From attosecond to zeptosecond 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 Garcia de Abajo, F. Carbone
FFrom attosecond to zeptosecond coherent control of free-electronwave functions using semi-infinite light fields
G. M. Vanacore, ∗ I. Madan, ∗ G. Berruto, K. Wang,
1, 2
E. Pomarico, R. J. Lamb, D. McGrouther, I. Kaminer,
1, 2
B. Barwick, F. Javier Garc´ıa de Abajo,
5, 6, † and F. Carbone ‡ Institute of Physics, Laboratory for UltrafastMicroscopy and Electron Scattering (LUMES),´Ecole Polytechnique F´ed´eral de Lausanne,Station 6, CH-1015 Lausanne, Switzerland Technion Department of Physics SUPA, School of Physics and Astronomy,University of Glasgow, Glasgow G12 8QQ, UK Ripon College, 300 W. Seward St., Ripon, WI 54971, United States ICFO-Institut de Ciencies Fotoniques,The Barcelona Institute of Science and Technology,08860 Castelldefels (Barcelona), Spain ICREA-Instituci´o Catalana de Recerca i Estudis Avan¸cats,Passeig Llu´ıs Companys 23, 08010 Barcelona, Spain (Dated: December 25, 2017) a r X i v : . [ phy s i c s . op ti c s ] D ec bstract Light-electron interaction in empty space is the seminal ingredient for free-electron lasers andalso for controlling electron beams to dynamically investigate materials and molecules. Pushing thecoherent control of free electrons by light to unexplored timescales, below the attosecond, wouldenable unprecedented applications in light-assisted electron quantum circuits and diagnostics atextremely small timescales, such as those governing intramolecular electronic motion and nuclearphenomena. We experimentally demonstrate attosecond coherent manipulation of the electronwave function in a transmission electron microscope, and show that it can be pushed down to thezeptosecond regime with existing technology. We make a relativistic pulsed electron beam interactin free space with an appropriately synthesized semi-infinite light field generated by two femtosec-ond laser pulses reflected at the surface of a mirror and delayed by fractions of the optical cycle.The amplitude and phase of the resulting coherent oscillations of the electron states in energy-momentum space are mapped via momentum-resolved ultrafast electron energy-loss spectroscopy.The experimental results are in full agreement with our theoretical framework for light-electroninteraction, which predicts access to the zeptosecond timescale by combining semi-infinite X-rayfields with free electrons. . INTRODUCTION The scattering of single photons by free electrons is extremely weak, as quantified bythe Thomson scattering cross-section, which for visible frequencies is of the order of 10 − m . Additionally, direct photon absorption or emission by a free-space electron is forbid-den due to energy-momentum mismatch. To circumvent these limitations and increase theprobability of electron-photon interaction, a variety of methods have been devised . Forexample, the Kapitza-Dirac effect involves a conceptually simple configuration in which anelectron intersects a light grating produced by two counter-propagating light beams of thesame frequency . The interaction is then elastic and requires the electron to undergo anequal number of virtual photon absorption/stimulated-emission processes. When the ab-sorbed and emitted photons differ in energy, the interaction results in frequency up- ordown-conversion , which is the basis of undulator radiation and free-electron lasers .A direct single-photon emission/absorption process can also bridge the energy-momentummismatch if either the electrons are not free (e.g., in photoemission from atoms/molecules and solid surfaces ) or when a scattering structure generates evanescent light fields inthe vicinity of the interaction volume. Such an electron-photon-matter interaction createsoptical field components with a frequency-momentum decomposition that lies outside thelight cone, allowing emission/absorption to take place. This type of interaction, which isforbidden in free space , is regularly exploited for generating radiation and for acceler-ating charged particles. Recently, it has also prompted the development of photon-inducednear-field electron microscopy (PINEM) . In PINEM, an energetic electron beam in-teracts with the evanescent near fields surrounding an illuminated material structure. Theinteraction is particularly strong when the structure supports surface-plasmon polaritons(SPP) that are excited by short light pulses . Optical near-fields then produce coherentsplitting of the electron wave function in energy space, giving rise to Rabi oscillations amongelectron quantum states separated by multiples of the photon energy . The microscopicdetails of the process are encoded in the electron wave function, which can be revealed viaultrafast electron energy-loss spectroscopy (EELS) and controlled using suitable illuminationschemes .In this work, we report on a different and more general method for controlling andmanipulating the strength