Supplementary Information

Figure 3.9:Experimental and calculated spectrograms and corresponding reconstructed Wigner functions. a Measured spectrograms after subtraction of the low-loss plasmon band with the full spectral resolution provided by the spectrometer. bCalculations employing coupling con-stants as given in the figure reproduce well the prominent phase-dependent spectral features, while minor differences are attributed to phase averaging effects not accounted for in Eq.3.2.

cWigner function reconstructed from experimental spectrograms. The increase of the coupling constant g_2ω from top to bottom is reflected in a growing amplitude of the sinusoidal phase modulation. Black solid lines according to Eq.3.1serve as guide to the eye.

3.6 Supplementary Information

Figure 3.10: Simulation of attosecond temporal reshaping a Simulated spectrogram assuming a pure state withg_pump=3.95 andg_probe=3.52, including a small timing jitter of 80as (3% of the optical period). These parameters correspond to the experi-mental values in Fig. 3.4.

b Corresponding Wigner function. c The temporal projection of the Wigner function exhibits density modulations with a FWHM pulse duration of 531as (after baseline subtrac-tion, rms pulse duration:

296as). d Corresponding electron energy spectrum (momentum projection).

Figure 3.11:Further measurements of attosecond temporal reshapinga,b,cExperimental spec-trograms recorded for pump and probe strengths (g_pump =1.98, g_probe =1.76), (3.42,3.05) and(3.95,3.52), from left to right. d,e,fCorresponding Wigner functions. Larger pump field strengths result in higher amplitudes of the sinusoidal phase modulation and shorter final at-tosecond pulse structures at the measurement position. g,h,i The temporal projection of the reconstructed Wigner function (black solid line) exhibits density modulations with a FWHM pulse duration (after baseline subtraction) as given in the Figure that decreases with increasing g_pump. Model calculations taking into account a timing jitter of 120as (dashed blue line) are in good agreement with the reconstruction.

Chapter 4

Discussion

Free electrons perhaps constitute one of the most simple and fundamental quantum sys-tems apart from photons, such that it practically suggests itself to transfer quantum optical methods from photons to free electrons. In the present work, inelastic electron-light scat-tering is introduced as a powerful means for the preparation, coherent manipulation and characterisation of free-electron quantum states, building on concepts from both attosec-ond physics and quantum optics. The underlying principle of optical phase-modulation together with the advanced capabilities ofin-situand sub-cycle light shaping renders all-optical control of free electrons highly versatile. Moreover, the interaction of free elec-trons and light does not suffer from dephasing mechanisms and thus preserves quantum coherence.

In the following, a summary of the work presented in the preceding Chapters will be given, pointing out the key results and setting them into perspective. Specifically, our approach for the reconstruction of quantum states will be discussed in a broader context, including related concepts from ultrashort pulse characterisation and coherent diffractive imaging. Potential future applications in time-resolved electron microscopy and free-electron quantum optics will be elucidated in more detail.

4.1 Summary

Chapter 2 reports the experimental implementation of a Ramsey-type electron-light in-terferometer based on sequential near-field interactions. Depending on the relative phase between two near-fields, the sinusoidal phase-modulation imprinted in the first interaction region can be enhanced or cancelled by the second interaction. This results from the

time-reversal symmetry of unitary operations, a fundamental property of quantum mechanical time evolution. Electron-light scattering can be interpreted as a continuous quantum ran-dom walk on an infinitely extended, equidistant energy ladder. In this picture, it becomes clear that the degree of reversibility is a measure of quantum coherence: In the incoherent limit of a classical random walk, the energy spectrum could not be recompressed to its initial width [165]. The observed near-perfect reversibility of the phase-modulation thus reconfirms the quantum coherence of the prepared free-electron quantum state, which was also demonstrated in Ref. [28] by the observation of Rabi-oscillations.

Going beyond the elementary case of sinusoidal phase modulations, two-colour near-fields are employed in Chapter 3. Although the light shaping capabilities of our setup fall short of spatial light modulators, merely changing the relative phase and intensities of two laser beams at commensurate frequencies proved to be a powerful approach for free-electron coherent control. Two-colour interactions actually present more than a first step towards more complex phase modulations: Quantum state tomography for free elec-trons based on phase-controlled multi-field interactions was successfully demonstrated.

This is a particularly important application, since state characterisation is a cornerstone technique for (free-electron) quantum optics. In Spectral QUantum Interference for the Regularised Reconstruction of free-ELectron States (SQUIRRELS), the density matrix is reconstructed in the longitudinal momentum basis from experimental spectrograms.

SQUIRRELS is not limited to the combination of fundamental and second harmonic pulses. Essentially any pulse pair at commensurate frequencies, including pulses at the same frequency, can be employed. To avoid phase-ambiguities in the reconstructed state in the case of differing frequency, the smaller frequency pulse must be used as the probe.

Moreover, reconstruction is possible irrespective of whether the interactions take place at the same or spatially separated positions. Consequently, SQUIRRELS is ideally suited for the quantitative characterisation of attosecond electron pulse trains, that occur after time-periodic phase modulations due to dispersive free-space propagation and thus require a means for probing at a position distant from the preparation region. The temporal enve-lope of the electron pulse is obtained from the marginal distribution of the reconstructed Wigner function, and in a first experiment, trains of sub-fs pulses were measured.