Attosecond coherent control of free-electron wave functions using semi-infinite light fields

G. M. Vanacore, I. Madan, G. Berruto, K. Wang, E. Pomarico, R. J. Lamb, D. McGrouther, I. Kaminer, B. Barwick, F. Javier García de Abajo and F. Carbone, Attosecond coherent control of free-electron wave functions using semi-infinite light fields, Nature Communications 9, article 2694, 2018. Reproduced under the Creative Commons Attribution 4.0 International licence stated in the article; the published version is at doi.org/10.1038/s41467-018-05021-x.

(On this site, the near-field method this paper generalises — photon-induced near-field electron microscopy, with Brett Barwick as a coauthor there too — is at /library/stm-fd038781e5, and the other line of engineered free-electron states, electron vortex beams carrying quantised orbital angular momentum, is at /library/stm-932991c317.)

Abstract

Light-electron interaction is the seminal ingredient in free-electron lasers and dynamical investigation of matter. Pushing the coherent control of electrons by light to the attosecond timescale and below would enable unprecedented applications in quantum circuits and exploration of electronic motions and nuclear phenomena. Here we demonstrate attosecond coherent manipulation of a free-electron wave function, and show that it can be pushed down to the zeptosecond regime. We make a relativistic single-electron wavepacket interact in free-space with a semi-infinite light field generated by two light pulses reflected from a mirror and delayed by fractions of the optical cycle. The amplitude and phase of the resulting electron-state coherent oscillations are mapped in energy-momentum space via momentum-resolved ultrafast electron spectroscopy. The experimental results are in full agreement with our analytical theory, which predicts access to the zeptosecond timescale by adopting semi-infinite X-ray pulses.

Introduction

The scattering of single photons by free-electrons is extremely weak, as quantified by the Thomson scattering cross-section, which for visible frequencies is of the order of ten to the minus twenty-ninth square metres. Additionally, direct photon absorption or emission by a free-space electron is forbidden due to energy-momentum mismatch. To circumvent these limitations and increase the probability of electron-photon interaction, a variety of methods have been devised. For example, the Kapitza-Dirac effect involves a conceptually simple configuration in which an electron intersects a light grating produced by two counter-propagating light beams of the same frequency. The interaction is then elastic and requires the electron to undergo an equal number of virtual photon absorption and stimulated-emission processes. When the absorbed and emitted photons differ in energy, the interaction results in frequency up- or down-conversion, which is the basis of undulator radiation and free-electron lasers.

A direct single-photon emission or absorption process can also bridge the energy-momentum mismatch if either the electrons are not free — for example, in photoemission from atoms and molecules and from solid surfaces — or when a scattering structure generates evanescent light fields in the vicinity of the interaction volume. Such an electron-photon-matter interaction creates optical field components with a frequency-momentum decomposition that lies outside the light cone, allowing emission or absorption to take place. This type of interaction, which is forbidden in free space, is regularly exploited for generating radiation and for accelerating charged particles. Recently, it has also prompted the development of photon-induced near-field electron microscopy, PINEM. In PINEM, an energetic electron beam interacts with the evanescent near-fields surrounding an illuminated material structure. The interaction is particularly strong when the structure supports surface-plasmon polaritons that are excited by short light pulses. Optical near-fields then produce coherent splitting of the electron wave function in energy space, giving rise to Rabi oscillations among electron quantum states separated by multiples of the photon energy. The microscopic details of the process are encoded in the electron wave function, which can be revealed via ultrafast electron energy-loss spectroscopy and controlled using suitable illumination schemes.

In this work, we adopt a more general method for controlling and manipulating the strength of electron-photon interaction. Instead of relying on localized near-fields, for example plasmons, which inevitably depend on the intrinsic cross-section associated with the optical excitation of confined optical modes, we make use of a spatially abrupt interruption of the light field in free space, also referred to as a semi-infinite field. Such a boundary condition can be attained by sending the electrons through a light beam that intersects a refractor, an absorber, or more efficiently, a reflecting mirror along the optical path. When the light wave extends only over half-space, the energy-momentum conservation constraint is relaxed and electron-photon interaction can take place with an efficiency exceeding that produced by a resonant plasmonic nanostructure. Using this configuration, we have been able to simultaneously observe the quantized exchange of both energy and transverse momentum between a free-electron and a light wave, revealing the primary role of the quantum nature of the electron-light coupling at optical frequencies and above, and we provide a full description of the strength of interaction within parameter space.