of electron-photon interaction. Instead of relying on localized3ear fields (e.g., plasmons), which inevitably depend on the intrinsic cross-section associatedwith the optical excitation of confined optical modes, we make use of a spatially abruptinterruption of the light field in free-space, also referred to as semi-infinite field (see Fig.1a). Such a boundary condition can be attained by sending the electrons through a lightbeam that intersects a refractor, an absorber, or more efficiently, a reflecting mirror alongthe optical path. When the light wave extends only over half-space, the energy-momentumconservation constraint is relaxed and electron-photon interaction can take place (see Fig.1b and Fig. S1 in Supporting Information (SI)) with an efficiency exceeding that producedby a resonant plasmonic nanostructure.Within this scenario, we demonstrate attosecond coherent control of the electron wavefunction by appropriately synthesizing a semi-infinite optical field using a sequence of twomutually-phase-locked light pulses impinging on a mirror and delayed in time by fractions ofthe optical cycle (see schematics in Fig. 1c). The profile of the field resulting from such a tem-poral combination of pulses changes the energy and momentum of an electron as it traversesthe interaction volume. The energy-momentum distribution of electron states is recordedas a function of the delay between the two photon pulses via momentum-resolved fs-EELSperformed in an ultrafast transmission electron microscope , revealing the light-inducedmodulation of both amplitude and phase of the electron wave function. Our experimentalresults are successfully described within a general theoretical framework for electron-lightinteraction, which is able to further predict the ability of this method to achieve coherentcontrol over the electron wave function down to the zeptosecond regime using semi-infinitex-ray fields. II. RESULTS AND DISCUSSION
The translational symmetry of a propagating electromagnetic wave is broken by refrac-tion, absorption, or reflection at a material interface. In our study, we use a Ag thin film (43nm) deposited on a Si N membrane (30 nm) acting as a mirror. As schematically depictedin Fig. 1a, the mirror is mounted on a double-tilt holder able to rotate around the x (angle α ) and y (tilting angle, ϑ ) axes. To demonstrate that electron-photon interaction can bestrongly enhanced by the semi-infinite field effect, we display EELS spectra recorded as afunction of laser field amplitude for a fixed orientation of the mirror (Fig. 2b), and as a func-4ion of mirror tilting angle ϑ for fixed field amplitude (Fig. 2a), using p-polarized light in allcases (incident field parallel to x axis). Following the interaction, the zero-loss peak (ZLP)at an energy E = 200 keV is redistributed among sidebands at multiples of the incidentphoton energy ± (cid:96) ¯ hω , corresponding to energy losses and gains by the electrons. At largevalues of both ϑ and the light field amplitude, the electron distribution is almost completelytransferred toward high-energy spectral sidebands ( | (cid:96) | (cid:29) E z ( z ) along the electron beam direction z . Following previous works ,the strength of the electron-photon interaction can be quantified in terms of the parameter(see Methods) β = ( eγ/ ¯ hω ) ˆ dz E z ( z ) e − i ωz/v . (1)In particular, the fraction of electrons transmitted in the (cid:96) th sideband is approximately givenby the squared Bessel function P (cid:96) = J (cid:96) (2 | β | ) . (2)The spectral distribution of the electron density can be thus changed either by tilting themirror (Fig. 2b) or by increasing the laser power (Fig. 2c), producing quantitatively similareffects. Considering the large permittivity ( ≈ −
30 + 0 . hω ≈ ≈
11 nm for 1 /e decay in intensity)compared with the silver layer thickness, the mirror reflects >
98% of the incident light.Thus, neglecting light penetration inside the material, the electric field along the electronpath can be considered to be made of incident (i) and reflected (r) components as E z ( z ) = (cid:16) E i z e i k i z z + E r z e − i k r z z (cid:17) θ ( − z ), where the step function θ ( − z ) limits light propagation to theupper part of the mirror and k i / r z is the projection of the incident/reflected light wave vectoralong z . Inserting this field into Eq. (1), we find β ≈ (i eγ/ ¯ hω ) (cid:20) E i z ω/v − k i z + E r z ω/v + k r z (cid:21) , (3)which makes the interaction strength finite and explicitly dependent on the field amplitudeand tilting geometry. We further present in the Methods section a detailed analytical theoryextended to deal with arbitrary pulse durations, two light pulses, and real material mirrors,5sed for comparison with the experimental results in the figures that follow. Nonetheless,Eq. (3) provides a satisfactory level of description that allows us to understand the data insimple terms, specially when the mirror is considered