Disentangling the Effect of Ionic Coupling and Multiple Interfering Terms in Attosecond Molecular Interferometry

Abstract: Attosecond interferometry in a two-color field is central to attosecond metrology and spectroscopy. In this technique, a photoelectron wave packet is released when a single photon from an extreme ultraviolet comb is absorbed. The wave packet then either emits or absorbs one or more near-infrared photons, leading to the formation of sidebands of the main photoelectron peaks. This picture applies well to atoms and assumes that the near-infrared laser pulse only acts on the photoelectron leaving the parent ion. The effect of the near-infrared pulse on the electronic structure of the cation is not considered, since the field usually cannot induce transitions between its electronic levels. Here, we demonstrate how dynamics induced by the near-infrared field in the cation can significantly impact the amplitude and phases of the sideband signal of the photoelectrons associated with specific dissociative channels of CO$_2$ molecules. This coupling of the near-infrared field with the molecular cation opens a third quantum pathway contributing to the signal measured in attosecond interferometry. Through angle- and energy-resolved characterization of the sideband oscillations, we observe reduction of interference amplitude over specific energy range upon angle integration. By comparison with theoretical predictions, we can isolate the contributions of specific interfering pathways to the two-color multi-pathway photoionization process. The scheme investigated in our work is general, and our observations highlight the importance of the additional pathway for accurately interpreting attosecond interferometry experiments involving molecules and more complex quantum systems.

• The paper identifies a third quantum pathway from IR-induced ionic coupling that alters sideband signals and time delays in RABBIT schemes.
• The paper employs multi-particle coincidence spectroscopy and partial-wave analysis to directly correlate experimental measurements with theoretical R-matrix simulations.
• The paper demonstrates that incorporating ionic coupling effects is essential for advancing attosecond metrology and accurately probing ultrafast molecular dynamics.

Disentangling Ionic Coupling and Multiple Interfering Terms in Attosecond Molecular Interferometry

Overview

This paper presents an integrated theoretical and experimental investigation of attosecond time delays in photoionization dynamics within CO$_2$ molecules, employing attosecond interferometry with an extreme ultraviolet (XUV) pulse train and synchronized infrared (IR) field. The central insight is the identification and characterization of a third quantum pathway, rooted in IR-induced ionic coupling between cationic states (specifically $B^2\Sigma^+_u$ and $C^2\Sigma^+_g$), which fundamentally modifies sideband signals and measured time delays in RABBIT schemes. By combining energy-, angle-, and ion-resolved measurements, the authors elucidate the origins and behavior of interfering terms in two-color molecular photoionization, highlighting new theoretical and practical implications for attosecond metrology and molecular spectroscopy.

Dissociation Spectroscopy and Fragmentation Analysis

Detailed spectroscopy is performed on dissociative channels of CO$_2$, with emphasis on the population and predissociation of various cationic states. The ground vibrational state in $C^2\Sigma^+_g$ is shown to dominate O$^+$ fragment formation via Franck-Condon overlap, while excited vibrational levels contribute to CO$^+$ channel formation. By measuring photoelectron spectra in coincidence with ionic fragments and kinetic energy release (KER), the authors unambiguously assign fragment pathways and vibrational states.

Figure 1: Energy cuts of PESs for CO$_2$ and CO$_2^+$, and vibrational levels of $C^2\Sigma^+_g$, showing fragmentation thresholds and vibronic mixing effects.

Experimental Coincidence Measurements and Angular-Resolved Spectroscopy

Employing a multi-particle coincidence spectrometer, angle- and KER-resolved photoelectron spectra are measured for XUV-only and XUV-IR cases. These data reveal state-selective ionization potentials, spectral widths, and angular asymmetry parameters ($B^2\Sigma^+_u$0), coinciding with literature and confirming the dominant fragmentation pathways inferred from Franck-Condon factors and branching ratios. Notably, angular distributions differ for O$B^2\Sigma^+_u$1 and CO$B^2\Sigma^+_u$2 channels, with perpendicular emission in the CO$B^2\Sigma^+_u$3 channel correlating to vibronic mixing into the $B^2\Sigma^+_u$4 state.

Figure 2: Angle-resolved XUV-only spectra for Ar$B^2\Sigma^+_u$5, O$B^2\Sigma^+_u$6, and CO$B^2\Sigma^+_u$7, and integrated photoelectron peak energies and widths.

Figure 3: XUV-only spectra for O$B^2\Sigma^+_u$8 and CO$B^2\Sigma^+_u$9, showing angular dependence and signature of vibronic interaction.