Within this scenario, we demonstrate attosecond coherent control of the electron wave function by appropriately synthesizing a semi-infinite optical field using a sequence of two mutually phase-locked light pulses impinging on a mirror and delayed in time by fractions of the optical cycle. The profile of the field resulting from such a temporal combination of pulses changes the energy and momentum of an electron as it traverses the interaction volume. The energy-momentum distribution of electron states is recorded as a function of the delay between the two photon pulses via momentum-resolved femtosecond electron energy-loss spectroscopy performed in an ultrafast transmission electron microscope, revealing the light-induced modulation of both amplitude and phase of the electron wave function. In our scheme, the coherent control of the electron wave function is mediated by two temporally delayed pulses at the same position, in contrast with recently reported configurations where the electron modulation was determined by the light interaction at different spatial positions along the electron pathway. This allows us to shift the interaction to the temporal domain instead of the spatial domain and thus to take full advantage of the intrinsic longitudinal coherence of the single-electron wave function, while allowing us to explore other interesting scenarios such as the attosecond-nanometer modulation of plasmonic near-fields. Our experimental results are successfully described within a general theoretical framework for electron-light interaction, which is able to further predict the ability of this method to achieve coherent control over the electron wave function down to the zeptosecond regime using semi-infinite X-ray fields.

Results

Free-electron interaction with a semi-infinite light field

The translational symmetry of a propagating electromagnetic wave is broken by refraction, absorption, or reflection at a material interface. In our study, we use a silver thin film, 43 nanometres, deposited on a silicon nitride membrane, 30 nanometres, acting as a mirror. As schematically depicted in Figure 1a, the mirror is mounted on a double-tilt holder able to rotate around two axes. To demonstrate that electron-photon interaction can be strongly enhanced by the semi-infinite field effect, we display electron energy-loss spectra recorded as a function of laser field amplitude for a fixed orientation of the mirror, and as a function of mirror tilting angle for fixed field amplitude, using p-polarized light in all cases, that is with the incident field parallel to the x axis. Following the interaction, the zero-loss peak at an energy of 200 kiloelectronvolts is redistributed among sidebands at multiples of the incident photon energy, corresponding to energy losses and gains by the electrons. At large values of both the tilt angle and the light field amplitude, the electron distribution is almost completely transferred toward high-energy spectral sidebands, leaving a nearly depleted zero-loss peak and revealing a high probability for multiphoton creation and annihilation.

The modulation of the spectra is determined by the integral of the optical electric field amplitude along the electron-beam direction. Following previous works, the strength of the electron-photon interaction can be quantified in terms of a single parameter, given as Equation 1 in the source: the electron charge divided by the reduced Planck constant times the angular frequency, multiplied by the integral along the beam direction of the field component along that direction, weighted by the complex exponential of the angular frequency times the coordinate divided by the electron velocity. In particular, the fraction of electrons transmitted in a given sideband is approximately given by Equation 2 in the source: the square of the Bessel function of that order, evaluated at twice the magnitude of the interaction parameter.

The spectral distribution of the electron density can thus be changed either by tilting the mirror or by increasing the laser power, producing quantitatively similar effects. Considering the large permittivity of silver at the employed photon energy of about 1.57 electronvolts, and the small optical skin depth of about 11 nanometres for decay in intensity compared with the silver layer thickness, the mirror reflects more than 98 percent of the incident light. Thus, neglecting light penetration inside the material, the electric field along the electron path can be considered to be made of incident and reflected components multiplied by a step function that limits light propagation to the upper part of the mirror. Inserting this field into the first equation, we find Equation 3 in the source: the interaction parameter is the imaginary unit times the electron charge over the reduced Planck constant times the frequency, multiplied by the sum of two terms — the incident field component divided by the frequency over the velocity less the incident wave-vector projection, and the reflected field component divided by the frequency over the velocity plus the reflected wave-vector projection. This makes the interaction strength finite and explicitly dependent on the field amplitude and tilting geometry. We further present in the Methods section a detailed analytical theory extended to deal with arbitrary pulse durations, two light pulses, and real material mirrors, used for comparison with the experimental results in the figures that follow. Nonetheless, this equation provides a satisfactory level of description that allows us to understand the data in simple terms, especially when the mirror is considered to be perfect.

