In this paper we present the full theoretical model of a modified Ives-Stillwell experiment where counter propagating lasers are used to form a narrow interference fringe when the lasers form a double resnonance. This narrow resoannce can be as small as 1 Hz wide and therefore provides a connection between the atomic resonance in its rest frame and the laser frequency in the lab frame. The current paper builds on a simplified approach suggested recently and presents a fully developed theory of the interaction within the Lorentz vilating electrodynamics of the Standard Model Extension.
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