Influence of molecular rotation on the generation of N$_2^+$ air lasing

N$_2^+$ air lasing has attracted considerable attention due to its promising applications in remote sensing and the debates surrounding its generation mechanisms. Here, we present a comprehensive theoretical investigation of the role of molecular rotation in N$_2^+$ lasing at 391 nm ($B^2 \Sigma _u^+(v''=0)\rightarrow X^2 \Sigma _g^+ (v=0)$). By solving the open-system density matrix and Maxwell-Bloch equations in a rovibronic-state basis, we examine both the formation of the N$_2^+$ gain medium induced by a femtosecond pump pulse and the subsequent spatial propagation of the seed pulse. During the pump stage, rotational dynamics are found to significantly modify the angle-dependent populations of ionic vibrational-electronic states within tens of femtoseconds. Furthermore, ionization-produced rotational coherences substantially enhance the population inversion between the $X^2 \Sigma _g^+ (v=0)$ and $B^2 \Sigma _u^+(v''=0)$ states. In the seed propagation stage, both population inversion and rotational coherence are found to contribute to the lasing process, with the latter playing a dominant role in amplifying the lasing signals. These findings reveal the crucial role of molecular rotation in N$_2^+$ air lasing and highlight its potential as a tunable parameter for controlling lasing dynamics.