Because light and electron beams in our apparatus are not collinear, the interaction strength for p-polarized light vanishes only when the tilt angles are set to zero degrees and to a critical angle of 12.9 degrees, in agreement with calculations based on the theory reported in the Methods. This corresponds to the condition that the incident and reflected amplitudes almost completely cancel each other, hence producing a negligible net effect. This result is also in agreement with the relation, derived in Supplementary Note 3, giving the critical angle as the arctangent of the sine of the light incidence angle divided by its cosine less the ratio of electron velocity to the speed of light, assuming a perfect mirror. Likewise, the interaction parameter cancels when the polarization is changed from p to s, a result that is clearly observed in polarization-dependent measurements.

To extract quantitative information on the measurements, we perform the corresponding simulations for the energy distribution of a pulsed electron beam after impinging on an illuminated silver and silicon nitride bilayer film, using the same layer thicknesses and geometrical arrangement as in the experiment. In particular, we consider p-polarized light incident with the second tilt fixed to the critical angle. Simulations are carried out incorporating realistic dielectric data for the involved materials. The ratio of electron-to-light pulse durations, about 410 femtoseconds to about 430 femtoseconds, is the same as estimated in experiment, long enough to ensure large temporal overlap between the electron and light pulses, thus enhancing the probability of interaction. The agreement between experiment and theory is rather satisfactory. Similar conclusions are also obtained from measurements and simulations for a short light pulse compared with the electron pulse.

We remark that, in contrast to previous studies of electron-photon interactions, the effect here observed is primarily due to electrons coupling directly to the light waves rather than to the near-field created around a nanostructure. The kinematic mismatch in the electron-light coupling is remedied by the formation of semi-infinite light plane-waves. As noted above, at a photon energy of about 1.57 electronvolts the silver skin depth of about 11 nanometres is much smaller than both the optical wavelength and the metal layer thickness, so the evanescent tail inside the silver film gives a negligible contribution, as confirmed by direct comparison with perfect-mirror simulations.

Quantized energy-momentum exchange in electron-photon coupling

Energy exchanges between light and electrons should also be accompanied by momentum transfers along the direction parallel to the film, where translational invariance guarantees momentum conservation. Measuring such momentum exchanges is quite challenging because of the small induced electron deflection, only a few microradians, which demands high transverse coherence that we achieve by operating the microscope in high-dispersion diffraction mode. In Figure 3a, we show the direct electron beam measured in the diffraction plane when no light is applied, whereas Figures 3b and 3c show the effect of light interaction for tilt angles of zero and 35 degrees, with the second tilt fixed at the critical angle. A clear streaking of the electron beam appears along one momentum direction for the 35-degree tilt as a result of the noted momentum exchange. As already observed in the electron energy spectra, the interaction vanishes at zero tilt and the critical angle for p-polarization, resulting in zero transverse-momentum exchange. The physical origin of this behavior is well described by the analytical expressions above, in which the electric field component along the beam axis modulates the interaction strength. This can also be experimentally probed by rotating the polarization of the light wave, which results in a corresponding modulation of the electron-beam streaking. In our experiment, because the transverse coherence of the electrons is comparable with the light wavelength, a coherent quantized interaction between the electron wave function and the light wave is expected, rather than a purely classical deflection as mediated by the Lorentz force, which is generally negligible under excitation at optical photon frequencies and above.

This can be demonstrated by the simultaneous visualization of inelastic energy and transverse-momentum exchanges, which we directly map using the reciprocal-space imaging ability of the electron spectrometer in our microscope. The streaking of the electron beam occurs along a line in energy-momentum space with slope given by the transverse component of the transferred momentum divided by the photon energy, and in the limit of small incidence and tilt angles that transverse momentum is approximately the frequency over the speed of light times the cosine of the tilt angle times its sine. For every photon absorption or emission event, the electron gains or loses a quantum of energy and a quantum of transverse momentum.

Such an experiment yields a direct observation of the simultaneous quantized exchange of energy and transverse momentum between a propagating light wave and a free-electron, and shows the unique ability of our technique to map transient energy exchanges in momentum space. It could prompt the development of new microscopy methods in which the limitation imposed by energy-loss spectroscopy resolution is lifted for large momentum transfers, such as in the dynamic imaging of low-energy phonons. Furthermore, it demonstrates the ability of external electromagnetic fields to modulate the linear momentum, and potentially the angular momentum, of a free-electron in a dynamic way.

