Dimensional reduction of cavities with axial symmetry: A complete analysis of when an optical fiber is approximately one-dimensional

Intuition dictates that a very long, very thin cavity (e.g., a fiber optic cable) could perhaps be modeled as an approximately one dimensional system. In this paper we rigorously explore the validity of such intuition from the perspective of a localized probe coupling to a quantum field inside a cavity (e.g., an atom or an Unruh-DeWitt particle detector in a fiber optic cable). To do so, we introduce the notion of subfield decomposition in which a $D+1$ dimensional quantum field in an axially-symmetric cavity can be reduced to an infinite collection of uncoupled, massive $1+1$ dimensional fields. We show that the ability to approximate a higher-dimensional scenario by a $1+1$ dimensional model is equivalent to making a certain change of the probe's shape in the higher-dimensional space. The approximation is justified whenever this change of shape is "small enough". In this light, we identify the dynamically relevant norm by which the magnitude of these changes in probe shape ought to be judged. Finally, we explore this approximation in particular setups corresponding to quantum optics and superconducting circuits.
Comment: 14 Pages, 4 Figures. RevTeX 4.1