  We study transient dynamics of hole carriers injected at a certain time into
a Mott insulator with antiferromagnetic long range order. This is termed
``dynamical hole doping" as contrast with chemical hole doping. Theoretical
framework for the transient carrier dynamics are presented based on the two
dimensional $t-J$ model. Time dependences of the optical conductivity spectra
as well as the one-particle excitation spectra are calculated based on the
Keldysh Green's function formalism at zero temperature combined with the
self-consistent Born approximation. At early stage after dynamical hole doping,
the Drude component appears, and then incoherent components originating from
hole-magnon scatterings start to grow. Fast oscillatory behavior due to
coherent magnon, and slow relaxation dynamics are confirmed in the spectra.
Time profiles are interpreted as that doped bare holes are dressed by magnon
clouds, and are relaxed into spin polaron quasi-particle states. Characteristic
relaxation times for Drude and incoherent peaks strongly depend on momentum of
a dynamically doped hole, and the exchange constant. Implications to the recent
pump-probe experiments are discussed.
