Highly enhanced steady-state optomechanical entanglement via cross-Kerr nonlinearity

Entanglement [1, 2] is one of the most intriguing feature of quantum mechanics, having myriads of potential applications in quantum information and communication [3]. So far, preparation and manipulation of entanglement has been successfully demonstrated in microscopic systems, such as atoms , photons, ions [4–6] etc. However, the validity of entanglement in the macroscopic realm is still a debatable fact. In this regard, optomechanical system [7, 8] where a macroscopic mechanical motion interacts with an optical field has emerged as a promising platform to realize entanglement at a macroscopic level. Several studies have been reported to generate entanglement between optical and mechanical mode or two optical modes or two mechanical modes [9–34]. These studies mostly explore entanglement in the steadystate regime of the system where the strength of the entanglement is strictly limited by the stability conditions. In particular, in this regime entanglement becomes maximum only when the system is driven close to the stability threshold, which demands a strong multi-photon optomechanical coupling or high external laser driving [11]. Recently, it has been proposed [35] and experimentally demonstrated [36] that a two-level system or a nonlinear inductive element (single-Cooper pair transistor) coupled to an optomechanical system, can significantly enhance the single-photon radiation-pressure coupling. More importantly, it induces an additional cross-Kerr type coupling between the optical and mechanical mode. In nonlinear optics, cross-Kerr effect describes the change in refractive index of one electromagnetic mode by the intensity of another. It has been extensively studied in the context of quantum information processing, such as nondemolition photon number detection [37], C-NOT gate [38] and continuous-variable entanglement concentration [39–42]. Concerning optomechanical system, the crossKerr coupling between the optical and mechanical mode, involves the change in the refractive index of the optical mode depending on the resonators phonon number