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Theory of femtosecond strong field ion excitation and subsequent lasing in N$_2^+$

Delayed cavity-free forward lasing at the wavelengths of 391 and 428 nm was observed in recent experiments in air or pure nitrogen pumped with an intense femtosecond laser pulse at wavelength of 800~nm. The mechanism responsible for the lasing is highly controversial. In this article we explain the delayed emission by the presence of long-lived polarizations coupling simultaneously ground state X$^2Σ_g^+$ to states A$^2Π_u$ and B$^2Σ_u^+$ of singly ionized nitrogen molecules N$_2^+$. Ionization of neutral nitrogen molecules in a strong laser field and subsequent ion excitation are described by a system of Bloch equations providing a distribution of ions in the ground and excited states A and B at the end of the laser pulse. The delayed signal amplification at the B-X transition wavelength is described by a system of Maxwell-Bloch equations with polarization coupling maintained by a weak laser post-pulse. Two regimes of signal amplification are identified: a signal of a few ps duration at low gas pressures and a short (sub-picosecond) signal at high gas pressures. The theoretical model compares favorably with experimental results.