Cooling of mechanical oscillators is an important direction of cavity optomechanics research. Cooling the mechanical oscillators to their quantum ground state is a prerequisite for a wide range of applications based on cavity optomechanics. Therefore, ground-state cooling of mechanical oscillators is the focus of cavity optomechanics at present, which attracts a large number of scholars to study it. However, due to noise interference from external environments, the mechanical oscillators cannot enter the quantum regime. The hybrid system-coupled optical parametric amplifier provides a unique platform to solve the above problem.

The hybrid optomechanical system consists of two fixed mirrors FM including a rotational mirror RM mounted on the support S which can rotate around the Z axis, and an OPA medium. Cavity 1 which couples the OPA medium is made up of partially transparent FM1 and perfectly reflecting RM while cavity 2 is composed of FM1 and another perfectly reflecting FM2. The cavity 1 is driven by the transmitted beam with charge 0 and a Laguerre-Gauss beam G of charge 0 is incident on FM1. The charge 0 beam reflected from the RM is charged to +2l and then returns to FM1, where a mode with charge 0 is generated and enters cavity 2. After the reflection of FM2, it is also charged to +2l. We study the problem of the intracavity-squeezed cooling in the optical parametric amplifier coupled by a double Laguerre-Gaussian-cavity optomechanical system by calculating the optical force noise spectrum and the steady-state final phonon number. In the weak coupling regime, the optical force noise spectrum of the system is obtained by the perturbation approximation method, and the analytical expression of the final phonon number is calculated by the Fermi Golden Rule theory.

When the OPA medium is considered in the hybrid optomechanical system, the heating rate of the optical noise spectrum SFFω at ω=-ω? is reduced to 0, with an unaffected cooling rate. In other words, A+ drops while A- remains the same, the net cooling rate Γ=A--A+ naturally becomes larger, and the cooling effect is improved (Fig. 2). Next, we proceed to study how the optical noise spectrum SFFω is affected by the coupling strength J between two cavities. The value of SFFω at ω/ω?=1 is greater in the presence of the auxiliary cavity (Fig. 3). We depict the variations of the optical noise spectrum SFFω with ω/ω? for a given coupling strength J when Δc1=-ω?, Δc1=-2ω?, Δc1=-2.5ω?, and Δc1=-3ω?. The right-hand peak of the optical noise spectrum SFFω is observed to move rightward with the decreasing effective detuning Δc1. As a result, a suitable set of effective detuning Δc1 and coupling strength J can be chosen to make sure that the location of the right peak of the optical noise spectrum is at ω=ω?, which can greatly enhance the cooling process as much as possible (Fig. 4). Fig. 5(a) illustrates the optical noise spectrum SFFω as a function of ω/ω? for three different decay rates κ2. As shown in Fig. 5(a), the value of the optical noise spectrum SFFω at ω=ω? notably rises with the reducing κ2, which means that the decay rate decrease of the auxiliary optical cavity helps promote the cooling process. Meanwhile, SFFω goes down to zero at ω=-ω?, which indicates that the heating is completely suppressed whether the decay rate κ2 is changed or not. As exhibited in Fig. 6, the influence of the different optical coupling strengths J on the net cooling rate Γ is plotted. With the increasing coupling strength J, the net cooling rate Γ first rises to a maximum value and then decreases. Additionally, the net cooling rate Γ is significantly reinforced when the OPA medium is added. Subsequently, we investigate the final phonon number nf versus the coupling strength J with or without the OPA medium. With the increasing coupling strength J, the final phonon number nf will first decrease and then increase. Notably, as the coupling strength J rises, the final phonon number nf of RM drops to markedly less than 1 in the presence of an OPA medium (Fig. 7). Meanwhile, the final phonon number can be less than 1 by regulating the detuning of the auxiliary cavity (Fig. 8) and the decay rate of the cavity field (Fig. 9) respectively.

We propose an intracavity-squeezed cooling scheme to achieve a quantum ground state of RM in a double-Laguerre-Gaussian cavity optomechanical system comprising of an OPA medium. We demonstrate that the quantum backaction heating can be completely suppressed by adding OPA and the cooling efficiency is improved by coupling the auxiliary cavity. Further, the perfect cooling effect can be remarkably accomplished by selecting appropriate coupling strength, effective detuning, and decay rate, respectively. The restriction on the auxiliary cavity in the hybrid system is considerably loosened with the help of OPA. These results may have potential applications for achieving the quantum ground-state of mechanical resonators and greatly promote the study of various quantum phenomena in mechanical systems.