Nonlinear interaction effects in a three-mode cavity optomechanical system

Phonon-phonon interaction and parametric down-conversion generation in multimode optomechanical systems

“…We consider an optomechanical system with two membranes inside a Fabry-Perot cavity under a single monochromatic laser drive [30]. Assuming both of the two membranes have high transmission and oscillate at their equilibrium positions with coupling to the intracavity photons [31], the system Hamiltonian is given as…”

“…The parameter g i is single-photon optomechanical coupling strengths, and α is the classical drive amplitude at the frequency ω L . Transforming into the rotating frame, displacing the cavity field by a = α + d and making the standard linearization, the system Hamiltonian can be divided into a linearized part and a nonlinear interacting term as [11][12][13]30]…”

“…Nevertheless, when the nonlinear coupling is sufficiently strong (g ∼ κ, as discussed later), the nonlinear term should be taken into consideration. To study these nonlinear effects, our general approach to discuss the nonlinear interactioninduced effects is to treat the nonlinear interaction H NL as perturbations [11,13,30,32]. To keep the linear Hamiltonian in a stable range to avoid optomechanical instability [1], we focus on the red detuning case (Δ > 0) and the large detuning limit Δ ?…”

“…To keep the linear Hamiltonian in a stable range to avoid optomechanical instability [1], we focus on the red detuning case (Δ > 0) and the large detuning limit Δ ? ω mi , G. If the cavity field is close to its ground state at the initial time, we may suggest the cavity emitting into the excited state via the optomechanical couplings will have small probabilities, the cavity mode d can be eliminated adiabatically from the full Hamiltonian [30,33], and will provide an effective phonon-phonon coupling…”

“…Using the resonant condition Δ ; 2G 2 /(2ω m1 − ω m2 ) [30], one can find the optimal value of ω m2 to maximize the ratio of…”