The atom a light wave opens like a wall

The electrons a field pulls from cold metal found that a strong, steady electric field tilts the step at a metal’s surface into a triangle, and electrons tunnel through it at a rate that rises by thirty orders of magnitude over a factor of ten in field. It ended by asking about emission from things that are not metals. The most important answer comes from an unexpected direction: from isolated atoms, in the oscillating field of a laser, at intensities that only became available in the 1980s, and from the question of whether a field that reverses a thousand trillion times a second can tilt anything into a barrier at all.

Light arrives in lumps stated the photoelectric rule: the frequency of light decides whether electrons come out, and the intensity decides only how many. Light whose photons carry less energy than the binding energy frees no electrons, however bright. The rule is exact for the light Einstein had in mind and fails completely for a focused femtosecond laser, which puts as much power through a spot a few micrometres across as a large power station produces, compressed into a few dozen oscillations of the field. The failure is not a single effect. There are two ways for an atom to ionise in light whose photons are individually too weak, and the question of which one is happening turns out to have a clean answer.

A field strong enough to tip the atom #

At an intensity of $10^{14}$ watts per square centimetre, the peak electric field of a laser beam is about $3\times10^{10}$ volts per metre — a twentieth of the field that holds hydrogen’s electron to its proton. That is not a perturbation. For the moment the field points one way, it tilts the atom’s Coulomb well as the static field tilted the metal’s step.

On the downhill side, the well and the field together make a barrier, and the bound electron sits behind it at its energy of −13.6 electronvolts. At $3\times10^{13}$ watts per square centimetre the barrier is still nearly a nanometre thick at the electron’s energy. At $10^{14}$ it has thinned to a third of a nanometre. At $1.37\times10^{14}$ the top of the barrier falls to the level itself — the barrier-suppression intensity, $I_p^2/4Z$ in atomic units — and above that the electron can simply flow out over the top, without tunnelling at all.

If the field were steady, this would be field emission from an atom, and the rate would follow from the barrier as it did for the metal. The difficulty is that the field is not steady. At a wavelength of 800 nanometres it reverses every 1.3 femtoseconds, and the barrier on the right becomes a barrier on the left, then disappears, then reappears. Whether the electron can make use of the barrier depends on whether it can get through before it moves.

How long the barrier holds still #

Leonid Keldysh compared two times. One is the time a tunnelling electron would spend crossing the barrier, estimated as the barrier’s width divided by the electron’s typical speed under it: the width at the level is about $I_p/eE$, and the speed associated with the binding energy is $\sqrt{2I_p/m}$. The other is the time over which the field stays roughly constant, about one over its angular frequency. Their ratio is the Keldysh parameter,

$$\gamma = \frac{\omega\sqrt{2mI_p}}{eE} = \sqrt{\frac{I_p}{2U_p}},$$

where $U_p = e^2E^2/4m\omega^2$ is the ponderomotive energy, the average kinetic energy of a free electron’s quiver in the oscillating field — the same quiver that the light that moves an electron and pays it nothing found a passing pulse returning in full, and whose spatial variation is the ponderomotive push that held up by a force that averages to nothing traced to a fast oscillation.

When $\gamma \ll 1$, the field changes slowly compared with the crossing time; the electron sees a barrier that is effectively static, tunnels through it near each peak of the field, and the ionisation rate follows the field’s instantaneous strength, as if the atom were in a sequence of static fields. When $\gamma \gg 1$, the barrier flips many times while the electron would be crossing it, and the tunnelling picture loses its meaning. The atom then ionises by absorbing several photons at once, each too weak alone, through a chain of virtual intermediate states that exist only for the brief time the energy uncertainty allows.

The crossover at $\gamma = 1$ comes at an intensity proportional to $\omega^2$: tunnelling sets in sixteen times more easily at a wavelength four times longer. That is why long-wavelength lasers, in the mid-infrared, are the natural tool for strong-field physics, and why the barrier-suppression intensity for hydrogen at 800 nanometres sits so close to the crossover that the classic experiments, done with titanium-sapphire lasers, live in the middle of the transition rather than in either clean limit.

Keldysh’s own derivation did not choose between the two. His 1964 paper computed the ionisation rate in a single formula that becomes the multiphoton rate for large $\gamma$ and the tunnelling rate for small $\gamma$, and it treated electrons in crystals, crossing a band gap, as well as electrons in atoms. Both limits are therefore one process seen from two sides — the same electron leaving the same atom, described in the language that suits how fast the field is changing.

Nine photons at once #

The multiphoton side of the divide was predicted long before anyone could test it. Maria Göppert-Mayer showed in 1931, in her doctoral thesis, that quantum mechanics allows an atom to absorb two photons in a single act, neither of which alone matches any transition, provided their energies together do. The probability of such an event is tiny at any intensity a lamp can produce, because the atom must find the second photon during the fleeting time it spends in a virtual intermediate state — a state that exists only for as long as the energy–time uncertainty tolerates its mismatch, about the time light takes to cross an atom. Two-photon absorption was first seen in 1961, a year after the laser was invented, and multiphoton ionisation of xenon by seven ruby-laser photons in 1965.

