Temperature gradient and asymmetric steady state correlations in dissipatively coupled cascaded optomechanical systems

The interaction between a light mode and a mechanical oscillator via radiation pressure in optomechanical systems is an excellent platform for a multitude of applications in quantum technologies. In this work we study the dynamics of a pair of optomechanical systems interacting dissipatively with a wave guide in a unidirectional way. Focusing on the regime where the cavity modes can be adiabatically eliminated, we derive an effective coupling between the two mechanical modes and explore the classical and quantum correlations established between the modes in both the transient and the stationary regime, highlighting their asymmmetrical nature due to the unidirectional coupling. Noteworthy, we find that a constant amount of steady correlations can exist at long times. Furthermore we show that this unidirectional coupling establishes a temperature gradient between the mirrors, depending on the frequencies' detuning. We additionally analyze the power spectrum of the output guide field and we show how, thanks to the chiral coupling, from such spectrum it is possible to reconstruct the spectra of each single mirror.