to be perfect (see Figs. S2 and S3 inSI).Because light and electron beams in our apparatus are not collinear, the interactionstrength described by β for p-polarized light vanishes only when the tilt angles are set to ϑ = 0 ◦ and α = α C = 12 . ◦ , in agreement with calculations based on the theory reportedon Methods. This corresponds to the condition that the incident and reflected amplitudesalmost completely cancel each other in Eq. (3), hence producing a negligible net effect(minimum | β | , see red curve in Fig. S4, SI). This result is also in agreement with the relation α C = tan − [sin δ/ (cos δ − v/c )] derived in the SI from Eq. (3) to yield β = 0 assuming aperfect mirror (blue curve in Fig. S4, SI). Likewise, β cancels when the polarization ischanged from p to s, a result that is clearly observed in polarization-dependent measurements(see Fig. S5 in SI).To extract quantitative information on the measurements presented in Fig. 2a-c, we per-form the corresponding simulations shown in Fig. 2d-f for the energy distribution of a pulsedelectron beam after impinging on an illuminated Ag/Si N bilayer film, using the same layerthicknesses and geometrical arrangement as in the experiment. In particular, we considerp-polarized light incident with α fixed to the critical angle α C . Simulations are carried outincorporating realistic dielectric data for the involved materials (see Methods). The ratioof electron-to-light pulse durations τ e /τ L ≈
410 fs /
430 fs ≈ .
95 is the same as estimated inexperiment (see Methods), long enough to ensure large temporal overlap between the elec-tron and light pulses, thus enhancing the probability of interaction. The agreement betweenexperiment and theory is rather satisfactory. Similar conclusions are also obtained frommeasurements and simulations for small τ L compared with τ e (see Figs. S3 and S6 in SI).We remark that, in contrast to previous studies of electron-photon interactions , theeffect here observed is primarily due to electrons coupling directly to the light waves ratherthan to the near-field created around a nanostructure. The kinematic mismatch in theelectron-light coupling is remedied by the formation of semi-infinite light plane-waves (seeFig. S1 in SI). As noted above, at a photon energy of ≈ .
57 eV the silver skin depth( ≈
11 nm) is much smaller than both the optical wavelength and the metal layer thickness,so the evanescent tail inside the Ag film gives a negligible contribution, as confirmed by6irect comparison with perfect-mirror simulations based on Eq. (3) (see Fig. S3 in SI).Overall, this experiment-theory framework is general and allows describing other inter-esting scenarios, such as the phase-controlled combination of two interactions arising fromsemi-infinite light fields and plasmon polaritons propagating on a metal film. This is illus-trated by measurements presented in Fig. S9 (SI), with SPPs generated at the edge of ananocavity carved in the Ag layer. The interference between the traveling plasmon wave andthe semi-infinite light field creates a standing wave distribution sampled by the electrons,which allows us to produce a snapshot of the SPP in real-space.Energy exchanges between light and electrons should be also accompanied by momentumtransfers along the direction parallel to the film, where translational invariance guaranteesmomentum conservation. Measuring such momentum exchanges is quite challenging becauseof the small induced electron deflection (only a few µ rad), which demands high transversecoherence that we achieve by operating the microscope in high dispersion diffraction mode.In Fig. 3a we show the direct electron beam measured in the k x - k y diffraction plane when nolight is applied, whereas Fig. 3b,c shows the effect of light interaction for tilt angles ϑ = 0 ◦ and ϑ = 35 ◦ , with fixed α = α C . A clear streaking of the electron beam appears along the k x direction for ϑ = 35 ◦ as a result of the noted momentum exchange. As already observed inthe electron energy spectra, the interaction vanishes at ϑ = 0 and α = α C for p-polarization,resulting in zero momentum exchange. The physical origin of this behavior is well describedby the analytical expressions in Eqs. (2) and (3), in which the electric field component alongthe z axis modulates the interaction strength. This can be also experimentally probed byrotating the polarization of the light wave, which results in a corresponding modulationof the electron beam streaking (see Fig. 3d). Particularly instructive is the simultaneousvisualization of inelastic energy and momentum exchanges, which we directly map using thereciprocal-space imaging ability of the electron spectrometer in our microscope (see Fig.3e-g). The streaking of the electron beam occurs along a line ¯ hω = ¯ hcq T ,x , where q T ,x isthe transverse component of the transferred momentum along x , which in the limit of smallangles δ and α admits the expression q T ,x ≈ ( ω/c ) cos ϑ sin ϑ (see SI for the full derivation).For every photon absorption/emission event the electron gains/loses a quantum of energy ¯ hω and momentum ¯ hq T ,x along x . The unique ability of our technique to map transient energyexchanges in momentum space could prompt the development of new microscopy methodsin which the limitation imposed by EELS energy resolution is lifted for large momentum7ransfers, such as in the dynamic imaging of low-energy phonons.These results provide a full characterization of electron-photon interaction at the mirrorinterface in energy-momentum space, which suggests using such interaction for the coherentmanipulation of the electron wave function. We implement this idea by engineering theparameter | β | (which can be thought of as a light-driven Rabi phase for transitions in theelectron multilevel quantum ladder with ¯ hω energy spacings ) through a three-pulse exper-iment in which the electron interacts with a properly shaped field distribution consisting ofa sequence of two mutually-phase-locked photon pulses, delayed by time intervals ∆ and∆ with respect to the electron pulse (see schematics in Fig. 1c and additional details inMethods). We change the relative phase between the two light pulses by varying ∆ − ∆ insteps of 500 attoseconds. The field distribution resulting from such a temporal combinationof pulses is then used to coherently manipulate the energy-momentum distributions of theelectrons.A sequence of EELS spectra measured as a function of ∆ − ∆ is shown in Fig. 4afor ϑ = 35 ◦ , α = α C , τ e ≈
350 fs electron pulses, τ L ≈
60 fs optical pulses, a light fieldamplitude of 21 . × V/m per pulse, and delays ∆ = 0 fs and ∆ ≈ −
115 fs. Thelarge values of ∆ enable fine modulation of the optical phase while considerably reducingthe intensity changes associated with light-pulse overlap. We observe periodic oscillationsof the spectral sidebands with a period ≈ . π/ω . This effectcannot be assimilated to a simple intensity variation of the impinging light, which staysat the ∼ × − level; in fact, the employed intensities lie in the saturation regime, asshown by the negligible electron-spectra changes observed for single light pulses at high fieldamplitude (see Fig. S7 in SI). Detailed inspection of the EELS spectra for two differentdelays (∆ = 109 fs and 110.5 fs in Fig. 4b, corresponding to the horizontal dashed lines inFig. 4a) reveals radically different distributions of the sidebands relative to the ZLP, whichare further quantified in Fig. 4c by plotting the (cid:96) = 9 and (cid:96) = 14 features as a functionof ∆ − ∆ . We observe significant intensity oscillations with a period of ≈ . ∼ π relative phase shift. We remark once more that measurements shown inFigs. 4a and 4b,c are well reproduced by our analytical simulations for two light pulses (seeMethods) plotted in Figs. 4d and 4e,f, respectively.This behavior is indicative of a continuous redistribution within the quantum electron-population ladder, periodically transferred back and forth between high- and low-energy8evels. Such an effect is the result of coherent modulation of the electron wave function via the coherent constructive and destructive modulation of | β | when changing the relativephase between the two driving optical pulses. The time-Fourier transform of the maps inFigs. 4a and 4d, presented in Figs. 5a and 5c, gives access to the spectral distribution withinthe quantum ladder at the modulation frequency 2 π/ (2 . ≈
385 THz. The amplitude andphase of such a modulation, shown in Figs. 5b and 5d, provide a complete picture of theoptically-manipulated electron wave function resolved for each electron energy level.The coherent control of ultrafast electron beams has recently attracted much attentionfor its potential application in novel ultrashort (attosecond) electron sources, as well aselectron imaging and spectroscopy. While semi-infinite light beams have been used forthe temporal streaking and compression of electron pulses , their potential to controlpurely quantum aspects of the electron energy-momentum distribution has not been fullyexplored. This approach allows us to develop new capabilities of coherent control of freeelectrons beyond the utilization of surface and localized plasmons employed so far to assistthe electron scattering . In our experiments, we synthesize a semi-infinite temporallymodulated field distribution (obtained by a sequence of two mutually-phase-locked lightpulses impinging on a mirror) to demonstrate coherent modulation of the electron wavefunction. A schematic representation of such modulation is shown in Fig. 5e, where snapshotsof the strong electron density redistribution in both energy and momentum, as observedexperimentally and calculated theoretically, are presented for different values of the opticalphase shift of the synthesized optical field distribution.This method, which does not involve evanescent near fields, is very general and