| The Casimir efect is a typical macroscopic quantum phenomenon arising from azero-point energy change of the electromagnetic field around material boundaries. Atypical manifestation of the efect is an attraction between two objects with a micron orsubmicron gap. Recently, many works are devoted to study the influences of the prop-erties and shape of the boundary materials, roughness of the surface and temperature onthe Casimir force. In particular, the Casimir force is not always attractive. For example,it is found theoretically that the Casimir force between two artificial bulk materials, i.e.the chiral metamaterials and the topological insulators, can be repulsive with a certaingap. However, in general the thickness of the artificial material is always finite. Thus,the influence of the finite thickness of the materials on the Casimir repulsion and attrac-tion needs to be investigated in detail. The study and application of the Casimir forcein a micro-and nano-mechanical system receive an extensive attention. In this thesis,we mainly investigate the attractive and repulsive Casimir force between two finite-thicktopological insulator slabs. Further, we investigate the application of the Casimir force inan optomechanical system and a thermally driven mechanical system. This thesis mainlyincludes the following three parts:In the first part, we numerically investigate the the Casimir force between two finite-thick topological insulator slabs by using the scattering formula. Two typical substratematerials including semi-ifinite vacuum and silicon are used in the study, in which theCasimir force can always change from attractive to repulsive when the gap size decreases.The gap width at transition is a function of the slab thickness and also depends stronglyon the electric permittivity and topological magnetoelectric polarizability of the slabs.The characteristic features of the Casimir force may be detected experimentally throughexploring its gradient with a certain dynamical method.In the second part, we establish a kind of the optomechanical system with theCasimir efect, where a dielectric nanosphere is trapped near the cavity mirror by externaldriving laser beams. We study the influence of the Casimir force between the nanosphreand the cavity mirror on the dynamics of the nanosphere’s center-of-mass (c.m.) motionand the oscillation frequency. In addition, the steady-state characteristics of the optome-chanical system and the optical spring efect of the nanosphere are investigated in detail. Based on the linearized dynamics of the quantum fluctuation of the cavity field and themechanical mode around the semiclassical fixed point, the steady-state optomechanicalentanglement between the mechanical and cavity field modes and the ground-state cool-ing of the levitated nanosphere are analyzed in detail. The Casimir efect leads to anoscillation frequency shift of the nanosphere in the optomechanical system, which couldbe used for estimating the magnitude of the Casimir force between a sphere and a plane.In the third part, we investigate a fluctuation-driven ratchet-oscillator system thatconsists of two ratchet pinions in contact with thermal baths at diferent temperatures.Coupling between noncontact parts of the system is mediated by the Casimir forcethrough a thin corrugated plate. Due to mutual rectification, the two ratchets achievedirected average motion in opposite directions. By using the stochastic dynamics sim-ulation, we numerically probe the average velocity, the Peclet number, and the thermalefciency of the system as functions of the efective temperatures of the thermal bath-s and other important dimensionless control parameters. The optimal values of somecontrol parameters obtained here may help in designing noncontact power devices in thefuture and probing the essential role of the Casimir efect in micro-and nano-mechanicalsystems. |
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