Attosecond coherent control of an electron wave function

These results provide a full characterization of electron-photon interaction at the mirror interface in energy-momentum space, which suggests using such interaction for the coherent manipulation of the electron wave function. We implement this idea by engineering the magnitude of the interaction parameter — which can be thought of as a light-driven Rabi phase for transitions in the electron multilevel quantum ladder with the photon energy as spacing — through a three-pulse experiment in which the electron interacts with a properly shaped field distribution consisting of a sequence of two mutually phase-locked photon pulses, delayed by two time intervals with respect to the electron pulse. We change the relative phase between the two light pulses by varying their difference in steps of 500 attoseconds. The field distribution resulting from such a temporal combination of pulses is then used to coherently manipulate the energy-momentum distributions of the electrons.

A sequence of spectra measured as a function of the delay difference is shown in Figure 4a for a 35-degree tilt at the critical angle, with electron pulses of about 350 femtoseconds, optical pulses of about 60 femtoseconds, a light field amplitude of 21.4 times ten to the seventh volts per metre per pulse, and delays of zero and about 100 to 115 femtoseconds. The large values of the second delay enable fine modulation of the optical phase while considerably reducing the intensity changes associated with light-pulse overlap. We observe periodic oscillations of the spectral sidebands with a period of about 2.6 femtoseconds, equal to the optical cycle. Detailed inspection of the spectra for two different delays, 109 and 110.5 femtoseconds, reveals radically different distributions of the sidebands relative to the zero-loss peak, which are further quantified by plotting the ninth and fourteenth sideband features as a function of the delay difference. We observe significant intensity oscillations with a period of about 2.6 femtoseconds and a well-defined relative phase shift of about pi. We remark once more that these measurements are well reproduced by our analytical simulations for two light pulses.

As described in detail in Supplementary Note 7, where we have included several control experiments, additional calculations, and further considerations on the intrinsic temporal coherence of the single-electron wave function, we demonstrate that this effect cannot be assimilated to a simple intensity variation of the impinging light, which stays at the level of about plus or minus five times ten to the minus second, and neither to an incoherent interaction between the electrons and the two temporally delayed pulses. In fact, in the latter case the modulation of the energy spectrum, and especially of the high-energy sidebands, would be only determined by the 5 percent optical interference and would be quantitatively in a similar range, in contrast with the experimental observations.

The measured oscillatory behavior is indicative of a continuous redistribution within the quantum electron-population ladder, periodically transferred back and forth between high- and low-energy levels. Such an effect is the result of coherent modulation of the electron wave function via the coherent constructive and destructive modulation of the interaction parameter when changing the relative phase between the two driving optical pulses. The time-Fourier transform of the maps gives access to the spectral distribution within the quantum ladder at the modulation frequency of about 385 terahertz. The amplitude and phase of such a modulation provide a complete picture of the optically manipulated electron wave function resolved for each electron energy level.

The coherent control of ultrafast electron beams has recently attracted much attention for its potential application in ultrashort, attosecond, electron sources, as well as electron imaging and spectroscopy. While semi-infinite light beams have been used for the temporal streaking and compression of electron pulses, here we demonstrate the simultaneous quantized exchange of energy and transverse momentum between electrons and light, which is the dominant mechanism at optical frequencies and above, and we provide a direct measurement of the strength of this quantum coherent interaction for controlling the electron energy-momentum distribution. In our experiments, we synthesize a semi-infinite temporally modulated field distribution, obtained by a sequence of two mutually phase-locked light pulses impinging on a mirror, to demonstrate coherent modulation of the electron wave function. A schematic representation of such modulation is shown in Figure 5e, where snapshots of the strong electron density redistribution in both energy and momentum, as observed experimentally and calculated theoretically, are presented for different values of the optical phase shift of the synthesized optical field distribution. This approach allows us to develop additional capabilities of coherent control of free-electrons beyond similar configurations adopted so far, where the electron wave function modulation is determined by the light interaction at different spatial positions along the electron pathway. In our scheme, the adoption of two temporally separated semi-infinite light fields on one flat and homogeneous thin layer allows us to employ a simpler experimental geometry and shift the two interactions temporally instead of spatially, thus taking full advantage of the intrinsic longitudinal coherence of the single-electron wave function.

