## 1 Introduction  
In X-ray spectroscopy, the photon absorption and emission processes are usually treated within a weak photon-matter interaction limit, which implies that photon-in and photon-out processes are linearly correlated. However, for strong enough X-ray fields, the nonlinear regime may be accessed and multi-photon processes are possible, leading to a nonlinear dependence of the X-ray photons in/out signals. Such a nonlinear regime, available for many years at optical frequencies in laser spectroscopy [1; 2; 3; 4], remained an unexplored area at X-ray wavelengths due to the lack of strong enough X-ray sources. The ability to access nonlinear light-matter interactions at X-ray wavelengths became possible only recently thanks to the development of X-ray Free Electron Lasers (XFELs) [5; 6]. In contrast to the optical laser wavelengths, the photon-atom interaction at hard X-ray wavelengths involves bound core-electrons, and leads to the excitation of intermediate electronic states with sub-femtosecond lifetimes [7; 8; 9; 10]. Consequently, at femtosecond-durations, the intense pulses made available by XFELs presently allow us to access a thus-far-uninvestigated area of physics, and to probe the physical mechanisms that drive the nonlinear interaction of X-rays with matter [11; 12; 13].  
With regard to applications, the two-photon absorption (TPA) process using hard X-rays allows access to different excitation states of matter, thanks to the specific selection rules for electronic transitions. In contrast to the one photon absorption (OPA) process, which is determined by dipole-allowed transitions, the TPA process requests a change in the electron orbital quantum number by \(\pm 2\) or \(0\), thus allowing access to quadrupole or forbidden excitations. This opens new avenues for studying unexplored electronic states of matter via two-photon absorption. However, the application of TPA at XFELs for studying electronic excitations is still problematic in practice. Because of low cross-sections, the TPA process is competing with other first- and second-order photon-atom interactions, which have to be considered, in particular, for ultrashort X-ray pulses. Consequently, knowledge of the time-evolution of the population of states, as well as on the subsequent X-ray emission process for ultrashort X-ray pulses, is crucial in order to interpret the experimental results from XFELs.  
In the present paper, we combine the results of a TPA/OPA experiment performed in the Linac Coherent Light Source (LCLS) [13], in which the X-rays indicating the TPA and OPA processes were measured, with time-dependent calculations allowing us to interpret the one- and two-photon absorption process for ultrashort hard X-ray pulses. In these calculations, we used a three-level atomic state model to describe the time-evolution of a population of excited electronic states during the course of a femtosecond-duration X-ray pulse. In this way, we could follow the time dependences of population states and subsequent X-ray emission for both linear and nonlinear atomic response. The predictions of the presented model are in good agreement with the measured X-ray rates corresponding to one- and two-photon absorption in copper near the K-absorption edge. We also demonstrate a narrowing effect in population states generated through the nonlinear two-photon absorption mechanism.