Extracting phase information on continuum-continuum couplings

In measurements of relative phases in photoionization processes, the contribution of continuum-continuum (CC) couplings to the atomic (Wigner) phases, corresponding to attosecond time delays, is currently accounted for only by theoretical models. Here, we introduce a measurement scheme that is sensitive to phase differences of CC transitions after photoionization in the presence of an optical field. The method allows us to separate CC transitions from the ionization process and can help in testing existing theoretical models. It is based on a multicolor interferometer, similar to the RABBITT (reconstruction of attosecond beating by interference of two-photon transitions) method. The key idea involves the comparison of two measurements with different orders of CC transitions. The scheme is illustrated by an ab initio calculation on atomic hydrogen.

Figure 1

Schematic representation of the multicolor RABBITT method: The purple arrows describe the one-photon ionization process by the high-order harmonics $\mathrm{HH}\left(q\right)$ and $\mathrm{HH}(q+2)$; $I_P$ labels the ionization threshold. Sidebands (SB) are generated either by a two-photon process with the second harmonic frequency (case “blue”) or by a three-photon process with the fundamental (case “red”). See text for details.

Figure 2

Simulation of RABBITT traces in an artificial few-level system using probe fields with frequency $2{\omega }_L$ (a) or ${\omega }_L$ (b). (a-1) and (b-1): Population of the continuum states as a function of the photoelectron energy and the delay between the APT and the probe field. (a-2) and (b-2): Phase information $\mathrm{\Delta }{\varphi }^{\text{(blue)}}$ and $\mathrm{\Delta }{\varphi }^{\text{(red)}}$ retrieved from the RABBITT data in (a-1) and (b-1). The phase difference $\mathrm{\Delta }{\varphi }^{\text{(blue)}}-\mathrm{\Delta }{\varphi }^{\text{(red)}}$ is shown in (c) for different sets of couplings (see text). Parameters for this simulation: ${\omega }_L=1.5$ eV; harmonics separation: $4{\omega }_L$; intensity of the APT: ${10}^{-4}\mathrm{W}/{\mathrm{cm}}^2$; intensity of the probe field: ${10}^7\mathrm{W}/{\mathrm{cm}}^2$; modulus of all couplings: $|{\mu }_{ij}|=5.95$ atomic units; CC coupling: ${\phi }_{\mathrm{CC}}(k_i,k_j)=-0.5{k_i}^2/2$; EWS contribution: ${\phi }_W\left(k_i\right)=-0.002{k_i}^2/2$; and XUV chirp: group delay dispersion $\mathrm{GDD}=-8.68$ atomic units.

Figure 3

Ab initio TDSE calculation of two RABBITT experiments in atomic hydrogen using probe fields with frequency $2{\omega }_L$ (a) or ${\omega }_L$ (b). (a-1),(b-1) and (a-2),(b-2) are analogs to Fig. 2. The phase difference $\mathrm{\Delta }{\varphi }^{\left(\mathrm{blue}\right)}-\mathrm{\Delta }{\varphi }^{\left(\mathrm{red}\right)}$ is shown in (c). Also shown is the reference curve (green line) based on an analytical approximation for $\mathrm{\Delta }{\varphi }_{\mathrm{CC}}$ (see text). The phase can be converted into a time via $(\mathrm{\Delta }{\varphi }^{\left(\mathrm{blue}\right)}-\mathrm{\Delta }{\varphi }^{\left(\mathrm{red}\right)})/\left(2{\omega }_p\right)$, as shown on the right axis in (c). Parameters for this simulation: ${\omega }_L=1.5$ eV; harmonics separation: $4{\omega }_L$; intensity of each harmonic field: ${10}^9\mathrm{W}/{\mathrm{cm}}^2$; and intensity of the probe field $I_p$: ${10}^{11}\mathrm{W}/{\mathrm{cm}}^2$. Green stars: Fourier-limited APT; red stars: chirped APT with an attosecond pulse GDD = $-0.8656$ in atomic units.)