Quantum noise in the mirror–field system: A field theoretic approach

Abstract

We revisit the quantum noise problem in the mirror–field system by a field-theoretic approach. Here a perfectly reflecting mirror is illuminated by a single-mode coherent state of the massless scalar field. The associated radiation pressure is described by a surface integral of the stress-tensor of the field. The read-out field is measured by a monopole detector, from which the effective distance between the detector and mirror can be obtained. In the slow-motion limit of the mirror, this field-theoretic approach allows to identify various sources of quantum noise that all in all leads to uncertainty of the read-out measurement. In addition to well-known sources from shot noise and radiation pressure fluctuations, a new source of noise is found from field fluctuations modified by the mirror’s displacement. Correlation between different sources of noise can be established in the read-out measurement as the consequence of interference between the incident field and the field reflected off the mirror. In the case of negative correlation, we found that the uncertainty can be lowered than the value predicted by the standard quantum limit. Since the particle-number approach is often used in quantum optics, we compared results obtained by both approaches and examine its validity. We also derive a Langevin equation that describes the stochastic dynamics of the mirror. The underlying fluctuation–dissipation relation is briefly mentioned. Finally we discuss the backreaction induced by the radiation pressure. It will alter the mean displacement of the mirror, but we argue this backreaction can be ignored for a slowly moving mirror.