  We investigate the condensation of charged $K^-$ meson and neutral $\bar K^0$
meson in dense neutron star matter. Calculations are performed in relativistic
mean field models in which both the baryon-baryon and (anti)kaon-baryon
interactions are mediated by meson exchange. It is found that $\bar K^0$
condensation is quite sensitive to the antikaon optical potential and depends
more strongly on the nucleonic equation of state. For moderate values of
antikaon potential and a rather stiff equation of state, a significant region
of maximum mass star will contain $\bar K^0$ meson. The critical density of
$\bar K^0$ condensation is always higher than that of $K^-$ condensation. With
the appearance of $K^-$ and $\bar K^0$ condensates, pairs of $p-K^-$ and
$n-\bar K^0$ are produced with equal proportion leading to a perfectly
symmetric matter of nucleons and antikaons in neutron stars. Along with $K^-$
condensate, $\bar K^0$ condensate makes the equation of state much softer
resulting in smaller maximum mass stars compared to the case without any
condensate.
