  Using time-, spin- and angle-resolved photoemission, we investigate the
ultrafast spin dynamics of hot electrons on the surface of the topological
insulator Bi$_2$Te$_3$ following optical excitation by fs-infrared pulses. We
observe two surface-resonance states above the Fermi level coexisting with a
transient population of Dirac fermions that relax in about $\sim$2 ps. One
state is located below $\sim$0.4 eV just above the bulk continuum, the other
one at $\sim$0.8 eV inside a projected bulk band gap. At the onset of the
excitation, both states exhibit a reversed spin texture with respect to that of
the transient Dirac bands, in agreement with our one-step photoemission
calculations. Our data reveal that the high-energy state undergoes spin
relaxation within $\sim$0.5 ps, a process that triggers the subsequent spin
dynamics of both the Dirac cone and the low-energy state, which behave as two
dynamically-locked electron populations. We discuss the origin of this behavior
by comparing the relaxation times observed for electrons with opposite spins to
the ones obtained from a microscopic Boltzmann model of ultrafast band cooling
introduced into the photoemission calculations. Our results demonstrate that
the nonequilibrium surface dynamics is governed by electron-electron rather
than electron-phonon scattering, with a characteristic time scale unambiguously
determined by the complex spin texture of excited states above the Fermi level.
Our findings reveal the critical importance of detecting momentum and
energy-resolved spin textures with fs resolution to fully understand the sub-ps
dynamics of transient electrons on the surface of topological insulators.
