Advanced computational methods for nonlinear spin dynamics

We survey methods for the accurate computation of the dynamics of spin in general nonlinear accelerator lattices. Specifically, we show how it is possible to compute high-order nonlinear spin transfer maps in SO(3) or SU(2) representations in parallel with the corresponding orbit transfer maps. Specifically, using suitable invariant subspaces of the coupled spin-orbit dynamics, it is possible to develop a differential algebraic flow operator in a similar way as in the symplectic case of the orbit dynamics. The resulting high-order maps can be utilized for a variety of applications, including long-term spin-orbit tracking under preservation of the symplectic-orthonormal structure and the associated determination of depolarization rates. Using normal form methods, it is also possible to determine spin-orbit invariants of the motion, in particular the nonlinear invariant axis as well as the associated spin-orbit tune shifts. The methods are implemented in the code COSY INFINITY [1] and available for spin-orbit computations for general accelerator lattices, including conventional particle optical elements including their fringe fields, as well as user specified field arrangements.