Coherent manipulation of quantum systems with light is expected to be a cornerstone of future information and communication technology, including quantum computation and cryptography. The transfer of an optical phase onto a quantum wavefunction is a defining aspect of coherent interactions and forms the basis of quantum state preparation, synchronization and metrology. Light-phase-modulated electron states near atoms and molecules are essential for the techniques of attosecond science, including the generation of extreme-ultraviolet pulses and orbital tomography. In contrast, the quantum-coherent phase-modulation of energetic free-electron beams has not been demonstrated, although it promises direct access to ultrafast imaging and spectroscopy with tailored electron pulses on the attosecond scale. Here we demonstrate the coherent quantum state manipulation of free-electron populations in an electron microscope beam. We employ the interaction of ultrashort electron pulses with optical near-fields to induce Rabi oscillations in the populations of electron momentum states, observed as a function of the optical driving field. Excellent agreement with the scaling of an equal-Rabi multilevel quantum ladder is obtained, representing the observation of a light-driven 'quantum walk' coherently reshaping electron density in momentum space. We note that, after the interaction, the optically generated superposition of momentum states evolves into a train of attosecond electron pulses. Our results reveal the potential of quantum control for the precision structuring of electron densities, with possible applications ranging from ultrafast electron spectroscopy and microscopy to accelerator science and free-electron lasers.

The active control of matter by strong electromagnetic fields is of growing importance, with applications all across the optical spectrum from the extreme-ultraviolet to the far-infrared. In recent years, phase-stable terahertz fields have shown tremendous potential for observing and manipulating elementary excitations in solids 1-3. In the gas phase, on the other hand, driving free charges with terahertz transients provides insight into ultrafast ionization dynamics 4,5. Developing such approaches for locally enhanced terahertz fields in nanostructures will create new means to govern electron currents on the nanoscale. Here, we use single-cycle terahertz transients to demonstrate extensive control over nanotip photoelectron emission. The terahertz near-field is shown to either enhance or suppress photocurrents, with the tip acting as an ultrafast rectifying diode 6. We record phase-resolved sub-cycle dynamics and find spectral compression and expansion arising from electron propagation within the terahertz near-field. These interactions produce rich spectro-temporal features and o er unprecedented control over ultrashort free electron pulses for imaging and di raction. Controlling electric charges with external fields is at the heart of modern information technology, with ultimate bandwidths limited by switching speeds in nanoscopic devices. The term light-wave electronics illustrates the anticipated application of optically fielddriven processes to solids, starting from schemes initially developed for atoms and molecules 7,8. In the terahertz range, strong tabletop sources have opened up the field of nonlinear terahertz optics and are enabling comprehensive control over electronic or structural dynamics, for example, in the manipulation of spin waves, the triggering of phase transitions, and the implementation of terahertz-driven scanning tunnelling microscopy 1-3,9-11. Completely new degrees of freedom are added by employing the localization of optical fields within nanostructures 12-15. Specifically, at metallic nanotips, photoelectron emission 16-18 with characteristic strongfield features is observed 19-21 , including carrier-envelope-phase sensitivity 22,23 , and sub-cycle electron acceleration at mid-infrared frequencies 12. A phase-resolved sampling of such processes may be achieved by so-called streaking spectroscopy, a method commonly applied in attosecond science, in which transient fields translate temporal information, for example, instances of ionization, into photoelectron energy or other degrees of freedom 24-26. The application of streaking spectroscopy to metallic nanostructures has been theoretically studied in-depth, aiming primarily at the full temporal characterization of near-infrared plasmonic fields 27-31. However, the prospects of transferring these concepts to the terahertz domain have not yet been investigated. Here, we show that the enhancement of terahertz fields in nanostructures allows for far-reaching electron trajectory control, spanning from phase-resolved streaking governed by...

Interference between multiple distinct paths is a defining property of quantum physics, 1 where "paths" may involve actual physical trajectories, as in interferometry, 2 or transitions between different internal (e.g. spin) states, 3 or both. 4 A hallmark of quantum coherent evolution is the possibility to interact with a system multiple times in a phasepreserving manner. This principle underpins powerful multi-dimensional optical 5 and nuclear magnetic resonance 3 spectroscopies and related techniques, including Ramsey's method of separated oscillatory fields 6 used in atomic clocks. Previously established for atomic, molecular and quantum dot systems, 7 recent developments in the optical quantum state preparation of free electron beams 8 suggest a transfer of such concepts to the realm of ultrafast electron imaging and spectroscopy.Here, we demonstrate the sequential coherent interaction of free electron states with two spatially separated, phase-controlled optical near-fields. Ultrashort electron pulses are acted upon in a tailored nanostructure featuring two near-field regions with anisotropic polarization response. The amplitude and relative phase of these two near-fields are independently controlled by the incident polarization state, allowing for constructive and destructive quantum interference of the subsequent interactions. Future implementations of such electron-light interferometers may yield unprecedented access to optically phase-resolved electronic dynamics and dephasing mechanisms with attosecond precision.A central objective of attosecond science is the optical control over electron motion in and near atoms, molecules and solids, leading to the generation of attosecond light pulses or the study of static and dynamic properties of bound electronic wavefunctions. 9-13 One of the most elementary forms of optical control is the dressing of free electron states in a periodic field, 14,15 which is observed, for example, in two-color ionization, 16,17 free-free transitions near atoms, 14,18 and in photoemission from surfaces. [19][20][21] Similarly, beams of free electrons can be manipulated by the interaction with standing waves 22,23 or optical near-fields. [24][25][26][27]8 In this process, field localization at nanostructures facilitates the exchange of energy and momentum between free electrons and light. In the past few years, inelastic electron-light scattering 25,26,28 found application in so-called "photon-induced near-field electron microscopy" or PINEM, 24,29,30,8 the characterization of ultrashort electron pulses, 26,27,30 or in work towards optically-driven electron accelerators. 32,33 Very recently, the quantum coherence of such interactions was demonstrated by observing multilevel Rabi-oscillations in the electron populations of the comb of photon sidebands. 8,25 Access to these quantum features, gained by nanoscopic electron sources of high spatial coherence, 34,35 opens up a wide range of possibilities in coherent manipulations, control schemes and interferometry with free electron stat...

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Internal quantum efficiency enhancement of GaInN/GaN quantum-well structures using Ag nanoparticles AIP Advances 5, 097169 (2015)