  There has been a growing evidence for the existence of magnetic fields in the
extra-galactic regions, while the attempt to associate their origin with the
inflationary epoch alone has been found extremely challenging. We therefore
take into account the consistent post-inflationary evolution of the magnetic
fields that are originated from vacuum fluctuations during inflation. In the
model of our interest, the electromagnetic (EM) field is coupled to a
pseudo-scalar inflaton $\phi$ through the characteristic term $\phi F\tilde F$,
breaking the conformal invariance. This interaction dynamically breaks the
parity and enables a continuous production of only one of the polarization
states of the EM field through tachyonic instability. The produced magnetic
fields are thus helical. We find that the dominant contribution to the observed
magnetic fields in this model comes from the modes that leave the horizon near
the end of inflation, further enhanced by the tachyonic instability right after
the end of inflation. The EM field is subsequently amplified by parametric
resonance during the period of inflaton oscillation. Once the thermal plasma is
formed (reheating), the produced helical magnetic fields undergo a turbulent
process called inverse cascade, which shifts their peak correlation scales from
smaller to larger scales. We consistently take all these effects into account
within the regime where the perturbation of $\phi$ is negligible and obtain
$B_{\rm eff} \sim 10^{-19}$G, indicating the necessity of additional mechanisms
to accommodate the observations.
