Attosecond time delay in the photoionization of endohedral atoms $A@{\text{C}}_{60}$: A probe of confinement resonances

The effects of confinement resonances on photoelectron group delay (Wigner time delay) following ionization of an atom encapsulated inside a ${\text{C}}_{60}$ cage have been studied theoretically using both relativistic and nonrelativistic random phase approximations. The results indicate clearly the resonant character of the confinement oscillations in time delay of the $4d$ shell of Xe@${\text{C}}_{60}$ and present a most direct manifestation of Wigner time delay. These oscillations were missed in a previous theoretical investigation of Ar@${\text{C}}_{60}$ [Phys. Rev. Lett. 111, 203003 (2013)].

Figure 1

Graphical representation of the RPA equations [26]. Left: noncorrelated dipole matrix element. Center: time-forward process. Right: time-reverse process.

Figure 2

Top: partial photoionization cross sections of the $4d$ shell in Xe (left) and Xe@${\text{C}}_{60}$ (right). The RRPA calculations for spin-orbit-resolved components $4d_{3/2}$ ($\times 5/2$), $4d_{5/2}$ ($\times 5/3$), and their sum are drawn with lines. The RPA calculations for the $4d\to \epsilon f$ and $4d\to \epsilon p$ (left panel only) cross sections are drawn with dotted lines. Bottom: time delay in photoionization of the $4d\to \epsilon f$ channel of Xe (left) and Xe@${\text{C}}_{60}$ (right). The RRPA calculations for various spin-orbit-resolved components are drawn with lines while the RPA calculation is drawn with dotted lines.

Figure 3

Normalized photoionization cross-section difference $\left[\sigma (\mathrm{Xe}@{\mathrm{C}}_{60})-\sigma \left(\mathrm{Xe}\right)\right]/\sigma \left(\mathrm{Xe}\right)$ (top) and time delay difference $\tau (\mathrm{Xe}@{\mathrm{C}}_{60})-\tau \left(\mathrm{Xe}\right)$ (bottom) as functions of photoelectron energy. The RPA calculation is shown with a solid line. The experimental data in the top panel are from Ref. [22].