Tunable competing optical excitation pathways in the topological surface states of Bi$_2$Te$_3$
Understanding coherent optical responses of topological surface states (TSSs) requires disentangling excitation pathways from the electronic band structure. Here, using angle-resolved two-photon photoemission spectroscopy, we identify two distinct excitation pathways in the TSSs of Bi$_2$Te$_3$: an off-resonant transition via virtual states and a resonant transition via unoccupied intermediate states. A pronounced modulation of the spectral response is observed, revealing a competition between the two coherent pathways. This competition is tunable via temperature-induced shifts of the chemical potential, which selectively modify the resonant channel. These results provide microscopic insight into the optical excitation mechanisms of TSSs and highlight the potential for controlling their optical responses, relevant for future spintronic devices.