The signature of an $N$-photon process is its dependence on intensity. The atom needs $N$ photons to be present at once within a region and time set by atomic scales, so the rate scales as the $N$-th power of the photon flux, $I^N$. For hydrogen at 800 nanometres $N$ is nine, and doubling the intensity raises the rate 512 times. A response that steep is closer to a switch than to an amplifier: across a laser focus, where the intensity falls smoothly from the centre, essentially all the ionisation happens in the hottest region, and as the pulse energy rises the ionised volume does not grow gradually but jumps from nothing to complete depletion of the centre. That steepness is the same nonlinearity that the oscillator that answers at three times the question found turning a sinusoidal drive into harmonics, carried to an order where the atom’s response to a field is no longer described by any short power series at all.

On the tunnelling side the dependence is different in kind. The tunnelling rate depends on the field exponentially, as $\exp(-2(2mI_p)^{3/2}/3\hbar eE)$, the same form that made a wall that a factor of two makes impassable so sensitive to its thickness; plotted against intensity on logarithmic axes it bends over, its slope falling as the intensity rises. The two pictures predict different slopes, and measured ion yields in xenon and argon show the steep multiphoton slope at low intensity bending over towards the tunnelling form near $\gamma = 1$, before saturation flattens everything.

Tunnelling also has a clock that multiphoton absorption lacks. The rate is exponentially largest at the peaks of the field, so electrons leave in bursts a fraction of a cycle long, twice per cycle, each burst emerging on the side the field is pointing at that moment. The electrons are released, in effect, by the wall that is not quite a wall opening and closing at the laser’s frequency. That timing is what the rest of the story depends on.

The threshold the field raises #

The multiphoton picture has a subtlety that the photoelectric rule does not. The number of photons needed is not simply the binding energy divided by the photon energy.

An electron freed inside the laser field does not emerge at rest. It is free, but it is quivering, and its quiver carries the energy $U_p$, which has to be supplied in the act of ionisation. So the effective threshold is $I_p + U_p$, and it rises in proportion to the intensity. At 800 nanometres $U_p$ is about six electronvolts at $10^{14}$ watts per square centimetre — almost four photons’ worth. Each time the rising threshold passes a whole number of photons, the lowest-order channel closes and the atom must absorb one more. The steps are not a curiosity: as the pulse’s intensity rises and falls over its duration, the threshold sweeps through several of them, and when it passes through a resonance with an excited state of the atom, the ionisation rate jumps, which is how such resonances were first identified in the 1980s.

The figure also shows why the multiphoton picture becomes awkward exactly where the tunnelling picture becomes natural. Near $\gamma = 1$ the threshold moves by more than a photon’s energy for a modest change in intensity. A description that counts photons has to keep recounting them; a description in terms of a field strength and a barrier has a single smooth variable.

A second consequence was discovered in 1979 by Pierre Agostini and his colleagues in Saclay, who measured the energies of the electrons released from xenon by an intense laser. They found not one peak, at the minimum number of photons, but a series of peaks separated by exactly one photon’s energy, extending to many photons beyond the minimum: above-threshold ionisation. An electron that has been freed can go on absorbing photons, because near the ion it is still not truly free. The peaks are the multiphoton picture’s natural language. In the tunnelling regime the same spectrum stretches into a long plateau, and explaining that plateau required a new idea.

The electron that comes back #

The idea, worked out by Paul Corkum and independently by Kenneth Kulander and colleagues in 1993, is almost entirely classical. An electron tunnels out near a peak of the field with nearly zero velocity. It is then a free charge in an oscillating field, and its motion is easy to compute. The field accelerates it away from the ion, slows it, reverses it, and — for electrons released in the right part of the cycle — drives it back to where it started.

An electron born exactly at the field’s peak returns with almost no energy; one born a quarter-cycle later never returns. In between, the return energy rises to a maximum of $3.17U_p$ for electrons born about 18 degrees after the peak. The number 3.17 is not adjustable. It comes out of solving the classical motion, $x(t) \propto \cos\omega t - \cos\varphi_0 + (\omega t - \varphi_0)\sin\varphi_0$, for the first return to $x = 0$, and maximising the kinetic energy there.

On returning, the electron can do three things. It can scatter elastically off the ion, picking up energy from the field as it leaves again; that is what makes the plateau in the above-threshold spectrum, out to about $10U_p$. It can knock out a second electron, which explains why atoms in intense fields lose two electrons far more often than independent ionisations would predict. Or it can recombine with the ion it left, giving up its kinetic energy and the binding energy together as a single photon, of energy up to $I_p + 3.17U_p$.

That last process is high-harmonic generation. Because the recollisions repeat every half-cycle of the laser, the emitted light is a comb of odd harmonics of the laser frequency, and because return energies are spread up to $3.17U_p$, the harmonics form a long plateau of nearly equal strength that ends abruptly at $I_p + 3.17U_p$. That cutoff had been measured before anyone could explain it — Anne L’Huillier’s group in Saclay, and Charles Rhodes’s in Chicago, saw harmonics extending to dozens of orders in the late 1980s, against every expectation from perturbation theory, in which each higher order should be weaker than the last. The three-step model explained the plateau, the cutoff and its scaling at once.