Attosecond-nanometer control of plasmonic near-fields

Overall, this experiment-theory framework is general and allows the description of other interesting scenarios, such as the phase-controlled combination of the interaction arising from both semi-infinite light fields and plasmon polaritons propagating on a metal film. This is illustrated by measurements presented in Supplementary Figure 11, with surface plasmon polaritons optically generated at the edge of a linear nanocavity carved in the silver layer. The interference between the traveling plasmon wave and the semi-infinite light field creates a standing wave distribution sampled by the electrons, which allows us to produce a snapshot of the propagating plasmon in real space. By using the two-pulse scheme described above, coherent control of these plasmonic near-fields can be achieved at attosecond-nanometer scale. This is demonstrated by using a nano-fabricated plasmonic Fabry-Perot resonator, while simultaneously adopting an experimental geometry that cancels the interaction with the semi-infinite field, allowing the resonant plasmon modes of the resonator to be solely imaged. Varying the delay between the two optical pulses in steps of 334 attoseconds allows us to control the relative phase between the optically excited plasmons, resulting in a time-dependent sequence of constructive and destructive interference between them. The plasmonic coherent control experiments presented here, which would have been unfeasible in other schemes involving spatially separated interactions, offer the unique opportunity to perform time-domain spectro-microscopy of plasmon resonances, where the energy resolution is obtained via Fourier transform of the temporal traces and is not limited by the overall electron energy-loss resolution, sub-electronvolt at best. In a complementary frequency-domain approach, the spectral response of the resonance was obtained by using a single optical pulse with a tunable wavelength and a resolution determined by the 20 millielectronvolt laser linewidth.

Zeptosecond coherent control of an electron wave function

As described above, when the electron scattering cannot be assisted by the plasmonic near-fields, a refracting, absorbing, or reflecting interface can be used for mediating the electron-light interaction. A particularly appealing consequence of such a condition consists in the possibility of controlling the electron wave function using photons of different energies, not restricted by the ability of materials to support localized plasmon resonances, but solely determined by the quality of the mirror surface at a specific frequency. Using high-energy photons all the way to the X-ray regime, our methodology would then allow us to control the electron wave function down to the zeptosecond timescale.

To verify the feasibility of this idea, we have designed a multilayer mirror composed of 30 layers of 1.6-nanometre-thick cobalt spaced by 1-nanometre-thick gold, for a total thickness of 78 nanometres, still transparent for 200-kiloelectronvolt electrons and capable of reflecting around 35 percent of 777-electronvolt light at an angle of incidence of 45 degrees. This type of mirror is routinely used in X-ray facilities, and combined with commonly employed sample preparation techniques for transmission electron microscopy, such as ion-milling and focused-ion-beam machining, it can be fabricated in the form of electron-transparent lamellas. We then simulated a three-pulse experiment with two 100-femtosecond, 777-electronvolt, 50-terawatt-per-square-centimetre X-ray pulses, similar to what is currently available from free-electron lasers, impinging on the multilayer along the same direction as a 300-femtosecond electron pulse. We carry out simulations within a 30-attosecond window starting from an initial delay of 150 femtoseconds. Electron sidebands are clearly discernible at energies of plus and minus 777 electronvolts relative to the zero-loss peak, originating in the same electron-ladder interaction as observed for near-infrared light. From our calculation, we infer that the first sideband has a relative intensity with respect to the zero-loss peak of about ten to the minus fifth. For a repetition rate of 300 kilohertz, such as used in the LCLS-II free-electron laser at SLAC, this translates to about 3 electrons per second in a single channel of the detector, whose measurement can be done using commercially available highly sensitive direct detector cameras.

The resulting spectrum as a function of the delay between the two X-ray pulses reveals a clear modulation by the optical cycle of the X-ray pulse, about 5.3 attoseconds, and an intensity change rate of about 1 percent per 511 zeptoseconds. It is worth noting that in our scheme phase fluctuations of the X-ray beam, whose main effect is the generation of a temporal jitter of a few femtoseconds between consecutive pulses, do not represent an issue. This is because our experiment uses two photon pulses originating from the same photon pulse by means of an interferometer, and thus the two X-ray pulses will be intrinsically phase-locked with an inherently zero jitter between them. Coherent manipulation of the electron wave function can be thus pushed to the zeptosecond regime using currently existing technology within our electron-light interaction scheme. Access to such timescales may open interesting perspectives for the observation of intramolecular electronic motions and nuclear processes such as fission, quasifission, and fusion.

