## 1 Introduction  
The dynamics of photo-excited states is one of the most fascinating topics in materials science. One reason is that the control of functionalities by light, such as a photoinduced phase transition, directly connects with the development of optical devices [1; 2; 3]. The recent remarkable progress of time- and energy-resolution in pump-probe spectroscopy helps the precise understanding of transient states after the photo-excitation [4; 5]. Information on the precise time-evolution of the photo-excited states with fine energy and momentum resolutions is of course important for their application. In addition, such precise information is now giving another stage for investigation of more fundamental problems. For example, it is well known that fermions in solids occupy the band states to make the total energy the lowest. As a result, the states below the Fermi energy are completely occupied, but those above the Fermi energy are unoccupied at the zero temperature. This is a well-established phenomenon called Fermi degeneracy. However, it is not clear how the fermions go to Fermi degeneracy and how long it takes. The Franck-Condon principle is another example. In this principle, an electron is adiabatically excited by light without any momentum exchange to phonons. However, this does not mean that phonons never contribute to the photo-excitation. What happens if the photo-excited states are initially coupled with phonons before excitation? This must be a question to be treated within the Franck-Condon principle. Experimentally, two-photon photoemission spectroscopy (2 PPES) can answer these fundamental questions. A macroscopic number of electrons are excited to the initially empty conduction band by the first light. Then, the second light works for photoemission spectroscopy on these photo-excited electrons relaxing in the conduction band. By changing the time-duration between the first and second lights, we can obtain information on how electrons form Fermi degeneracy in the conduction band. Similarly, the intervalley transition of the photo-excited electrons can be observed by 2 PPES. We suggest that this intervalley transition can be caused by the electron-phonon coupling in the initial electronic states before photo-excitation within the framework of the Franck-Condon principle. Thus, one of the most significant highlights in the recent progress of the optical measurement techniques is the realization of time- and momentum-resolved 2 PPES. As mentioned above, 2 PPES can give qualitatively different data of the photo-excited carriers from the ordinary optical measurements such as reflectance and transmittance. In fact, the recent 2 PPES experiments have revealed the novel dynamics of photo-excited electrons, such as Fermi degeneracy formation and the intervalley transition within the Franck-Condon principle [6; 7; 8; 9; 10]. In this paper, following the state-of-the-art 2 PPES measurements, we review the recent progress on the theoretical study of the real-time dynamics of the photo-excited electrons in the conduction band of semiconductors. Especially, we focus on two major channels of the carrier dynamics, namely intravalley and intervalley transitions. The former is closely related to the Fermi degeneracy formation [11], while the latter is related to the Franck-Condon principle in the photo-excitation [12]. Although we are mainly concerned with GaAs, the basic results and concepts would be widely applicable to a variety of other materials. We have succeeded in describing the 2 PPES experiments and giving reasonable physical pictures to them. In both cases, the electron-phonon coupling plays a very important role. In the intravalley transition, for example, photo-excited electrons can change their energies and momenta by the electron-phonon scattering, and they can occupy the lower parts of the conduction band very quickly.