<PERSON> spectroscopy of atoms -- <PERSON> Picture
I was reading this thesis on the <PERSON> spectroscopy, which deals with <PERSON> spectroscopy of a two level system using the <PERSON> picture. This topic is dealt with in the Appendix A.1 of the given thesis.
I think I understand how <PERSON> spectroscopy works in the <PERSON> picture -- A $\pi/2$ pulse is given (but the $\pi/2$ is not resonant with the atomic transition frequency, hence the atom enters an improperly mixed ground and excited state). A free time evolution is done for a time $T$, and then the system is given another $\pi/2$ pulse (again, non-resonant but with the same frequency). By measuring the relative number of atoms in the excited state and ground state after the pulse-evolve-pulse sequence, we are able to calculate the number of atoms in the ground and excited states, and hence the phase difference and eventually the difference in frequency of the source and the resonant frequency of the atom ($\omega-\omega_0$).
However, in this thesis, as I mentioned earlier, the topic of <PERSON> spectroscopy is dealt with in terms of the <PERSON> picture, which I find a bit confusing. 346
They define angular momentum $J_x$, $J_y$ and $J_z$ operators based on the energy levels $|e>$ and $|g>$, which refer to the excited and ground state energies respectively. It is after this part that it gets confusing. I would like to mention that in this thesis, they have considered a $\pi/2$-evolve-$(-\pi/2)$ sequence instead.
They proceed to say that the $\pi/2$ pulse rotates the angular momentum vectors by $\pi/2$ about the x-axis, the free evolution makes the net vector rotate about the z-axis with an angular velocity which is $\omega-\omega_0$ and another $(-\pi/2)$ pulse.
My questions about the explanation are:
i. Isn't the $\pi/2$ pulse only approximate, and that the entire reason for the phase difference comes from the fact that it is not an exact $\pi/2$ pulse? How can they assume that the original pulse exactly rotates the angular momentum operators by $\pi/2$?
ii. How does the time evolution of the pulse when it is propagating freely without any interaction, in the evolve phase, depend on the frequency of the probe light used ($\omega$)? Since the Hamiltonian is just the free Hamiltonian with no interaction between the levels, why does the probe light come in? Shouldn't they just evolve with a frequency $\omega_0$? 101