Emergence of classical behavior from the quantum spin

A classical Hamiltonian system of a point moving on a sphere of fixed radius is shown to emerge from the constrained evolution of quantum spin. The constrained quantum evolution corresponds to an appropriate coarse graining of the quantum states into equivalence classes, and forces the equivalence classes to evolve as single units representing the classical states. The coarse-grained quantum spin with the constrained evolution in the limit of the large spin becomes indistinguishable from the classical system.

Figure 1

Illustration of the constrained (dotted line) and classical (solid line) evolution for $J=$ (a) 5, (b) 10, (c) 20, and (d) 30. Time series $J_z\left(t\right)=\langle \Omega \left(t\right)|{\widehat{J}}_z|\Omega \left(t\right)\rangle$ are shown. All quantities are dimensionless. Parameters are $\lambda =\mu =1,\epsilon =0$ and the initial state is the coherent state $|{\Omega }_0\rangle =(\theta =\pi /2,\phi =0)$.