The Dark State And Quanum Many-Body Simulation With Quadratic Optomechanics

Cavity optomechanics, which describes the radiation pressure between cavity field mode and mechanical oscillator mode, has attracted great interest in recent years. On one hand, it provides a promising platform for manipulating quantum states of a me-chanical system in a quantum region, e.g., the cooling of a mechanical oscillator, prepa-ration of non-classical states. All of them are the basic steps for applications in quantum information processing. On the other hand, the optomechanical systems are versatile in building hybrid quantum devices that combine different physical systems, which offer a chance for investigating kinds of phenomenon of quantum mechanics in the mechanical system. This thesis is divided into three parts. In part one, we will introduce the situa-tion of cavity optomechanics. Then in part two, we will discuss the nonlinear dark state with on mechanical system. In part three, we will present a scheme with N oscillators for simulation the extended Bose-Hubbard model.Part two consider a hybrid system consisting of a cavity optomechanical device with nonlinear quadratic radiation pressure coupled to an atomic ensemble. By consid-ering the collective excitation, we show that this system supports nontrivial, nonlinear dark states. The coupling strength can be tuned via the lasers that ensure the population transfer adiabatically between the mechanical modes and the collective atomic excita-tions in a controlled way. In addition, we show how to detect the dark-state resonance by calculating the single-photon spectrum of the output fields and the transmission of the probe beam based on two-phonon optomechanically induced transparency. Possible application and extension of the dark states are also discussed.In part three, We present a scheme to realize (extended) Bose-Hubbard model in an N-coupled optomechanical system. By treating the cavities as intermediary and eliminating them adiabatically with condition of large detuning or fast decay, we can obtain the effective Hamiltonian for the N oscillators, with the regular terms in the Bose-Hubbard model, the pair tunnelings and the density-density interactions. Then we verify and provide the condition for our approximation with numerical results. Due to the existence of the pair tunnelings and the density-density interactions, we can investigate the density wave and supersolid phases in our model. Moreover, we also discuss the competition between the regular tunneling and the pair tunneling.