Tracking attosecond electronic coherences using phase-manipulated extreme ultraviolet pulses

The recent development of ultrafast extreme ultraviolet (XUV) coherent light sources bears great potential for a better understanding of the structure and dynamics of matter. Promising routes are advanced coherent control and nonlinear spectroscopy schemes in the XUV energy range, yielding unprecedented spatial and temporal resolution. However, their implementation has been hampered by the experimental challenge of generating XUV pulse sequences with precisely controlled timing and phase properties. In particular, direct control and manipulation of the phase of individual pulses within an XUV pulse sequence opens exciting possibilities for coherent control and multidimensional spectroscopy, but has not been accomplished. Here, we overcome these constraints in a highly time-stabilized and phase-modulated XUV-pump, XUV-probe experiment, which directly probes the evolution and dephasing of an inner subshell electronic coherence. This approach, avoiding any XUV optics for direct pulse manipulation, opens up extensive applications of advanced nonlinear optics and spectroscopy at XUV wavelengths.


Supplementary Note 1: Phase retrieval using heterodyned detection:
The full characterization of an electron WP in amplitude and phase is a non-trivial problem and a major goal in attosecond metrology. The issue of dissecting the amplitude ( ) and phase function ( ) of an arbitrary signal ( ) = ( ) ( ) is well-known from e.g. optical pulse characterization, where many methods have been developed to solve this problem. However, it is difficult to apply these methods in the XUV spectral range, which makes the phase retrieval of WPs excited/probed in the XUV domain generally a challenging task.
We solve this problem by introducing heterodyned detection with a known reference waveform, a common method in signal processing. Instead of using optical heterodyning, which is challenging at XUV wavelengths, the imprinted phase modulation effectively shifts the signal down to the low kHz-frequency regime where standard lock-in electronics can be used for heterodyned detection with a known electronic waveform. The quadrature demodulation with the known reference then yields in-phase and in quadrature signal components from which amplitude and phase are readily reconstructed (see also Methods section).
We note that our approach for retrieving amplitude and phase of a WP is universal and does not require energy-resolved detection. This permits its implementation with arbitrary detection types like ion time-of-flight detection, velocity map imaging or reaction microscopes.

Supplementary Note 2: Contribution of the Fano profile to the ion/electron count rate:
In quantum interference experiments, typically the pathway interference between the pump excitation and the delayed probe excitation is probed. For the presented study of a Fano resonance, the situation is slightly more complex, as each pulse can excite two coherent pathways leading to the same final state (labeled = 1, 2 in supplementary Fig. 2a). For pathway amplitudes the ion/electron signal is accordingly ∝ | 1 ( ) + 2 ( ) + 1 ( + ) + 2 ( + )| 2 . (1) Here, the interference term of pathway i excited by the pump with pathway j excited by the 3 delayed probe ( , = 1,2, ≠ ) contributes the phase difference between 1 and 2 to the signal which is hence encoded in the measured ion/electron count rates.
For a quantitative derivation of the signal we apply time-dependent perturbation theory and adapt the calculations from supplementary Ref. 1 to a one-dimensional quantum interference experiment. The experimental observable is the ion/electron yield which is proportional to the population probability of the quasi-bound eigenstates | ⟩ of the diagonalized Hamiltonian (supplementary Fig. 2b). Expanding the signal to second-order in the optical field and assuming delta-like excitation pulses yields: where kg = cg ( + )/( + ) denotes the | ⟩ → | ⟩ transition dipole moment with Fano parameter and reduced energy = ( kg − eg )/ e . It is jg the | ⟩ → | ⟩ transition frequencies, kg the dephasing rate of the k-g coherence, e ∝ 2 the dissipation/tunneling rate from the | ⟩ state into the continuum and Θ(t) the Heaviside step function. For simplicity we omitted the phase modulation term.
A Fourier transform yields the spectral signal ( ) ∝ ∫ | kg | 2 1 − kg − kg . (3) For the dilute gas-phase sample probed in our study, we can assume kg ≪ e . The real part of supplementary Eq. 3 is then which describes the well-known Fano profile.