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Theory of optomechanics: Oscillator-field model of moving mirrors

In this paper we present a model for the kinematics and dynamics of optomechanics which describe the coupling between an optical field, here modeled by a massless scalar field, and the internal (e.g., determining its reflectivity) and mechanical (e.g., displacement) degrees of freedom of a moveable mirror. As opposed to implementing boundary conditions on the field we highlight the internal dynamics of the mirror which provides added flexibility to describe a variety of setups relevant to current experiments. The inclusion of the internal degrees of freedom in this model allows for a variety of optical activities of mirrors from those exhibiting broadband reflective properties to the cases where reflection is suppressed except for a narrow band centered around the characteristic frequency associated with the mirror's internal dynamics. After establishing the model and the reflective properties of the mirror we show how appropriate parameter choices lead to useful optomechanical models such as the well known Barton-Calogeracos model [G. Barton and A. Calogeracos, Ann. Phys. 238, 227 (1995)] and the important yet lesser explored nonlinear models (e.g., $Nx$ coupling) for small photon numbers $N$, which present models based on side-band approximations [H. Kimble et al., Phys. Rev. D 65, 022002 (2001)] cannot cope with. As a simple illustrative application we consider classical radiation pressure cooling with this model. To expound its theoretical structure and physical meanings we connect our model to field-theoretical models using auxiliary fields and the ubiquitous Brownian motion model of quantum open systems. Finally we describe the range of applications of this model, from a full quantum mechanical treatment of radiation pressure cooling, quantum entanglement between macroscopic mirrors, to the backreaction of Hawking radiation on black hole evaporation in a moving mirror analog.