Reconstruction of the vibronic-state density matrix based on pump-probe state-resolved energy spectra

We propose a scheme based on perturbation theory to reconstruct a molecular density matrix (DM) using the pump-probe state-resolved energy spectra. By applying the scheme to an O+2 system produced via strongfield ionization, we demonstrate that the reconstruction scheme can retrieve the populations and coherences of vibrational-electronic (vibronic) states while maintaining good robustness. We further extend the strong-field transient ionization model to include the rotational degree of freedom. The evolution of the vibronic-state DM can be recovered, benefiting from the population stabilities between adjacent fractional revivals. The retrieved signals quantitatively capture the distinguishing decay features of the vibronic coherences. Our strategy paves the way toward understanding elaborate vibronic-state DMs in molecular systems.