Quantum Octahedral Fractal via Random Spin-State Jumps

Quantum jumps of spin 1/2 states on the Bloch sphere are realized by Möbius transformations. Chaos games produced by a random sequence of six noncommuting transformations centered at the vertices of the octahedron result in a fractal pattern on the sphere. The parameter , , defines the sharpness of the spin-direction measurements.

The algorithm for computing Platonic quantum fractals in general: define vertices of a regular polyhedron (there are five Platonic solids). Choose a fixed parameter , , which defines the "sharpness" of the fractal. Then select a random starting point on the unit sphere. Randomly select a vertex index ; let (one of the possible vertices). Apply the transformation

.

This is a Möbius transformation. It gives a new starting point on the sphere. Iterate the algorithm 10000 or more times.

Plot the first two coordinates of each point. The Möbius transformation theoretically maps the unit sphere onto itself. Yet after many iterations due to numerical inaccuracies, the norm can start to diverge. Therefore, it is better to normalize after each transformation.

For more details about Platonic quantum fractals, see [1] and [2].