# A method for solving gyroscopic equations with operators

## TL;DR

This paper develops a comprehensive formalism to solve gyroscopic equations involving operator-valued Larmor vectors, revealing how initial spin states are partially preserved and how final polarization depends on coupling and spin nature.

## Contribution

A general solution method for gyroscopic equations with operator Larmor vectors applicable to all spin values, including detailed applications and insights into spin state memory.

## Key findings

- Initial fully polarized states retain partial memory after evolution.
- Final polarization can be tuned by adjusting the coupling constant.
- The robustness of initial states depends on fermionic or bosonic nature and system size.

## Abstract

The dynamics of a set of identical spins interacting with another one through a time-dependent coupling gives rise to a gyroscopic equation with a variable Larmor frequency and, more importantly, with an operator playing the role a Larmor vector. The subsequent technical complexity is due to non-trivial algebraic relations between multiple inner products coming from the non-commutative algebra of the angular momenta. A general formalism is derived giving the integrated solution valid for all values of the involved spins, and several applications of the formalism are treated in details. Among other results, it is seen that, starting from a fully polarised state for the set of identical spins, their total spin can at most only partially flip (in the mean); this somewhat surprising fact means that the memory of the initial state is kept for ever but varying the coupling constant allows to adjust at will the possible polarisation of the final state. The robustness of the initial state is shown to depend on the nature fermionic or bosonic of the perturbing spin and also on the size of the collection of identical spins.