We explore spin-orbit thermal entanglement in rare-earth ions, based on a witness obtained from mean energies. The entanglement temperature $T_{E}$, below which entanglement emerges, is found to be thousands of kelvin above room temperature for all l
ight rare earths. This demonstrate the robustness to environmental fluctuations of entanglement between internal degrees of freedom of a single ion.
In this paper we use the Nuclear Magnetic Resonance (NMR) to write eletronic states of a ferromagnetic system into a high-temperature paramagnetic nuclear spins. Through the control of phase and duration of radiofrequency pulses we set the NMR densit
y matrix populations, and apply the technique of quantum state tomography to experimentally obtain the matrix elements of the system, from which we calculate the temperature dependence of magnetization for different magnetic fields. The effects of the variation of temperature and magnetic field over the populations can be mapped in the angles of spins rotations, carried out by the RF pulses. The experimental results are compared to the Brillouin functions of ferromagnetic ordered systems in the mean field approximation for two cases: the mean field is given by (i) $B=B_0+lambda M$ and (ii) $B=B_0+lambda M + lambda^prime M^3$, where $B_0$ is the external magnetic field, and $lambda, lambda^prime$ are mean field parameters. The first case exhibits second order transition, whereas the second case has first order transition with temperature hysteresis. The NMR simulations are in good agreement with the magnetic predictions.
