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Spin Echo in Synchrotrons

As a polarized beam is accelerated through a depolarization resonance, its polarization is reduced by a well-defined calculable reduction factor. When the beam subsequently crosses a second resonance, the final beam polarization is considered to be reduced by the product of the two reduction factors corresponding to the two crossings, each calculated independently of the other. This is a good approximation when the spread of spin precession frequency {Delta}{nu}{sub spin} of the beam (particularly due to its energy spread) is sufficiently large that the spin precession phases of individual particles smear out completely during the time {tau} between the two crossings. This approximate picture, however, ignores two spin dynamics effects: an interference effect and a spin echo effect. This paper is to address these two effects. The interference effect occurs when {Delta}{nu}{sub spin} is too small, or when {tau} is too short, to complete the smearing process. In this case, the two resonance crossings interfere with each other, and the final polarization exhibits constructive or destructive patterns depending on the exact value of {tau}. Typically, the beam's energy spread is large and this interference effect does not occur. To study this effect, therefore, it is necessary to reduce the beam energy spread and to consider two resonance crossings very close to each other. The other mechanism, also due to the interplay between two resonance crossings, is spin echo. It turns out that even when the precession phases appear to be completely smeared between the two crossings, there will still be a sudden and short-lived echo signal of beam polarization at a time {tau} after the second crossing; the magnitude of which can be as large as 57%. This echo signal exists even when the beam has a sizable energy spread and when {tau} is very large, and could be a sensitive (albeit challenging) way to experimentally test the intricate spin dynamics in a synchrotron. After giving an analysis of the interference and the echo effects, two possible experiments to explore them are suggested.