onlyrequires a refracting, absorbing, or reflecting interface. The experimental requirements arethus simplified and do not necessitate nanofabricated structures for the excitation of plasmaresonances, while potentially enabling electro-optical tunability through graphene gating.A particularly appealing possibility consists in controlling electron-light interactions usingphotons of different energies, not restricted by the ability of materials to support localizedresonances such as plasmon polaritons, but solely determined by the quality of the mirrorsurface 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 thezeptosecond timescale . To verify the feasibility of this idea, we have designed a multilayermirror composed of 30 layers of 1.6-nm-thick cobalt spaced by 1-nm-thick gold (total thick-9ess is 78 nm), still transparent for 200 KeV electrons and capable of reflecting around 35%of 777 eV light at an angle of incidence of 45 ◦ (see Fig. S10 in the SI). This type of mirroris routinely used in x-ray facilities . We then simulated a three-pulse experiment with two100 fs, 777 eV, 50 TW/cm x-ray pulses, similar to what is currently available from free-electron lasers , impinging on the multilayer along the same direction as a 300 fs electronpulse (see schematics in Fig. 6a). We carry out simulations within a 30-attosecond windowstarting from an initial delay ∆ − ∆ = 150 fs. Electron sidebands are clearly discernable atenergies of ±
777 eV relative to the ZLP (see Fig. 6c), originating in the same electron-ladderinteraction as observed for near-infrared light. The resulting EELS spectrum as a functionof the delay between the two x-ray pulses is displayed in Fig. 6b, while the relative intensitychange for the first sideband is shown in Fig. 6d, revealing a clear modulation by the opticalcycle of the x-ray pulse ( ≈ . ≈ zs . Coherentmanipulation of the electron wave function can be thus pushed to the zeptosecond regimeusing currently existing technology within our electron-light interaction scheme. Access tosuch timescales may open new perspectives for the observation of intramolecular electronicmotions and nuclear processes such as fission, quasifission, and fusion .10 ETHODS
Materials and Experiment.
A sketch of our experiment is depicted in Fig. 1a. Weused an ultrafast transmission electron microscope (a detailed description can be foundin Ref. 26) to focus femtosecond electrons and light pulsed beams on an optically-thickmirror. The mirror was thin enough to transmit the electrons while producing large lightreflection. Specifically, it was made of a 43 nm-thick ( ± N membrane placed on a Si support with a 80 × µ m window, which was inturn mounted on a double-tilt sample holder that ensured rotation around the x (angle α )and y (angle ϑ ) axes over a ± ◦ range. Electron pulses were generated by photoemissionfrom a UV-irradiated LaB cathode, accelerated to an energy E = 200 keV along the z axis, and focused on the specimen surface. The mirror was simultaneously illuminated withfemtosecond laser pulses of ¯ hω = 1 .
57 eV central energy and variable duration, intensity,and polarization. The light pulses were focused on the sample surface (spot size of ∼ µ mFWHM). The light propagation direction lied within the y - z plane and formed an angle δ ∼ − ◦ with the z axis, as shown in Fig. 1a. The delay between electrons and photons wasvaried via a computer-controlled delay line. For the three-pulse experiment, we implementeda Michelson interferometer along the optical path of the infrared beam, incorporating acomputer-controlled variable delay stage on one arm.The transmission electron microscope was equipped with EELS capabilities, coupledto real-space and reciprocal-space imaging. Energy-resolved spectra were acquired usinga Gatan imaging filter (GIF) camera operated with a 0.05 eV-per-channel dispersion set-ting and typical exposure times of the CCD sensor from 30 s to 60 s. Multiple photonabsorption and emission events experienced by the electrons were analyzed as a functionof relative beam-mirror orientations by recording EELS spectra and diffraction patterns inhigh-dispersion-diffraction mode. During post-acquisition analysis, the EELS spectra werealigned based on their ZLP positions using a differential-based maximum intensity alignmentalgorithm.Special care was taken in evaluating the temporal width of the light and electron pulses.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 EELS spectra as a function of the delay time11etween electrons and the infrared light. In the low-excitation regime, the measured tempo-ral width of the (cid:96) th sideband is roughly τ (cid:96) ≈ (cid:112) ( τ + ( τ ) /(cid:96) ) (i.e., the convolution of electronand optical pulses of durations τ e and τ L , respectively). For infrared pulses with τ L = 60 fs,175 fs, and 430 fs FWHM, we derived electron pulse durations τ e = 350 fs, 395 fs, and 410 fsFWHM, respectively ( <
5% estimated error).
