Rebuilding the vibrational wavepacket in TRAS using attosecond X-ray pulses

The study addresses a key challenge in ultrafast spectroscopy: when X-ray probe pulses are shortened to the attosecond regime, the broad bandwidth of such pulses in time-resolved X-ray photoelectron spectroscopy (TXPS) hampers the resolution of vibrational structures. Resonant Auger-electron spectroscopy (RAS), governed by the Auger lifetime broadening, preserves vibrational features even with ultrashort X-ray pulses. However, fully reconstructing a coherent vibrational wavepacket—determining both populations and relative phases of constituent vibrational states—from time-resolved RAS (TRAS) has remained unresolved. The purpose of this work is to show, through theory and simulation, that TRAS with attosecond X-ray pulses contains sufficient information to reconstruct the complete vibrational wavepacket in a molecular system. This capability is important for tracking coupled nuclear–electronic dynamics, including motion through conical intersections, advancing femtochemistry and attosecond science and providing a powerful complement to TXPS at both tabletop and XFEL facilities.

The authors review established ultrafast pump–probe techniques for probing vibronic dynamics: time-resolved photoelectron spectroscopy and high-harmonic interferometry to track coupled nuclear–electronic motion, and attosecond transient absorption spectroscopy (ATAS) to map non-adiabatic dynamics. With XFEL advances, multiple time-resolved X-ray methods have emerged, including X-ray absorption, TXPS, photoelectron diffraction, Raman, and scattering techniques, enabling studies of ultrafast dynamics. They note that as X-ray pulses enter the attosecond regime, TXPS loses vibrational resolution due to bandwidth broadening. In contrast, RAS retains vibrational structures because resolution is set by Auger lifetimes; this has been shown for transient X-ray absorption of phototriggered molecules using core-to-valence transitions. Prior reconstructions of wavepackets include ATAS-based reconstructions of valence-electron motion (Kr+), two-electron dynamics in He, and rotational wavepackets in H2/D2, as well as retrieval of ionic vibrational wavepacket phases via delay-dependent kinetic energy release, and time-resolved photoelectron spectroscopy from molecular autoionization for electronic wavepacket reconstruction. High-harmonic generation spectroscopy has reconstructed attosecond charge migration in iodoacetylene. Despite progress, reconstructing full vibronic wavepackets at femto–attosecond timescales remains a central goal, motivating the present TRAS-based approach.

System and pulses: The target is CO. A UV pump (duration 8.0 fs, central frequency 8.0 eV, Rabi frequency 0.002 a.u.) excites a coherent nuclear wavepacket on the valence-excited state 5σ−1π* A′Π (state I). A delayed resonant attosecond X-ray probe pulse (duration τ = 1 fs, photon energy ω = 280.0 eV) promotes a core electron (C 1s−1 π1 Π core-excited state, state N), initiating resonant Auger decay to the ionic final state 1π−1 2Π (state F). The pump–probe delay Δt is scanned; Δt = 0 is set at 24 fs after the pump center. The total evolution (~100 fs) is much shorter than the CO rotational period (~8.64 ps), so molecular rotation is neglected. Potential curves and decay widths: Potential energy curves and Auger width Γ = 0.08 eV are taken from literature. An additional case with increased Auger width to 0.4 eV is studied in Supplementary Note 2. Computation: Time-dependent quantum wavepacket simulations of RAS/TRAS are performed including lifetime–vibrational interference, solving the time-dependent Schrödinger equation with the Heidelberg MCTDH package. The formalism follows few-level models for resonant Auger processes with local approximation. Signal and theory: The TRAS signal σ(ε, Δt) is computed as the asymptotic norm square of the Auger amplitude summed over final states. For an initial coherent vibrational wavepacket with amplitudes a_i(t) = c_i e^{−i E_i t + i φ_i}, the TRAS signal can be written (Kramers–Heisenberg-consistent) as σ(ε, Δt) = Σ_i c_i^2 σ_{ii}(ε) + Σ_{i>j} 2 Re[c_i c_j^* σ_{ij}(ε)] cos(Δφ_{ij} − ΔE_{ij} Δt) + Σ_{i>j} 2 Im[c_i c_j^* σ_{ij}(ε)] sin(Δφ_{ij} − ΔE_{ij} Δt), where ΔE_{ij}=E_i−E_j and Δφ_{ij}=φ_i−φ_j. The σ_{ij}(ε) terms depend on X-ray parameters but not on initial wavepacket coefficients. Model identification via summed signal: To robustly extract the number of vibrational states and their energy spacings from data at all ε, the summed signal S(Δt)=Σ_ε σ(ε, Δt) is fit with a simplified formula containing cos(ΔE_{ij} Δt) and sin(ΔE_{ij} Δt) components. Two-, three-, and four-state models are compared to determine N and ΔE_{ij}. Fitting procedures: Vibrational energies are first obtained by fitting S(Δt) using two-, three-, and four-state models. Then, using these ΔE_{ij}, the full TRAS σ(ε, Δt) is fit with the same equation to retrieve the initial wavepacket’s populations c_i and relative phases Δφ_{i0}. The needed σ_{ii}(ε) are computed by initializing each vibrational state |φ^{(i)}⟩ individually. Cross terms σ_{ij}(ε) are extracted by simulating two specific coherent superpositions φ_1=(|i⟩+|j⟩)/√2 and φ_2=(|i⟩+i|j⟩)/√2 and solving linear combinations to obtain Re[σ_{ij}(ε)] and Im[σ_{ij}(ε)]. Optimization: Nonlinear regression uses Levenberg–Marquardt and Universal Global Optimization algorithms (1stOpt software). Fit quality is evaluated with the coefficient of determination R^2. Robustness to experimental instabilities: To emulate experimental noise (energy resolution limits, timing jitter, etc.), σ(ε, Δt) is perturbed by a random mask R_k(ε, Δt) uniformly sampled in [1−k, 1+k], producing blurred spectra σ_k and corresponding summed signals S_k. Cases with k = 1% and 5% are analyzed (higher k = 10%, 20% in Supplementary Note 1). Fits are repeated to assess sensitivity of retrieved ΔE_{ij}, c_i, and Δφ_{i0}.

