Exploring the quantum frontier of spin dynamics

Our familiar classical concept of a \textit{spin} is that of a system characterized by the \textit{direction} in which the spin is \textit{pointing.} In this picture, we may think of the dynamics of a spin as the motion of a \textit{classical gyroscope}, wich we can aptly describe the spin dynamics as the motion of a point on a sphere. This classical description of the spin dynamics, formalized in the Landau-Lifshits-Gilbert equation, has proved extremely successful in the field micro- and nanomagnetism. However, as the size of the system is further decreased (e.g., when considering molecular magnets such as the Fe$_{8}$ or Mn$_{12}$ systems, which have a spin $S$=10), \textit{quantum} effects such as tunneling, interference, entanglement, coherence, etc., play an essential role, and one must adopt a fully quantum mechanical description of the spin system. The landscape in which the system evolves is then no longer a mere sphere, but rather it is the projective Hilbert space (wich is the projective complex space $\le $P$^{2S}$ for a spin $S)$, as space of considerably greater richness and complexity than the sphere of classical spin dynamics. A very appealing tool to describe a quantum spin system is Majorana's stellar representation, which is the extension for a spin $S$ of the Bloch sphere description of a spin $\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $. I shall discuss how this representation can help us in improving our understanding of fundamental quantum processes and concept such as Landau-Zener transitions, Rabi oscillations, Berry phase, diabolical points and illustrate this on the example of spin dynamics of molecular magnets.