Theory of ultrafast electron-light interaction.
Following previous works ,we describe an electron wave-packet exposed to an optical field through the Schr¨odingerequation ( H + H ) ψ = i¯ h∂ψ/∂t , where ψ ( r , t ) is the electron wave function, H is thefree-space Hamiltonian, and H = ( − i e ¯ h/m e c ) A ( r , t ) · ∇ represents the minimal-couplinginteraction involving the optical vector potential A ( r , t ) in a gauge in which the scalarpotential and ∇ · A are both zero. We consider an expansion of the electron wave functionin terms of components e i( k · r − E k t/ ¯ h ) of momentum ¯ h k piled near a central value ¯ h k with k = ¯ h − (cid:112) (2 m e E )(1 + E / m e c ), corresponding to an electron kinetic energy E . Eachof these components is an eigenstate of H with energy E k ≈ E + ¯ h v · ( k − k ), where v =(¯ h k /m e ) / (1+ E /m e c ) is the central electron velocity. This approximation is valid for smallmomentum spread (i.e., | k − k | (cid:28) k ). Under these conditions, we can also approximate H ≈ E − ¯ h v · (i ∇ + k ), as well as ∇ ≈ i k in H . Now, it is convenient to separatethe fast evolution of the wave function imposed by the central-momentum component as ψ ( r , t ) = e i( k · r − E t/ ¯ h ) φ ( r , t ), where φ ( r , t ) then displays a slower dynamics. Putting theseelements together, the Schr¨odinger equation reduces to( v · ∇ + ∂/∂t ) φ = − i eγ v ¯ hc · A φ, where γ = 1 / (cid:112) − v /c , which admits the rigorous solution φ ( r , t ) = φ ( r − v t ) exp (cid:20) − i eγ v ¯ hc · ˆ t −∞ dt (cid:48) A ( r + v t (cid:48) − v t, t (cid:48) ) (cid:21) . (4)Here, φ ( r − v t ) is the electron wave function before interaction with the optical field.In practice, we consider illumination by an optical pulse with a narrow spectral distri-bution centered around a frequency ω , so the vector potential can be approximated as A ( r , t ) ≈ ( − i c/ω ) (cid:126) E ( r , t )e − i ωt + c . c . , where the electric field amplitude (cid:126) E ( r , t ) describesa slowly-varying pulse envelope that changes negligibly over an optical period. Insert-ing this expression into Eq. (4), we find the solution φ ( r , t ) = φ ( r − v t ) e −B + B ∗ , where12 ( r , t ) = eγ v ¯ hω · ´ t −∞ dt (cid:48) (cid:126) E ( r + v t (cid:48) − v t, t (cid:48) ) e − i ωt (cid:48) . Finally, using the Jacobi-Anger expansione i u sin ϕ = (cid:80) ∞ (cid:96) = −∞ J (cid:96) ( u )e i (cid:96)ϕ (see Eq. (9.1.41) of Ref. 38) with | u | = 2 |B| and ϕ = arg {−B} ,we obtain φ ( r , t ) = φ ( r − v t ) (cid:80) ∞ (cid:96) = −∞ J (cid:96) (2 |B| ) e i (cid:96) arg {−B} . This expression has general appli-cability under the assumptions of small energy spread in both electron and optical pulses.For monochromatic light (i.e., when (cid:126) E ( r ) depends only on position), considering withoutloss of generality v along ˆ z , we find B = β ( r )e − i ω ( z/v − t ) with β ( r ) = eγ ¯ hω ˆ z −∞ dz (cid:48) E z ( x, y, z (cid:48) ) e − i ωz (cid:48) /v , (5)and the electron wave function then becomes φ ( r , t ) = φ ( r − v t ) ∞ (cid:88) (cid:96) = −∞ J (cid:96) (2 | β | ) e i (cid:96) arg {− β } +i (cid:96)ω ( z/v − t ) , (6)where the last term in the exponential shows a change in the energy and momentum of the (cid:96) wave-function component given by (cid:96) ¯ hω and (cid:96) ¯ hω/v .For a Gaussian light pulse (cid:126) E ( r , t ) = (cid:126) E ( r )e − t /σ , corresponding to a FWHM-intensityduration τ L = ( √ σ L ≈ . σ L , under the assumption that the time needed by theelectron to cross the interaction region is small compared with σ L , we recover the result ofEq. (6) with β (Eq. (5)) replaced by e − ( z/v − t ) /σ β .We now calculate the electron probability at the detector as the integral ´ d r | φ ( r , t ) | fora large time t . Assuming a Gaussian electron pulse φ ( r − v t ) ∝ e − ( t − z/v − ∆ ) /σ normalizedto one electron ( ´ d r | φ | = 1), with FWHM-intensity duration τ e = ( √ σ e , and adelay ∆ relative to the light pulse, we find the probability that the electron has exchangeda net number of photons (cid:96) to be P (cid:96) = (cid:114) π σ e ˆ dt e − t /σ J (cid:96) (cid:16) | β | e − ( t +∆ ) /σ (cid:17) , (7)with β evaluated in the z → ∞ limit of Eq. (5). From the identity (cid:80) (cid:96) J (cid:96) ( u ) = 1, wereassuringly obtain (cid:80) (cid:96) P (cid:96) = 1. In the derivation of this expression, we have assumed thatdifferent (cid:96) electron channels have well separated energies, a condition that is guaranteed bythe assumption of small energy spread in both pulses (i.e., E σ e (cid:29) ¯ h and ωσ L (cid:29) J (cid:96) ( u ) = (cid:80) ∞ j =0 ( − j ( u/ (cid:96) +2 j /j !( (cid:96) + j )! for the Bessel functions ,the time integral in Eq. (7) can be readily performed term by term to yield P (cid:96) = ∞ (cid:88) j =0 ∞ (cid:88) j (cid:48) =0 C (cid:96)j C (cid:96)j (cid:48) √ λ e − n (∆ /σ ) /λ , (8)13here n = (cid:96) + j + j (cid:48) , λ = 1 + n ( σ e /σ L ) , and C (cid:96)j = ( − j | β | (cid:96) +2 j /j !( (cid:96) + j )! In the monochro-matic limit σ L (cid:29) σ e , we trivially obtain P (cid:96) = J (cid:96) (2 | β | ) (i.e., Eq. (2)).Under illumination with two identical light pulses delayed by ∆ i ( i = 1 ,
2) relative tothe electron and with their amplitudes scaled by real factors A i , a similar analysis can becarried out, using the Newton binomial expansion, to yield P (cid:96) = ∞ (cid:88) j =0 ∞ (cid:88) j (cid:48) =0 N (cid:88) s =0 N (cid:88) s (cid:48) =0 C (cid:96)j C (cid:96)j (cid:48) (cid:18) ns (cid:19)(cid:18) ns (cid:48) (cid:19) A n − s − s (cid:48) A s + s (cid:48) cos [( s − s (cid:48) ) ω (∆ − ∆ )] (9) × √ λ e − n (∆ /σ ) /λ e − [1 − s − s (cid:48) ) / n ]( s − s (cid:48) ) n (∆ − ∆ ) /σ ) , where ∆ = [(2 n − s − s (cid:48) )∆ + ( s + s (cid:48) )∆ ] / n .In our numerical simulations, we use Eqs. (8) and (9) with the electric field obtained bya standard transfer-matrix approach for a bilayer formed by Ag and Si N , with the permit-tivities of these materials taken as − . .
39i and .
04, respectively. Calculations forx-ray pulses at 777 eV photon energy are performed for multilayers of Au and Co, describedby their permittivities 0 .
97 + 0 . .
