  Open-system density matrix methods typically employ incoherent population
injection to investigate the postionization dynamics in strong laser fields.
The presence of coherence injection has long been a subject of debate. In this
context, we introduce a coherence injection model based on the adiabatic
strong-field approximation (ASFA). This model effectively predicts ionic
coherence resulting from directional tunnel ionization. With increasing field
strength, the degree of coherence predicted by the ASFA model gradually
deviates from that of the SFA model but remains much milder compared to the
results of the simple and partial-wave expansion models. The impact of
coherence injection on the postionization molecular dynamics is explored in
O$_2$ and N$_2$. We find that the ionization-induced vibrational coherence
strongly enhances the population inversion of $X^2 \Sigma _g^+ -B^2 \Sigma _u^+
$ in N$_2^+$ and the dissociation probability of O$_2^+$. Conversely, the
ionization-induced vibronic coherences have inhibitory effects on the related
transitions. These findings reveal the significance of including the
vibronic-state-resolved coherence injection in simulating molecular dynamics
following strong-field ionization.