External control of nuclear excitations

Very recently, nuclear excitation by electron capture has been experimentally demonstrated. In such a process, an electron is captured by an ionized atom while simultaneously inducing the excitation of the nucleus. In that experimental design, ions were produced by stripping electrons away from an atomic beam going through a thin foil. The electronic levels of the ionized atoms were redistributed with respect to the equilibrium atoms, so that electrons interacting with them randomly sampled one of these configurations.

On a different approach, multiple atomic ionization can also be produced by interaction with ultrashort intense laser pulses. In one report, a surprising resonant effect is reported when tuning the energy of the ionizing X-ray laser pulse. During this process, atomic levels transit several intermediate configurations whose lifetimes lie in the range between a few zeptoseconds and a few attoseconds.

In this scenario, we propose that by synchronizing the carrier of a properly tuned X-ray pulse with ultrashort electron pulses at the attosecond or zeptosecond level, the nuclear excitation can be controlled coherently with an ad hoc removal and insertion of electrons from and into the atom. A train of zeptosecond electron pulses is synchronized to the optical cycle of an X-ray pulse. By varying their relative delay time, different out-of-equilibrium configurations of the ionized atoms may be sampled in a push-pull-like approach. Given the degrees of freedom that such an experiment can provide in choosing both the electron and X-ray energies, their relative timing, light polarization and intensity, we expect that interesting resonance effects can be discovered in the excitation of the nuclei.

Controlling nuclear phenomena via external parameters is an extremely interesting perspective. Ideally, one would like to induce instabilities in an otherwise stable or metastable nucleus to prompt energy-producing decays, or to generate radiation. However, accessing nuclei is difficult and energetically costly because of the protective shell of electrons surrounding it. Thus, external parameters such as pressure, magnetic field or chemical environment have little or no effect on decay rates and nuclear properties in general. Our scheme would offer a further perspective for the control of nuclear reactions with potential implications in various fields, from fundamental physics to energy-related applications.

Methods

Materials and experiment

We used an ultrafast transmission electron microscope to focus femtosecond electron and light pulsed beams on an optically thick mirror. The mirror was thin enough to transmit the electrons while producing large light reflection. Specifically, it was made of a 43-nanometre-thick silver thin film, plus or minus 5 nanometres, sputtered on a 30-nanometre silicon nitride membrane placed on a silicon support with an 80 by 80 micrometre window, which was in turn mounted on a double-tilt sample holder that ensured rotation around both axes over a range of plus or minus 35 degrees.

Electron pulses were generated by photoemission from an ultraviolet-irradiated lanthanum hexaboride cathode, accelerated to an energy of 200 kiloelectronvolts along the beam axis, and focused on the specimen surface. The mirror was simultaneously illuminated with femtosecond laser pulses of 1.57 electronvolts central energy and variable duration, intensity, and polarization. The light pulses were focused on the sample surface with a spot size of about 58 micrometres full width at half maximum. The light propagation direction lay within the plane containing the beam axis and formed an angle of about 4 to 5 degrees with it. The delay between electrons and photons was varied via a computer-controlled delay line. For the three-pulse experiment, we implemented a Michelson interferometer along the optical path of the infrared beam, incorporating a computer-controlled variable delay stage on one arm.

The transmission electron microscope was equipped with electron energy-loss spectroscopy capabilities, coupled to real-space and reciprocal-space imaging. Energy-resolved spectra were acquired using a Gatan imaging filter camera operated with a 0.05 electronvolt-per-channel dispersion setting and typical exposure times of the sensor from 30 to 60 seconds. Multiple photon absorption and emission events experienced by the electrons were analyzed as a function of relative beam-mirror orientations by recording energy-loss spectra and diffraction patterns in high-dispersion-diffraction mode. During post-acquisition analysis, the spectra were aligned based on their zero-loss peak positions using a differential-based maximum intensity alignment algorithm.

