Coherent phonons and the interplay between charge density wave and Mott phases in 1$T$-TaSe$_{2}$

1$T$-TaSe$_{2}$ is host to coexisting strongly-correlated phases including charge density waves (CDWs) and an unusual Mott transition at low temperature. Here, we investigate coherent phonon oscillations in 1$T$-TaSe$_{2}$ using a combination of time- and angle-resolved photoemission spectroscopy (TR-ARPES) and time-resolved reflectivity (TRR). Perturbation by a femtosecond laser pulse triggers a modulation of the valence band binding energy at the $\Gamma$-point, related to the Mott gap, that is consistent with the in-plane CDW amplitude mode frequency. By contrast, TRR measurements show a modulation of the differential reflectivity comprised of multiple frequencies belonging to the distorted CDW lattice modes. Comparison of the temperature dependence of coherent and spontaneous phonons across the CDW transition shows that the amplitude mode intensity is more easily suppressed during perturbation of the CDW state by the optical excitation compared to other modes. Our results clearly identify the relationship of the in-plane CDW amplitude mode with the Mott phase in 1$T$-TaSe$_{2}$ and highlight the importance of lattice degrees of freedom.