Putting Humpty-Dumpty back together: characterizing coherent recombination in Stern-Gerlach interferometers

Abstract

I show that superficially similar implementations of Stern-Gerlach Interferometers (SGIs) are expected to differ dramatically in their sensitivity to fields transverse to the primary acceleration direction. These transverse fields unavoidably accompany any static magnetic or electric field gradients, and have been shown by Comparat [Phys. Rev. A101, 023606 (2020)] to limit the precision application of SGIs. As a concrete example, I consider SGIs with ultracold Rb Rydberg atoms accelerated by spatially-varying electric fields and find that the deleterious effect of transverse fields imply that only some implementations (sequences of field gradients, internal state swaps and so-on) may exhibit fringes with high visibility. I further show that these differences are not strongly dependent on the form of the initial state.

I provide a derivation of the Humpty-Dumpty equation for a general initial state and show that it holds for any interferometry sequence where the force as a function of time is piecewise constant. A modified version of the equation is shown to hold for any general sequence with a linear potential. I then extend this analysis to the transverse components of the described SGI, and give a form for the time evolution operator that is analogous to the one used in deriving the Humpty-Dumpty equation.