01 + 0 . Acknowledgements
This work has been supported in part by the NCCR MUST of the Swiss National ScienceFoundation and the Spanish MINECO (MAT2014-59096-P and SEV2015-0522), AGAUR(2014 SGR 1400), Fundaci´o Privada Cellex, and the Catalan CERCA program.The authorsacknowledge Dr P. Baum and Dr C. Ropers for useful discussions.14
IGURE CAPTIONS
FIG. 1:
Schematic representation of the experiment used to probe the interaction offree-electrons with semi-infinite light fields. (a) Ultrashort 200 keV electron pulses travelalong the z axis and impinge on the surface of a Ag/Si N thin bilayer, which is mounted on adouble-tilt holder able to rotate around the x (angle α ) and y (tilt angle ϑ ) axes. Light propagateswithin the y - z plane, incident with an angle δ ∼ − ◦ relative to the z axis and then reflectedfrom the Ag surface. The resulting electron-photon interaction is probed by monitoring electronenergy-loss spectra as a function of geometrical parameters and light properties. (b) Description ofthe electron-light interaction here explored. The breaking of translational invariance produced bylight reflection enables photon absorption or emission by the electron corresponding to a quantizedenergy and momentum exchange. (c) Description of the three-pulse experiment used for coherentmodulation of the electron wave function. Electrons interact with an appropriately synthesizedoptical field distribution produced by two mutually-phase-locked photon pulses whose relativephase is changed by varying their relative delay ∆ − ∆ . IG. 2:
Energy exchange during electron-light interaction. (a) Sequence of measured EELSspectra (color map) plotted as a function of increasing angle ϑ . We use p-polarized light (incidentfield along x axis), α = α C , a peak field amplitude of 12 . × V/m, and light and electron pulsedurations τ L = 430 fs and τ e = 410 fs. Sidebands at energies ± (cid:96) ¯ hω relative to the zero-loss peak(ZLP) are visible, where (cid:96) is the net number of exchanged photons. (b) Sequence of EELS spectrameasured for increasing light field amplitude with fixed tilt angle ϑ = 35 ◦ . (c) Spectra selectedfrom (a), measured at ϑ = 9 ◦ (blue curve) and ϑ = 30 ◦ (red curve), showing a strong redistributionof the electron density toward the high-energy sidebands for large tilt angle. (d)-(f) SimulatedEELS spectra corresponding to the experimental conditions of (a)-(c) (see Methods for details ofcalculations). IG. 3:
Momentum exchange during electron-light interaction. (a) Direct electron beammeasured in the diffraction plane as a function of transversal momentum ( k x , k y ) when no lightis applied. (b)-(c) Same as (a) under illumination with 560 fs laser pulses of 11 . × V/m peakfield amplitude and α = α C . The ϑ tilt angle is 0 ◦ in (b) and 35 ◦ in (c). A clear streaking of theelectron beam appears along the k x direction in (c) as a result of momentum exchanges betweenlight and electrons. (d) Electron beam profile along k x as a function of light polarization (sin φ = 0for s-polarization and sin φ = ± k x - E in the absence of optical illumination. (f)-(g) Measured (f) andsimulated (g) momentum-energy maps for illumination under the conditions of (c). IG. 4:
Attosecond coherent control of free-electrons.
The electron beam interacts witha semi-infinite temporally modulated optical field distribution produced by a sequence of twomutually-phase-locked light pulses impinging on the mirror. (a) Measured EELS spectra as afunction of relative delay ∆ − ∆ between the two optical pulses. The tilt angles are ϑ = 35 ◦ and α = α C , the optical pulses are 60 fs long with a peak field amplitude of 21 . × V/m each, andthe delays are ∆ = 0 and ∆ ≈ −
115 fs, with ∆ initial2 = 100 fs. (b) EELS spectra taken attwo different time delays (marked by horizontal dashed lines in panel (a)). (c) Relative intensity(full circles) of the (cid:96) = 9 and (cid:96) = 14 sidebands plotted as a function of time delay between the twooptical pulses, exhibiting a periodic modulation of period ≈ . π/ω )and a relative π phase shift. Solid curves are least-square fits to the data. (d)-(f) Simulated EELSspectra and resulting intensity change corresponding to the experimental conditions of (a)-(c) (seeMethods for details of calculations). IG. 5:
Amplitude and phase modulation of the electron wave function. (a) Two-dimensional Fourier transform of the energy-time map plotted in Fig. 4a. (b) Complex spectraldistribution of the electron-wave-function amplitude (bottom) and phase (top), extracted at themodulation frequency of 2 π/ (2 . ≈
385 THz. (c)-(d) Two-dimensional Fourier transform ex-tracted from the calculated energy-time map plotted in Fig. 4d. (e) Schematic representation ofelectron-wave-function modulation, showing snapshots of the strong energy-momentum electrondensity redistribution for different values of the phase shift between the two optical pulses. IG. 6:
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