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.

We introduce a framework for the preparation, coherent manipulation and characterization of free-electron quantum states, experimentally demonstrating attosecond pulse trains for electron microscopy. Specifically, we employ phaselocked single-color and two-color optical fields to coherently control the electron wave function along the beam direction. We establish a new variant of quantum state tomography -"SQUIRRELS" -to reconstruct the density matrices of freeelectron ensembles and their attosecond temporal structure. The ability to tailor and quantitatively map electron quantum states will promote the nanoscale study of electron-matter entanglement and the development of new forms of ultrafast electron microscopy and spectroscopy down to the attosecond regime.Optical, electron and x-ray microscopy and spectroscopy reveal specimen properties via spatial and spectral signatures imprinted onto a beam of radiation or electrons. Leaving behind the traditional paradigm of idealized, simple probe beams, advanced optical techniques increasingly harness tailored probes, or even their quantum properties and probe-sample entanglement. The rise of structured illumination microscopy 1 , pulse shaping 2 , and multidimensional 3 and quantum-optical spectroscopy 4 exemplify this development. Similarly, electron microscopy explores the use of shaped electron beams exhibiting particular spatial symmetries 5 or angular momentum 6,7 , and novel measurement schemes involving quantum aspects of electron probes have been proposed 8,9 . Ultrafast imaging and spectroscopy with electrons and x-rays are the basis for an ongoing revolution in the understanding of dynamical processes in matter on atomic scales [10][11][12][13] . The underlying technology heavily rests on laser science for the 2 generation and characterization of ever-shorter femtosecond electron 10,14 and xray [15][16][17] probe pulses, with examples in optical pulse compression 18 and streaking spectroscopy [19][20][21] . The temporal structuring of electron probe beams is facilitated by time-dependent fields in the radio-frequency [22][23][24] , terahertz 18,25 or optical domains. Promising a further leap in temporal resolution, recent findings suggest that ultrafast electron diffraction and microscopy with optically phasecontrolled and sub-cycle, attosecond-structured wave functions may be feasible 8,[26][27][28][29][30] . Specifically, light-field control may translate the temporal resolution of ultrafast transmission electron microscopy (UTEM) 31,32 and electron diffraction (UED) 10,33 , currently at about 200 fs 34 and 20 fs 14,23 , respectively, to the range of attoseconds 26,27,35 . However, such future technologies call for means to both prepare and fully analyze the corresponding quantum states of free electrons.Here, we demonstrate the coherent control and attosecond density modulation of free-electron quantum states using multiple phase-locked optical interactions. Moreover, we introduce quantum state tomography for free electrons, providing crucial e...

Nonlinear photoelectron emission from metallic nanotips is explored in the strong-field regime. The passage between the multiphoton and the optical field emission regimes is clearly identified. The experimental observations are in agreement with a quantum mechanical strong-field model.

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...

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