\f5. Consider a two-state system driven by an optical light field, under the dipole approximation the time-dependent Hamiltonian is given by H (t) - Aoz - H . E(t).. where E(t) = 2E. cos(wt) is the electric field. . Write the equations of motion for the density matrix, p and give the conditions under which it can be described using a time independent or effective Hamiltonian. . Add to your equations of motion a term describing the dephasing and show that at long time the populations relax to some steady-state values. Use pil = 1 and p22 = 0 for the initial conditions. Show that if the pumping is strong enough that one can achieve a population inversion in a two-level system