Quantum theory of a one-dimensional optical cavity with output coupling. II. Thermal radiation field and the fluctuation-dissipation theorem

Abstract

The normally ordered correlation function and the power spectrum (cross spectral tensor) of a thermal radiation field existing inside and outside a one-dimensional optical cavity which transmits radiation at one of its end surfaces are calculated on the basis of a multimode formulation of the field developed in a previous paper. The correlation function is found to consist of a series of trigamma functions resulting from retarded correlations due to reflections at the cavity end surfaces. The power spectrum is a product of the Planck distribution and the mode functions of the universe which have a periodic factor with the period being equal to the cavity mode separation. For the purpose of obtaining a description of the cavity which allows one to deduce its thermal noise property, a form of fluctuation-dissipation theorem is derived which connects, in a one-dimensional space with arbitrary structure, the normally ordered correlation function with the admittance for the electric field probed by a current. Application of this theorem to the model cavity leads to an explicit and simultaneous description of the fluctuation and dissipation associated with all the cavity modes. The dissipation is shown to be due to radiation of electromagnetic energy into the free space. The above relation is reduced, under the assumption of high cavity quality factor and of single-cavity-mode selection, to that of a Markovian noise associated with a single decay constant. The consistency of these results with the field commutation relation is examined.