Quantum and quasi-classical effects in the strong field ionization and subsequent excitation of nitrogen molecules

The processes leading to the N₂⁺ lasing are rather complex and even the population distribution after the pump laser excitation is unknown. In this paper, we study the population distribution at electronic and vibrational levels in N₂⁺ driven by ultra-short laser pulse at the wavelengths of 800 nm and 400 nm by using the quantum-mechanical time-domain incoherent superposition model based on the time-dependent Schrödinger equation and the quasi-classical model assuming instantaneous ionization injection described by density matrix. It is shown that while both models provide qualitatively similar results, the quasi-classical instantaneous ionization injection model underestimates the population inversions corresponding to the optical transitions at 391 nm, 423 nm and 428 nm due to the assumption of quantum mixed states at the ionization time. A fast and accurate correction to this error is proposed. This work solidifies the theoretical models for population at vibrational states in N₂⁺ and paves the way to uncover the mechanism of the N₂⁺ lasing.