# Excitation probability for 2 Level Atom

I currently started learning Quantum Optics using the book "Atom Physics" of Christopher Foot. Currently I am working through Chapter 7 and a question occured concerning the intuitively physical understanding:

Assume a two level atom with energy levels $E_{1}$ and $E_{2}$ (with $E_{1}$<$E_{2}$ is given). The eigenstates of the system are given by $|1\rangle$ and $|2\rangle$ respectively. Now assume that we perturbe the system by using a time dependent Electric field of the form $$\vec{E} = \vec E_{0} \cos\left(kz - \omega t\right)$$ and expand the solution of the Schroedinger Equation in terms of the unperturbed eigenstates $$\Psi(\vec r,t) = c_{1}(t) \ e^{-i (E_{1}/\hbar) t} \ |1\rangle + c_{2}(t) \ e^{-i (E_{2}/\hbar) t} \ |2\rangle$$ Let's assume that $c_{1}(0)=1$ and $c_{2}(0)=0$. Using the dipole approximation as well as the rotating wave approximation as discussed for example in www.physics.ox.ac.uk/Users/lucas/BIII/QuantumNotes.pdf‎, we obtain that $$|c_{2}(t)| \sim t^{2} \ \frac{\sin^{2}(x)}{x^{2}}$$ with $x=(\omega-\omega_{0})t/2$ and $\hbar \omega_{0}:=E_{2}-E_{1}$. Plotting this as a function of $\omega-\omega_{0}$ we obtain a "Fraunhofer-diffraction"-like graph.

My Question:

1) As the interaction time increases the width of the maxima is decreasing. What is from a physical / intuitive perspective the reason for this.

2) There are also certain minima occuring for certain frequencies of the light. What is the physical interpretation (intuitively) for this?

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The atom is interacting with rectangular impulse of length t, modulated with frequency of $\omega$. The more energy is pumped to the atom on its resonance frequency, the more probability of its excitation. Spectral density (i.e, the energy density) of the impulse is $$S(\omega_0) = \left| \int_{-t/2}^{t/2} dt' e^{-i\omega_0 t} \cos\omega t \right|^2 \approx \frac{sin^2 {\frac{\omega-\omega_0}2t}}{(\omega-\omega_0)^2}$$ where I've used the RWA dropping the high-frequency part of the spectra.