.6 GeV electrons into a terawatt laser at $a_0$ below one: the field was weak but $\chi$ approached a third, and the experiment saw both the emission of several laser photons at once and the production of electron–positron pairs from light, the first time matter had been made from photons alone. Radiation reaction itself was first seen in 2017 and 2018 at the Gemini laser at the Rutherford Appleton Laboratory, by colliding electrons of one to two gigaelectronvolts from a laser-plasma accelerator with a pulse of $a_0$ between ten and twenty: the electrons that met the pulse came out with less energy, and the gamma-ray yield rose, in amounts consistent with radiation reaction and better described with quantum corrections than without, at the edge of statistical significance. The difficulty was alignment: the electron bunch and the laser focus are each a few micrometres across and a few tens of femtoseconds long, both produced afresh on every shot, and on most shots they simply miss. The analyses selected the shots on which the gamma-ray signal showed that a collision had happened, and compared the electrons’ spectra with and without it.

What the plane-wave solution leaves out #

The figures use an idealisation that makes the problem exactly solvable: a plane wave, infinite in its transverse extent, meeting an electron exactly head-on. A real laser is focused to a spot a few micrometres across, and an electron passing through it sees a peak field that depends on where in the spot it crosses; most of a real electron beam misses the focus, and the measured energy loss is an average over a distribution of peak fields. The pulse is taken to be ten cycles long with a smooth envelope; longer pulses at the same peak field take more energy, in proportion to the integral of $a^2$.

The electron is a single point charge, so the collective effects of a dense beam — the electrons’ own fields, the coherent emission of a bunch — are left out, as are the pairs that the gamma rays themselves create in the laser field once $\chi$ exceeds one, which at higher fields start a cascade of photons and pairs. And the quantum correction drawn is the averaged one, applied to a classical trajectory, which captures the reduced emission and nothing of its randomness. The domain of the drawings is head-on collisions of single electrons of 0.1 to 100 GeV with near-infrared plane-wave pulses of $a_0$ from one to a few hundred.

Still open: which equation the electron obeys #

The Landau–Lifshitz equation is one of several classical equations proposed to replace the original Lorentz–Abraham–Dirac equation, whose runaway solutions the force a charge exerts on itself described. In the regime of present experiments the candidates give nearly the same answers, and the differences between them are smaller than the quantum corrections, so the experiments cannot yet tell them apart; what they can test is whether a classical equation with a quantum correction describes the electron at all, or whether a fully quantum description, with photon emission as random events, is needed even for the average. The experiments being prepared at DESY’s European XFEL and at SLAC, with stable electron beams of ten to fifteen gigaelectronvolts and lasers of $a_0$ up to ten, aim at the regime where $\chi$ is near one and the theory of strong-field quantum electrodynamics, untested for a century, makes predictions that can be checked precisely. Beyond $a_0$ of a few hundred with $\chi$ of a few, a further question is open: the standard strong-field calculations assume that the light can be treated as a fixed background, and are thought to fail altogether at $\alpha\chi^{2/3}$ of order one, where nothing is known about how an electron behaves.

The arithmetic of the classical regime is short. An electron meeting light head-on sees its field raised by γ(1 + β), and its light-front momentum, conserved by the light alone, falls as $1/\bigl(1 + \tfrac{2}{3} r_e k_0 u_{-0} \int a^2\,d\varphi\bigr)$: a 511 MeV electron keeps 76 per cent of its energy after a ten-cycle pulse at $a_0 = 30$ and 22 per cent at $a_0 = 100$ — and where the field in its frame nears the critical field, χ ≈ 1, quantum emission radiates five times less than the classical formula says. The force that is a millionth of the push on a resting electron becomes the whole of the story for one that runs into the light.