Dimensional Reduction in Quantum Optics

One-dimensional quantum optical models usually rest on the intuition of large scale separations associated with the different spatial dimensions, for example when studying quasi one-dimensional atomic motion, potentially resulting in the violation of 3+1D Maxwell's theory. Here, we present a rigorous foundation for this approximation by means of the light-matter interaction. We show how the quantized electromagnetic field can be decomposed - without approximation - into an infinite number of one-dimensional subfields when studying axially symmetric setups, such as a fiber cavity, a laser beam or a wave guide. The dimensional reduction approximation then corresponds to a truncation in the number of such subfields that in turn, when considering the interaction with for instance an atom, corresponds to an approximation to the atomic spatial profile. We explore under what conditions the standard dimensional reduction approximation of a single subfield is justified, and when corrections are necessary in order to account for the dynamics due to the neglected spatial dimensions. In particular we will examine what role vacuum fluctuations play in the validity of the approximation.