Attosecond Resolved Electron Release in Two-Color Near-Threshold Photoionization of ${\mathrm{N}}_2$

We have simulated two-color photoionization of ${\mathrm{N}}_2$ by solving the time-dependent Schrödinger equation with a simple model accounting for the correlated vibronic dynamics of the molecule and of the ion ${\mathrm{N}}_2^{+}$. Our results, in very good agreement with recent experiments [Haessler et al., Phys. Rev. A 80, 011404 (2009)], show how a resonance embedded in the molecular continuum dramatically affects the phases of the two-photon transition amplitudes. In addition, we introduce a formal relation between these measurable phases and the photoelectron release time, opening the way to attosecond time-resolved measurements, equivalent to double-slit experiments in the time domain.

Figure 1

(a) Electron-nuclei potential $V_e(x,\rho )$ for the $X$ channel, cut at $\rho =2\mathrm{a}.\mathrm{u}.$ (black full curve). Horizontal lines indicate the ground ($G$) and resonant ($R$) electronic energies as well as the ionization threshold ($X$). (b) Relevant molecular energies of the model treated in the BO approximation, for the $X$ and $A$ ionization channels. Horizontal lines indicate the first few vibrational energies of each curve. Vertical arrows indicate the photon energies corresponding to harmonics 11 and 13 of ${\omega }_L=1.56\mathrm{eV}$.

Figure 2

$v^{\prime }$-resolved rabbit phases associated with the sidebands 12 to 18 at ${\omega }_L=1.56\mathrm{eV}$: (a) $A$ channel, (b) $X$ channel. Values corresponding to $v^{\prime }=0$, 1, and 2 are indicated by squares, circles, and triangles, respectively. Full symbols connected with straight lines correspond to theoretical results, while empty symbols correspond to the experimental data taken from 14.

Figure 3

(a) $v^{\prime }$-resolved probability to photoionize the model molecule in the $X$ channel with the 11th harmonic against the laser frequency, in the vicinity of the resonance. (b) $X$ channel $v^{\prime }$-resolved ${\mathrm{SB}}_{12}$ rabbit phases in the same range of frequencies. In both frames, each curve is labeled by the corresponding final vibrational state $v^{\prime }$.

Figure 4

(a) $v^{\prime }$-resolved ${\mathrm{SB}}_{12}$ formation delay in the $X$ channel versus ${\omega }_L$. (b) $v^{\prime }$-resolved ${\mathrm{SB}}_{12}$ rabbit phases derivative with respect to the corresponding ${\omega }_L$-dependent photoelectron energy [see Eq. (5)]. Labels indicate the final vibrational state $v^{\prime }$ as in Fig. 3.