Theoretical framework for enhancing or enabling cooling of a mechanical resonator via the anti-Stokes or Stokes interaction and zero-photon detection

We develop a theoretical framework to describe how zero-photon detection may be utilized to enhance optomechanical laser cooling via the anti-Stokes interaction and, somewhat surprisingly, enable cooling via the Stokes interaction commonly associated with heating. Our description includes both pulsed and continuous measurements as well as optical detection efficiency and open-system dynamics. For both cases, we discuss how the cooling depends on the system parameters such as detection efficiency and optomechanical cooperativity, and we study the continuous-measurement-induced dynamics, contrasting with single-photon-detection events. For the Stokes case, we explore the interplay between cooling and heating via optomechanical parametric amplification, and we find the minimum efficiency required to cool a mechanical oscillator via zero-photon detection. This work serves as a companion article to our recent experiment [E. A. Cryer-Jenkins et al., Phys. Rev. Lett. 134, 073601 (2025)], which demonstrated enhanced laser cooling of a mechanical oscillator via zero-photon detection on the anti-Stokes signal. The cooling techniques developed here can be applied to a wide range of areas including nonclassical state preparation, quantum thermodynamics, and avoiding the often unwanted heating effects of parametric amplification.