Fig. 3

Evolution of \(\left\langle J^{2}(n) \right\rangle _w\) with time n for the disordered system with kicks \((p=\pi /2)\) and without kicks \((p=0)\) for two different values of disorder strength \(w=1.5,~2.6\), initial state \({|{\theta ,\phi }\rangle }={|{2.25,1.1}\rangle }\), \(k=1\), and \(N=12\). The points correspond to numerical data and the dotted lines correspond to the analytical expression given in Eq. (11) for the unkicked Hamiltonian. The black horizontal lines is \(\langle J^2 _{RMT} \rangle =3N/4=9\) which is well below the saturation value. For \(p=0\), the saturation value clearly is independent of the strength of disorder, but depends on the initial state, as also predicted by Eq. (11). On the other hand, when \(p=\pi /2\), saturation value depends upon w and initial state.