The shortest flashes of light #

The recollision happens in a fraction of an optical cycle, and the harmonic light it emits comes out as a burst a few hundred attoseconds long, once per half-cycle. With a laser pulse only a few cycles long, or with tricks that let only one recollision radiate, the burst can be isolated. Ferenc Krausz’s group in Vienna produced isolated attosecond pulses in 2001, and Agostini and colleagues measured trains of them the same year. L’Huillier, Agostini and Krausz shared the Nobel Prize in Physics in 2023. Attosecond pulses now time the motion of electrons inside atoms, molecules and solids — including the photoemission delays and tunnelling times that how long the crossing takes found so hard to define.

The duration of such a pulse is set by its bandwidth, as sharpness has to be paid for insisted: a burst of a hundred attoseconds needs a spectrum about twenty electronvolts wide. The plateau supplies exactly that, and the cutoff sets how short a pulse can be made.

The cutoff grows with the ponderomotive energy, which grows as the intensity times the square of the wavelength. Raising the intensity soon ionises the gas completely and ends the process, so the route to higher photon energies has been longer wavelengths. A titanium-sapphire laser at 800 nanometres drives harmonics in the extreme ultraviolet; mid-infrared drivers at a few micrometres reach the water window, where carbon absorbs and water does not, and in 2012 a 3.9-micrometre driver produced harmonics beyond a kiloelectronvolt. The cost is efficiency. An electron that spends longer away from its ion spreads out quantum mechanically, so less of it comes back to recombine, and the harmonic yield falls roughly as the fifth or sixth power of the wavelength.

Emission from things that are not metals #

Keldysh’s 1964 paper was written as much for solids as for atoms, and the solid case has become a technology. A transparent dielectric such as fused silica has a band gap of about nine electronvolts, six times an 800-nanometre photon, so it does not absorb the light at all at ordinary intensities. Focused tightly inside the glass, a femtosecond pulse ionises it anyway, by multiphoton absorption and tunnelling across the gap, but only in the tiny focal volume where the intensity is high enough. The energy is deposited there and nowhere else, which is how waveguides are written inside glass, how eye surgeons cut flaps inside the cornea without touching its surface, and how microchannels are machined in three dimensions. The strong dependence on intensity, the steep $I^N$ of the multiphoton side, is what confines the damage.

Solids also make high harmonics. In 2011 a group at Stanford drove a zinc-oxide crystal with a mid-infrared laser and saw harmonics extending past its band gap, and since then harmonics from semiconductors, from monolayer materials and from the surfaces of dielectrics have been used to map their band structures. In a solid the freed electron does not fly away into vacuum: it moves through a band, and its recollision is with the hole it left in another. The physics is related to the atomic case and not identical to it.

Hydrogen, one electron and a classical return #

The figures treat a single electron in hydrogen’s potential, in a field that is perfectly linearly polarised, uniform over the atom, and described classically. The return energies come from the classical three-step model, which neglects the Coulomb pull of the ion on the departing and returning electron; including it shifts the return times and energies by a few per cent and focuses the returning electron towards the ion. The barrier picture is drawn at the field’s peak and ignores the field’s change during tunnelling, which is exactly the approximation that $\gamma$ measures. The channel-closing figure assumes the quiver energy of an electron in a uniform field; in a real focus the intensity varies across the beam, so different atoms see different thresholds and the steps are smeared. The harmonic cutoff is the single-atom value; in a gas, the harmonics from different atoms must add in phase, and the phase matching, not the single-atom physics, usually limits the measured flux. The domain is intensities from about $10^{13}$ to a few times $10^{14}$ watts per square centimetre, wavelengths from 400 nanometres to a few micrometres, and atoms with one loosely bound electron in the role that matters; above about $10^{18}$ the laser’s magnetic field pushes the electron forward and it misses its ion.

Still open: how a solid’s electron finds its way back #

In an atom the three steps are clean: tunnel, fly, return. In a crystal the electron is never free, and two mechanisms compete to make harmonics. One is the analogue of recollision, between an electron in the conduction band and the hole it left in the valence band. The other is the electron’s motion within a single band whose energy is not a simple parabola in momentum, which makes its current anharmonic and radiates harmonics even without any recollision. Measurements on different materials have been interpreted as evidence for each, and both certainly contribute; which dominates, in which materials, at which wavelengths, and how the harmonics should be read back into a band structure, is being argued experiment by experiment.

The atomic picture underneath is settled. An intense laser tips an atom’s potential into a barrier; whether the electron tunnels through it or instead absorbs many photons at once is decided by $\gamma = \sqrt{I_p/2U_p}$, about one at $1.1\times10^{14}$ W/cm² for hydrogen at 800 nm; the photon threshold rises by the quiver energy $U_p$ as the field strengthens; and a tunnelled electron driven back to its ion arrives with up to $3.17U_p$, which recombination turns into harmonics reaching $I_p + 3.17U_p$ — 1437 eV for argon at 4 µm. The light that cannot free an electron one photon at a time frees it as a field, and the field then throws it back.