Two-Center Interference in the Photoionization Delays of ${\mathrm{Kr}}_2$

We present the experimental observation of two-center interference in the ionization time delays of ${\mathrm{Kr}}_2$. Using attosecond electron-ion-coincidence spectroscopy, we simultaneously measure the photoionization delays of krypton monomer and dimer. The relative time delay is found to oscillate as a function of the electron kinetic energy, an effect that is traced back to constructive and destructive interference of the photoelectron wave packets that are emitted or scattered from the two atomic centers. Our interpretation of the experimental results is supported by solving the time-independent Schrödinger equation of a 1D double-well potential, as well as coupled-channel multiconfigurational quantum-scattering calculations of ${\mathrm{Kr}}_2$. This work opens the door to the study of a broad class of quantum-interference effects in photoionization delays and demonstrates the potential of attosecond coincidence spectroscopy for studying weakly bound systems.

Figure 1

(a) Measured ionic distribution as a function of the mass-over-charge ratio and the hit position on the detector along the direction of the supersonic molecular beam. The counts are shown on a logarithmic scale and displayed in false color. (b),(c) RABBIT spectrograms for electrons detected in coincidence with undissociated ${\mathrm{Kr}}^{+}$ and ${\mathrm{Kr}}_2^{+}$, corresponding to the sharp distributions labeled with dashed ellipses in (a), respectively. Counts are normalized and shown in false color. In (b), the red and green arrows indicate the energy positions ionized by harmonic 11 for the two spin-orbit-coupling split states ${}^2{P}_{3/2}$ and ${}^2{P}_{1/2}$ of ${\mathrm{Kr}}_2^{+}$, respectively.

Figure 2

Schematic illustration of the outermost valence molecular orbitals (a) for ${\mathrm{Kr}}_2$ and the corresponding potential energy curves (b) for ${\mathrm{Kr}}_2^{+}$. In (b), the data is taken from Ref. [37] and the vertical dashed line indicates the equilibrium internuclear distance (7.578 a.u.) for the ground state of neutral ${\mathrm{Kr}}_2$. Spin-orbit coupling has been neglected for clarity.

Figure 3

(a) Photoionization cross section of the 1D system calculated with Eq. (6). (b) Corresponding Wigner delay calculated with Eq. (7). (c) Potential used in the calculations (black line) with the initial-state wave functions (blue and red lines). The ionization energies of the two states are 14.0 eV and 13.06 eV, respectively, and their vertical separation is added arbitrarily for clarity.

Figure 4

(a) Ionization delays $\mathrm{\Delta }{\tau }_{{\mathrm{Kr}}_2-\mathrm{Kr}}$ from experimental data (open circles), where values show the phase differences extracted from Figs. 1, 1 and the error bars represent the uncertainty of the phase extraction. The relative ionization delays from the MCSCI calculations are shown as a solid line. Theory for ${\mathrm{Kr}}_2$ includes the cross-section weighted delays for the two bound states of ${\mathrm{Kr}}_2^{+}$, i.e., ${}^2{\mathrm{\Sigma }}_u^{+}$ and ${}^2{\mathrm{\Pi }}_g$. (b) State-resolved relative delays from the MCSCI calculations for the ${}^2{\mathrm{\Sigma }}_u^{+}$ and ${}^2{\mathrm{\Pi }}_g$ states. Note that the solid line in (a) is the cross-section-weighted average of the state-resolved delays in (b). For both experiment and theory the electron emission angles range from ${\mathrm{\Theta }}_{\text{Lab}}=0–25°$ with respect to the XUV polarization, and the average over the molecular axis orientation in the lab frame is included in the calculations.