Figure 4: KER-resolved photoelectron spectra for O$C^2\Sigma^+_g$0 and CO$C^2\Sigma^+_g$1, illustrating conservation of energy and vibrational population assignments.

RABBIT Measurements and Angular Integration Effects

RABBIT (Reconstruction of Attosecond Beating By Interference of Two-photon Transitions) traces are recorded as a function of relative XUV-IR delay with angular resolution. The sideband oscillation amplitude ($C^2\Sigma^+_g$2) and phase ($C^2\Sigma^+_g$3) exhibit a pronounced decrease and abrupt phase jump in specific energy and angular windows (notably 4–7 eV), which is not predicted by the conventional two-pathway model. The location and magnitude of these features vary with angular integration, directly correlating with the theoretical expectation for three-path interference.

Figure 5: Delay-averaged angular distributions for O$C^2\Sigma^+_g$4 and CO$C^2\Sigma^+_g$5 in XUV-IR cases, highlighting sideband energy ranges.

Figure 6: RABBIT traces for O$C^2\Sigma^+_g$6 ions at distinct angular intervals, visualizing oscillation amplitude minima and phase jumps.

Theory: Multi-Pathway Interference and Ionic Coupling

The multiphoton R-matrix formalism is employed, simulating full two-photon amplitudes for three distinct pathways:

• Path 1: Absorption of XUV followed by IR absorption by the electron (conventional).
• Path 2: Absorption of XUV followed by IR emission by the electron (conventional).
• Path 3: Absorption of XUV ionizes to the $C^2\Sigma^+_g$7 cation; subsequent IR photon induces ionic transition to $C^2\Sigma^+_g$8, populating the same final state.

  Figure 7: Schematic of three-path RABBIT mechanism, depicting direct and ionic-coupling pathways.

Path 3’s amplitude and phase depend critically on the dipole coupling between cationic states and their energy separation relative to the IR photon. The interference of all three pathways is responsible for the angular and energy-dependent modulation and phase jumps.

Figure 8: Comparison of simulated and measured photoionization delays for different angular intervals, demonstrating jump at 4–7 eV due to ionic coupling.

Figure 9: Contributions from different pathways to sideband photoelectron signal, revealing angular-energy regions dominated by ionic coupling interference.

Quantum Interference Structure and Partial-Wave Analysis

Quantitative analysis shows that regions of vanishing modulation and strong phase jumps in $C^2\Sigma^+_g$9 and $_2$0 correspond to energetic and angular thresholds where the amplitudes of paths 1–2 and 2–3 become comparable and out-of-phase. Detailed partial-wave analysis attributes these effects to the interplay between $_2$1 and $_2$2 partial waves in the $_2$3 and $_2$4 channels, with a characteristic $_2$5 phase difference. The delay jump matches the period $_2$6, confirming the theoretical prediction.

Figure 10: Simulated two-dimensional maps of $_2$7 and $_2$8, highlighting locations of amplitude minima and phase transition due to ionic coupling.

Figure 11: Experimental angle- and energy-resolved analysis of RABBIT traces, confirming theoretical predictions for amplitude and phase structure.

Figure 12: Magnitude and phase of orientation-averaged interference terms ($_2$9 and $C^2\Sigma^+_g$0), correlating with delay jump.

Figure 13: Partial-wave resolved one-photon ionization dipole magnitudes and phases, demonstrating partial-wave origin of phase shift.

Implications and Future Directions

These findings have direct implications for attosecond molecular interferometry. Accurate interpretation of molecular RABBIT signals requires explicit consideration of ionic coupling mechanisms whenever IR photon energies match cationic state energy separations, a condition likely met in more complex polyatomic or cluster systems. Practically, this advances the precision of attosecond metrology for probing ultrafast electron dynamics and entanglement effects in molecules. Theoretically, it establishes a clear connection between multi-pathway quantum interference, partial-wave structure, and delay measurements, suggesting avenues for control and manipulation of ion-electron entanglement.

Future directions include extending these investigations to other molecular systems, possibly exploiting ionic resonances for tailored time-delay control or for sensitive probes of correlation and vibronic dynamics.

Conclusion

This work offers a comprehensive framework for understanding and disentangling the effects of ionic coupling and multiple interfering terms in attosecond molecular interferometry. By combining precise coincidence spectroscopy with advanced theoretical modeling, the authors attribute strong energy- and angle-dependent variations in attosecond time delays to multi-pathway quantum interference involving IR-induced ionic transitions. The necessity of accounting for cationic state coupling is established for the rigorous interpretation of two-color molecular photoionization delays, with broad implications for attosecond science and molecular dynamics.