• TRAS spectra of CO exhibit clear delay-dependent modulations with distinct revival patterns. The expectation value of R_CO(Δt) on state I shows a revival period of ~22 fs with slightly decreasing amplitude, indicating that more than two vibrational states are involved.
• Sensitivity to nuclear motion: Auger spectra taken at left/right turning points of R_CO(Δt) show subtle but discernible differences, confirming TRAS sensitivity to wavepacket motion.
• Model selection via summed signal S(Δt):
  • Two-state model poorly reproduces S(Δt) (ratio difference ~5×10^−3; R^2 = 0.986329).
  • Three- and four-state models fit excellently: ratio differences ~10^−4 and ~10^−6, respectively; R^2 = 0.9999697 (N=3) and 0.99999999895 (N=4).
  • Reconstructed vibrational energies (eV) for state I (original: 0.183378, 0.179847, 0.166515) are recovered with high accuracy:
    • Three-state: 0.182283, 0.180372 (errors ~10^−3 eV).
    • Four-state: 0.183542, 0.179707, 0.166836 (errors down to 10^−4 eV).
• Full TRAS fits σ(ε, Δt): Ratio differences are ~1% (N=3) and ~0.1% (N=4) over the 2D spectrum; R^2 = 0.9996943 (N=3) and 0.9999909561 (N=4).
• Retrieved wavepacket parameters (N=4) match the original with high fidelity:
  • Populations: c1=0.8431, c2=0.5267, c3=0.1077, c4=0.0108 vs original 0.8431, 0.5269, 0.1075, 0.0107 (population accuracy to ~0.01%).
  • Relative phases: Δφ10=2.7371, Δφ20=−0.6967, Δφ30=2.5961 vs original 2.7276, −0.7024, 2.5790 (phase differences at the second–third decimal place).
• Robustness to noise (blurred spectra):
  • Summed signals S_k(Δt) remain well fit for k=1% and 5% (R^2 ≈ 0.99979–0.99982 at 1%; ≈0.9957–0.9963 at 5% depending on N). Reconstructed ΔE_{ij} differ from originals by ~10^−3–10^−4 eV (Table 3).
  • Full σ_k(ε, Δt) fits: For k=1%, ratio differences ~0.1–1%; for k=5%, up to ~10%. Four-state model R^2 = 0.9999038 (1%) and 0.9981426 (5%). Retrieved phases are more sensitive to noise and model choice than populations but remain close to reference values (Table 4).
• Beyond vibrational resolution: Even when Auger width (0.4 eV) exceeds vibrational spacings (vibrational structure unresolved), TRAS delay dependence still enables reconstruction of initial wavepacket parameters (Supplementary Note 2), consistent with the signal model that does not explicitly depend on the Auger width.

The results demonstrate that attosecond TRAS retains and encodes sufficient information to reconstruct a coherent nuclear wavepacket, thereby addressing the core challenge posed by bandwidth-limited vibrational resolution in TXPS. By exploiting the Auger-lifetime-determined resolution of RAS and the explicit time-delay dependence of interference terms, the approach retrieves both the energy spacings and the complex amplitudes (populations and phases) of the vibrational states. The strategy of fitting the summed signal S(Δt) stabilizes extraction of the number of states and ΔE_{ij}, which then enables accurate global fits of σ(ε, Δt). High R^2 values and sub-0.01% population errors validate the reconstruction. Moreover, the method is resilient to moderate experimental instabilities (1–5% fluctuations), and remains informative even when vibrational lines are broadened beyond resolution by large Auger widths, underscoring its practical applicability. This positions TRAS as a powerful complementary probe to TXPS for femtosecond and attosecond studies of coupled electron–nuclear dynamics and potentially vibronic dynamics near conical intersections.

The work introduces and validates a practical framework to fully reconstruct a coherent vibrational wavepacket from time-resolved resonant Auger-electron spectroscopy using attosecond X-ray pulses. Applying the method to CO, the authors recover the number of vibrational states, their energy spacings, populations, and relative phases with high accuracy, even under realistic measurement instabilities. Crucially, the reconstruction remains feasible when Auger broadening exceeds vibrational level spacings. These results establish TRAS as a robust alternative and complement to TXPS for accessing ultrafast vibronic dynamics. Future directions include extending the methodology to polyatomic molecules, multi-mode normal-mode descriptions or fully correlated vibronic eigenstates, and exploring attosecond electronic dynamics, where careful modeling and fitting strategies will be needed in the presence of high state densities.

• Fitting with models including more than four vibrational states (N > 4) led to instability and breakdown when states are closely spaced or nearly degenerate, indicating practical limits in resolving many neighboring levels from the current data and model.
• Retrieved relative phases are more sensitive than populations to both model choice and experimental noise; higher instability (e.g., 5% or more) degrades accuracy, with ratio differences reaching ~10% in σ(ε, Δt) fits.
• For worse energy resolution (larger effective instability, k = 10–20%; Supplementary), accuracy of extracted wavepacket parameters decreases.
• Extension to polyatomic molecules with dense vibronic manifolds may be challenging; separability assumptions or reduced models may be required, and fitting complexity increases with state density.