Abstract
Solutions to the Thomas-Bargmann-Michel-Telegdi spin equation for spin $1/2$ particles have to date been confined to the single-resonance crossing. However, in reality, most cases of interest concern the overlapping of several resonances. While there have been several serious studies of this problem, a good analytical solution or even an approximation has eluded the community. We show that this system can be transformed into a Hill-like equation. In this representation, we show that, while the single-resonance crossing represents the solution to the parabolic cylinder equation, the overlapping case becomes a parametric type of resonance.
Highlights
Two-level systems have been much studied in various branches of physics and have presented themselves early on in the development of quantum mechanics
In the accelerator physics community, similar problems emerge from the Thomas-BMT (Bargmann, Michel, and Telegdi) equation used to model the spin dynamics which particles undergo in a beam line
In 2004, with some limited degree of success, Mane [5] developed an approximation based on the first-order Magnus expansion, and applied, ad hoc, a modified resonance strength to the Froissart-Stora formula
Summary
Two-level systems have been much studied in various branches of physics and have presented themselves early on in the development of quantum mechanics. A well-developed theoretical apparatus exists to handle the various spin-depolarizing resonances It rests upon our understanding of solutions to the single-resonance model via the FroissartStora formula [1] for accelerating particles and harmonic oscillations for the stationary case. Experience with a lattice designed to accommodate the phase advance necessary for the operation of the electron lens beam-beam compensation appears to show a reduced polarization transmission during the acceleration ramp This new lattice has raised the strength of the neighboring resonances even while reducing the strong intrinsic resonance [6]. Might be useful to develop a single turn spin map with snakes which can provide solutions for the cases with rational tunes and perhaps irrational tunes following Mane’s approach
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