Multimode cold-damping optomechanics with delayed feedback

We investigate the role of time delay in cold-damping optomechanics with multiple mechanical resonances. For instantaneous electronic response, it was recently shown by C. Sommer and C. Genes [Phys. Rev. Lett. 123, 203605 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.203605] that a single feedback loop is sufficient to simultaneously remove thermal noise from many mechanical modes. While the intrinsic delayed response of the electronics can induce single-mode and mutual heating between adjacent modes, we propose to counteract such detrimental effects by introducing an additional time delay to the feedback loop. For lossy cavities and broadband feedback, we derive analytical results for the final occupancies of the mechanical modes within the formalism of quantum Langevin equations. For modes that are frequency degenerate collective effects dominate, mimicking behavior similar to Dicke super- and subradiance. These analytical results, corroborated with numerical simulations of both transient and steady state dynamics, allow us to find suitable conditions and strategies for efficient single-mode or multimode feedback optomechanics.