  Quantifying electron-phonon interactions for the surface states of
topological materials can provide key insights into surface-state transport,
topological superconductivity, and potentially how to manipulate the surface
state using a structural degree of freedom. We perform time-resolved x-ray
diffraction (XRD) and angle-resolved photoemission (ARPES) measurements on
Bi$_2$Te$_3$ and Bi$_2$Se$_3$, following the excitation of coherent A$_{1g}$
optical phonons. We extract and compare the deformation potentials coupling the
surface electronic states to local A$_{1g}$-like displacements in these two
materials using the experimentally determined atomic displacements from XRD and
electron band shifts from ARPES.We find the coupling in Bi$_2$Te$_3$ and
Bi$_2$Se$_3$ to be similar and in general in agreement with expectations from
density functional theory. We establish a methodology that quantifies the
mode-specific electron-phonon coupling experimentally, allowing detailed
comparison to theory. Our results shed light on fundamental processes in
topological insulators involving electron-phonon coupling.
