Coherence theory and coherence phenomena in a closed spin-1/2 system

A simplified Heisenberg spin model is studied in order to examine the idea of decoherence in closed quantum systems. For this purpose, we present a quantifiable definition to quantum coherence $\Xi$, and discuss in some detail a general coherence theory and its elementary results. As expected, decoherence is understood as a statistical process that is caused by the dynamics of the system, similar to the growth of entropy. It appears that coherence is an important measure that helps to understand quantum properties of a system, e.g., the decoherence time can be derived from the coherence function $\Xi(t)$, but not from the entropy dynamics. Moreover, the concept of decoherence time is applicable in closed and finite systems. However, in most cases, the decay of off-diagonal elements differs from the usual $\exp(-t/\tau_{\rm d})$ behaviour. For concreteness, we report the form of decoherence time $\tau_{\rm d}$ in a finite Heisenberg model with respect to the number of particles $N$, density $n_{\rho}$, spatial dimension $D$ and $\epsilon$ in a $\eta/r^{\epsilon}$-type of potential.