0.9 Exercise 9. Consider a three-level atom with two (degenerate) low-lying states (0) and (1) with zero energy, and a high level (e) (the "excited" state) with energy ħw. The low levels are coupled to the excited level by optical fields Omega 0 cos omega 0 t and Omega 1 cos omega 1 t. respectively. (a) Give the (time-dependent) Hamiltonian H for the system. (b) The time dependence in H is difficult to deal with, so we must transform to the rotating frame via some unitary transformation U(t) Show that H^ prime = U(t) * H U^ dagger (t)-i hbar U dU^ dagger dt . You can use the Schrödinger equation with | psi rangle=U^ dagger | psi' rangle. (c) Calculate H' if U(t) is given by U(t) = [[1, 0, 0], [0, e ^ (- i * (omega_{0} - omega_{1}) * t), 0], [0, 0, e ^ (- i * omega_{0}*t)]] Why can we ignore the remaining time dependence in H' ? This is called the Rotating Wave Approximation. (d) Calculate the lambda = 0 eigenstate of H' in the case where omega_{0} = omega_{1} (e) Design a way to bring the atom from the state (0) to (1) without ever populating the state (e).