Special care was taken in modulating and evaluating the temporal width of the light and electron pulses. We varied the duration of the optical pulses by modifying the temporal chirp of the laser amplifier output using a pair of tunable glass prisms. An infrared auto-correlator was used for measuring the duration of the infrared pulses. For electrons, the pulse duration was estimated by measuring the electron-photon cross-correlation as obtained by monitoring the spectra as a function of the delay time between electrons and the infrared light. In the low-excitation regime, the measured temporal width of a given sideband is roughly the square root of the squared electron pulse duration plus the squared light pulse duration divided by the sideband order — that is, the convolution of electron and optical pulses. For infrared pulses of 60, 175 and 430 femtoseconds full width at half maximum, we derived electron pulse durations of 350, 395 and 410 femtoseconds full width at half maximum respectively, with an estimated error below 5 percent.

Theory of ultrafast electron-light interaction

Following previous works, we describe an electron wavepacket exposed to an optical field through the Schrödinger equation with a free-space Hamiltonian plus a minimal-coupling interaction term built from the optical vector potential, in a gauge in which the scalar potential and the divergence of the vector potential both vanish. Expanding the electron wave function in momentum components piled near a central value, each component is an eigenstate of the free Hamiltonian, and separating out the fast evolution imposed by the central-momentum component reduces the Schrödinger equation to a first-order equation for the slowly varying part, Equation 4 in the source, whose rigorous solution is an exponential of the time integral of the vector potential along the electron trajectory.

For illumination by an optical pulse with a narrow spectral distribution, the vector potential is written in terms of a slowly varying envelope, and applying the Jacobi-Anger expansion turns the solution into a sum over sidebands weighted by Bessel functions of twice the magnitude of the interaction parameter. For monochromatic light this gives Equation 5 in the source, the definition of the interaction parameter as an integral of the field component along the beam axis weighted by the electron phase factor, and Equation 6 in the source, the wave function as a Bessel-weighted sum in which each component carries a change in energy and momentum given by the sideband order times the photon energy and by that same order times the photon energy divided by the electron velocity. For a Gaussian light pulse the same result holds with the interaction parameter multiplied by a Gaussian envelope.

The electron probability at the detector is then the spatial integral of the squared wave function at large times. Assuming a Gaussian electron pulse normalized to one electron, with a delay relative to the light pulse, the probability that the electron has exchanged a given net number of photons is Equation 7 in the source: a Gaussian-weighted time integral of the squared Bessel function of twice the magnitude of the interaction parameter, itself carrying the light-pulse envelope. Summing that probability over all sideband orders reassuringly gives one. The remaining derivations, together with the two-pulse and real-mirror extensions, the dielectric data used, and the Supplementary Notes and Figures, are at the source.

Figure captions

Figure 1. Experiment probing free-electron interaction with semi-infinite light fields. a, Ultrashort 200-kiloelectronvolt electron pulses travel along the beam axis and impinge on the surface of a silver and silicon nitride thin bilayer, which is mounted on a double-tilt holder able to rotate around two axes. Light propagates within the plane containing the beam axis, incident at an angle of about 4 to 5 degrees relative to it and then reflected from the silver surface. The resulting electron-photon interaction is probed by monitoring electron energy-loss spectra as a function of geometrical parameters and light properties. b, Description of the electron-light interaction here explored. The breaking of translational invariance produced by light reflection enables photon absorption or emission by the electron corresponding to a quantized energy and momentum exchange. c, Description of the three-pulse experiment used for coherent modulation of the electron wave function. Electrons interact with an appropriately synthesized optical field distribution produced by two mutually phase-locked photon pulses whose relative phase is changed by varying their relative delay.

Figure 2. Energy exchange during electron-light interaction. a, Sequence of measured energy-loss spectra plotted as a function of increasing tilt angle. We use p-polarized light with the incident field along the x axis, the second tilt at its critical value, a peak field amplitude of 12.8 times ten to the seventh volts per metre, and light and electron pulse durations of 430 and 410 femtoseconds. Sidebands at multiples of the photon energy relative to the zero-loss peak are visible, the multiple being the net number of exchanged photons. b, Sequence of spectra measured for increasing light field amplitude with the tilt angle fixed at 35 degrees. c, Spectra selected from a, measured at 9 and 30 degrees, showing a strong redistribution of the electron density toward the high-energy sidebands for large tilt angle. d to f, Simulated spectra corresponding to the experimental conditions of a to c.

