  Scanning tunneling microscopy and spectroscopy have been used to investigate
the femtosecond dynamics of Dirac fermions in the topological insulator
Bi$_2$Se$_3$ ultrathin films. At two-dimensional limit, bulk electrons becomes
quantized and the quantization can be controlled by film thickness at single
quintuple layer level. By studying the spatial decay of standing waves
(quasiparticle interference patterns) off steps, we measure directly the energy
and film thickness dependence of phase relaxation length $l_{\phi}$ and
inelastic scattering lifetime $\tau$ of topological surface-state electrons. We
find that $\tau$ exhibits a remarkable $(E-E_F)^{-2}$ energy dependence and
increases with film thickness. We show that the features revealed are typical
for electron-electron scattering between surface and bulk states.
