FIG. 3: (Color online). a) Calculated photoelectron spectra in He as a function of delay between the attosecond pump pulse and the IR probe pulse. Interference fringes are clearly seen at all delays where the attosecond pulse comes before the IR probe. b) Fourier transform of the photoelectron spectrum allowing the identification of the states that form the bound WP. The beat signals from the 2 p , 3 p and 4 p states can be seen as vertical lines around 2 eV while the interference between direct and indirect pathways give rise to contributions at an angle of 45 degrees. A lineout of the transform at 4 eV is presented in c. 3 Delay, t (fs) Energy, E (eV) (a) QB 0 5 10 15 20 25 30 0 5 10 15 Fourier frequency, E ′ (eV) (b) 2 p 3 p 4 QB 0 5 10 15 Fourier frequency, E ′ (eV) (c) 0 5 10 15 5.6 and 7.4 eV in Fig. 3c are due to the direct-indirect interferences involving the 4 p , 3 p and 2 p states respec- tively. When the observation energy E is varied, the Fourier frequency of the quantum beats does not change and they appear as vertical lines in Fig. 3b. In contrast, the Fourier frequency E ′ of the direct-indirect interfer- ences increases with E , since the accumulated phase dif- ference between the direct and indirect ionization path- ways is proportional to it. This linear relationship results in lines tilted at 45 degrees in Fig. 3b. The energies of the bound intermediate states (4 p , 3 p , 2 p ) in the WP can be read directly from the intersections of the 45 degree lines with the horizontal zero energy line. In addition, the relative strengths of the 45 degree lines are directly related to the contributions from each bound state to the ionization signal. Turning to the experimental data, we find that quan- tum beats and direct-indirect interferences are also present. Figure 4 presents an extended analysis of our experimental results. Fig. 4a is a zoom of the low energy region in Fig. 2, while Fig. 4b presents the Fourier trans- form of the experimental data. The observed 45-degrees lines which are characteristic for the direct-indirect in- terferences allow us to identify the composition of the bound WP and identify the contributions from the 3 p , 4 p and 5 p states. A weak indication of the 4 p -5 p quan- tum beat can also be seen as a beating with a 13 fs pe- riodicity in Fig. 4a and as a vertical line near 0.5 eV in Fig. 4b. Additional information is encoded in the an- gular distributions since the two interference processes are fundamentally different and involve different angular momentum states. Fig.4 c-f presents a more complete analysis making use of the full angular-resolved photo- electron distribution F ( E, θ, t ), which can be expanded as a sum of Legendre polynomials ence fringes appear as soon as the excitation of the bound states is separated in time from their ionization by the probe field, meaning that we have two clearly delineated routes into the continuum. The hyperbolic shape of the fringes is visible in the calculation and the fringe spacing in energy decreases as the delay increases. A quantum beat signal [22, 23] is also visible in Fig. 3a, due to simultaneous excitation of several bound states ( i , j ). Interferences between quantum paths leading to the same final energy, for example E$_{i}$ + ¯ hω = E$_{j}$ + ¯ hω ′ , ω , ω ′ being within the IR laser bandwidth, give rise to periodic structures in the ionization probability. The ac- cumulated phase difference between the quantum paths, equal to ( E$_{j}$ − E$_{i}$ ) t/ ¯ h , is independent on the observation energy and the quantum beat signal appears as periodic vertical structures. The beat signal observed in Fig. 3 has a main periodicity of about 2 fs, corresponding to the beating between the outgoing electrons from the 2 p and the 3 p states. The periodicity of the beat signal car- ries information on the relative energy separation of the states involved, while the absolute timing of the beat- ing depends on the relative phase of the pairs of states contributing to the signal. As we now discuss, analy- sis of the “direct-indirect” interferences, involving bound states and a reference continuum state allow us to go well beyond quantum beat spectroscopy. To begin with, the different components of the excited WP can be extracted by Fourier analysis of the delay- dependent photoelectron signal S$_{theo}$ ( E, t ). We first an- alyze the simulation shown in Fig. 3. The Fourier trans- form at all the possible observation energies yields a two dimensional function of the observation energy E and the Fourier frequency (represented as an energy E ′ ). This function, S$_{theo}$ ( E, E ′ ), is presented as a color plot in Fig. 3b as a function of E and E ′ . Fig. 3(c) shows a li- neout at E = 4 eV. It exhibits six prominent peaks. The three lowest peaks (at approximately 1, 2 and 3 eV) are from the quantum beat between 3 p -4 p , 2 p -3 p , and 2 p -4 p pairs of states respectively while the other peaks at 4.9, where P$_{J}$ is the Legendre polynomial of J th order, β$_{J}$ F ( E, θ, t ) = 2 L$_{max}$ ∑ j =0 β$_{j}$ ( E, t ) P$_{J}$ [cos( θ )] , (1)