  We report on the nonequilibrium quasiparticle dynamics in BaFe$_2$As$_2$ on
both the hole doped (Ba$_{1-x}$K$_x$Fe$_2$As$_2$) and electron doped
(BaFe$_{2-y}$Co$_y$As$_2$) sides of the phase diagram using ultrafast
pump-probe spectroscopy. Below $T_c$, measurements conducted at low
photoinjected quasiparticle densities in the optimally and overdoped
Ba$_{1-x}$K$_x$Fe$_2$As$_2$ samples reveal two distinct relaxation processes: a
fast component whose decay rate increases linearly with excitation density and
a slow component with an excitation density independent decay rate. We argue
that these two processes reflect the recombination of quasiparticles in the two
hole bands through intraband and interband processes. We also find that the
thermal recombination rate of quasiparticles increases quadratically with
temperature in these samples. The temperature and excitation density dependence
of the decays indicates fully gapped hole bands and nodal or very anisotropic
electron bands. At higher excitation densities and lower hole dopings, the
dependence of the dynamics on quasiparticle density disappears as the data are
more readily understood in terms of a model which accounts for the
quasiequilibrium temperature attained by the sample. In the
BaFe$_{2-y}$Co$_y$As$_2$ samples, dependence of the recombination rate on
quasiparticle density at low dopings (i.e., $y=0.12$) is suppressed upon
submergence of the inner hole band and quasiparticle relaxation occurs in a
slow, density independent manner.
