  We investigate theoretically the dynamics of two quasidegenerate mechanical
modes coupled through an open quantum two-level system. A mean-field approach
shows that by engineering the retarded response of the two-level system with a
coherent drive, the non-Hermitian mechanical spectrum exhibits an exceptional
degeneracy point where the two modes coalesce. We show that this degeneracy can
be exploited to manipulate the vectorial polarization of the mechanical
oscillations. We find that adiabatically varying the detuning and the intensity
of the drive induces a rotation of the mechanical polarization, which enables
the topological and chiral actuation of one mode from the other. This
topological manifestation of the degeneracy is further supported by
quantum-jump Monte Carlo simulations to account for the strong quantum
fluctuations due to the spontaneous emission of the two-level system. Our
presentation focuses on a promising realization based on flexural modes of a
carbon-nanotube cantilever coupled to a single-molecule electric dipole
irradiated by a laser.