Figure 3. Momentum exchange during electron-light interaction. a, Direct electron beam measured in the diffraction plane as a function of transverse momentum when no light is applied. b, c, The same under illumination with 560-femtosecond laser pulses of 11.1 times ten to the seventh volts per metre peak field amplitude at the critical second tilt. The tilt angle is 0 degrees in b and 35 degrees in c. A clear streaking of the electron beam appears along one momentum direction in c as a result of momentum exchanges between light and electrons. d, Electron-beam profile along that direction as a function of light polarization. e, Direct electron beam measured in the momentum-energy plane in the absence of optical illumination. f, g, Measured and simulated momentum-energy maps for illumination under the conditions of c.

Figure 4. Attosecond coherent control of free-electrons. The electron beam interacts with a semi-infinite temporally modulated optical field distribution produced by a sequence of two mutually phase-locked light pulses impinging on the mirror. a, Measured energy-loss spectra as a function of relative delay between the two optical pulses. The tilt angles are 35 degrees and the critical value, the optical pulses are 60 femtoseconds long with a peak field amplitude of 21.4 times ten to the seventh volts per metre each, and the delays are zero and about 100 to 115 femtoseconds, with an initial value of 100 femtoseconds. b, Spectra taken at two different time delays. c, Relative intensity of the ninth and fourteenth sidebands plotted as a function of time delay between the two optical pulses, exhibiting a periodic modulation of period about 2.6 femtoseconds, equal to the optical cycle, and a relative phase shift of pi. Solid curves are least-squares fits to the data. d to f, Simulated spectra and resulting intensity change corresponding to the experimental conditions of a to c.

Figure 5. Amplitude and phase modulation of the electron wave function. a, Two-dimensional Fourier transform of the energy-time map plotted in Figure 4a. b, Complex spectral distribution of the electron wave function amplitude and phase, extracted at the modulation frequency of about 385 terahertz. c, d, Two-dimensional Fourier transform extracted from the calculated energy-time map plotted in Figure 4d. e, Schematic representation of electron wave function modulation, showing snapshots of the strong energy-momentum electron density redistribution for different values of the phase shift between the two optical pulses.

Figure 6. Attosecond-nanometer modulation of plasmonic near-fields. a, Energy-filtered image of the plasmonic interference pattern created in the designed plasmonic Fabry-Perot resonator. The tilt angles are set to 0 and 12.9 degrees, while the optical pulses are 60 femtoseconds long. b, Spatiotemporal mapping of the plasmonic coherent control at attosecond-nanometer scale. c, Measured temporal modulation of the plasmon-mode amplitude. d, Fourier transform of the measured temporal trace.

Figure 7. Zeptosecond coherent control of free-electrons. a, An electron beam interacts with a semi-infinite temporally modulated X-ray field of 777 electronvolts photon energy produced by a sequence of two mutually phase-locked pulses partially reflected by a gold and cobalt multilayer. b, Calculated spectra as a function of relative delay between the two X-ray pulses. The tilt angles are set to 45 and 0 degrees, while the X-ray pulses are 100 femtoseconds long with a peak field amplitude of 9.4 times ten to the ninth volts per metre each. Simulations are performed within a 30-attosecond window starting from an initial delay of 150 femtoseconds. c, Calculated spectrum showing the electron sidebands at energies of plus and minus 777 electronvolts with respect to the zero-loss peak. d, Relative intensity change of the first sideband plotted as a function of time delay between the two X-ray pulses, exhibiting a periodic modulation of period about 5.3 attoseconds, equal to the X-ray cycle, and an intensity change rate of about 1 percent per 511 zeptoseconds. e, Schematic description of a thought experiment for external control of nuclear excitations. A train of zeptosecond electron pulses is overlapped with the optical cycle of an X-ray pulse. Their relative delay is externally varied while monitoring the change in the gamma-ray emission as a result of multiple ionization and electron capture events. The atomic levels for the equilibrium and ionized atom are pictorially represented.

Affiliations as printed: Institute of Physics, Laboratory for Ultrafast Microscopy and Electron Scattering, École Polytechnique Fédérale de Lausanne, Switzerland; Department of Electrical Engineering, Technion — Israel Institute of Technology, Haifa, Israel; SUPA, School of Physics and Astronomy, University of Glasgow, United Kingdom; Ripon College, Ripon, Wisconsin, United States; ICFO — Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels, Barcelona, Spain; and ICREA — Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain.

(Reference-number markers, running heads and page furniture have been dropped, display equations rendered in words with their source equation numbers, and the figures themselves not reproduced; the forty-five references, the Supplementary Information and the equations in their original form are at the source. Sections of the Methods are condensed for length; the complete text is at the source.)