#### 2.1 Temporal scale and mechanisms

Interestingly, the story of ultrafast magnetization dynamics for the past 15 years has been a constant swing between results involving the concepts of either “thermal processes” associated to the non equilibrium electrons, spins and phonons excited by a short laser pulse and “coherent processes” associated to the orbital and spin angular momenta. Let us trace back some works which stress these two fundamental notions.

The archetypal observation of an ultrafast change of the magnetization induced by femtosecond laser pulses is the one that we reported in 1996 in a 22 nm thick ferromagnetic Nickel film excited by pulses of 60 fs duration, using a pump-probe time resolved Magneto-Optical Kerr (TR-MOKE) configuration [1]. Figure 1, shows the corresponding time dependent change of the magnetization at remanence observed in this experiment. Soon confirmed by several other works [2-5], this result triggered the interest regarding an important question: How fast can a magnetic change occur in a thin film or structure displaying a magnetic order? Obviously the experiment was showing that spins are involved on the sub-picosecond time scale which is typically the time for electrons to equilibrate with the lattice in a metal excited with ultrashort laser pulses [47-49]. It shows the importance of “thermal processes” which can be explained qualitatively using a three temperature model, where the electrons (charges), the spins and the phonons have transient different temperatures, without further justification of the coupling between the charges and the spins. Soon later, an interesting experiment showed that one could take advantage of the laser induced demagnetization to initiate a motion of precession of the magnetization in a ferromagnetic/antiferromagnetic exchange/coupled system [20]. This second result showed that the exchange bias between magnetic layers is modified by the ultrafast demagnetization to the point that the spin angular momentum is affected, leading to the precession of the magnetization on a time scale of a few hundreds of picoseconds. That mechanism has been used since then as a way to optically induce the ferromagnetic resonance without the need of an external radio-frequency magnetic field [50, 51]. In addition, it has been shown to be strongly connected to the dynamical change of the material anisotropy, either via the shape or magneto-crystalline anisotropy [9].

If the angular momentum is re-oriented on such long time scale then what about the initial demagnetization? This question has become very important because it has been shown that the ultimate demagnetization time can be as fast as the thermalization time of the charges which, in the case of a CoPt_{3} thin film excited with low intensity 20 fs laser pulses, occurs within 50 fs [52]. Two consequences can be drawn. First, “thermal” processes are able to induce a full demagnetization without a significant heating of the lattice. Second, TR-MOKE experiments correctly reflect the spins dynamics. The first point has been further confirmed, by two other types of experiments establishing the importance of two main mechanisms which play a major role in reducing the magnetic order during the thermalization of the electrons. One of them is the spin-orbit interaction, as shown using the technique of time resolved X-ray Magnetic Circular Dichroism in CoPd ferromagnetic thin films [7]. The second mechanism is the reduction of the exchange splitting between majority and minority spins, as shown using the time resolved photo-emission technique on Gd films [53] or permalloy [54]. Regarding the pertinence of TR-MOKE for probing the ultrafast spins dynamics, it has been questioned after a question mark was put forward in 2000 about the importance of coherent optical processes [55]. It is now a solved problem as discussed further hereafter in the next section. Importantly, another experiment has shown that TR-MOKE does represent the spins dynamics. It consists in measuring the time dependent magnetization in a Fe-Ni ferromagnetic patterned structure both at the M edge of Nickel, using extreme UV light pulses, and in the visible, using TR-MOKE [56]. The thermalization of the spins is identical in both cases [57].

The role of the thermalization of the spins being established for inducing an ultrafast demagnetization, researchers have focused back on coherent processes. Three types of experiments have revived the idea that, in addition to “thermal” effects, one could use coherent processes for manipulating the magnetization of ordered spins systems in ultrashort times. It has been shown that one can switch a ferromagnetic Co_{70}Fe_{30} film by inducing an anisotropy with a large electric field of , with a duration of 70 fs, produced by a relativistic electron bunch at the Stanford Linear Accelerator [58]. This experiment shows that the magnetization dynamics can be modified in a coherent way by inducing an anisotropy due to the distortion of the cloud of valence electrons which couple to the spins via the spin-orbit interaction. In another experiment, our group used single optical pulses of 50 fs duration to show that the electric field of the laser pulse can modify the magneto-optical response of CoPt_{3} and Ni thin films [31]. We have attributed the effect to the electromagnetic potential of the laser field which contributes to the spin-orbit interaction in addition to the potential of the ions [32]. Finally, the coherent control of the magnetization found a nice substantiation by switching GdFeCo thin films using circularly polarized femtosecond light pulses [6]. As a result, a consensus has now been reached to consider that both coherent and “thermal” processes can be used for manipulating the magnetization in the femtosecond time scale.

To summarize the variety of mechanisms involved in the temporal aspects of the ultrafast magnetization dynamics, let us consider the time scale associated to the various interactions. Figure 2 shows several sketched interaction mechanisms. On the short times, the coherent spin-photon interaction is mediated by the dipolar transitions which selectively excite energy bands with a given orbital momentum coupled with the spins via the spin-orbit interaction. The resulting process is coherent and decays within the characteristic dephasing time *T*_{2e} of the electronic levels, which in metals is of the order of the damping of the electron plasma (∼0.1fs for the bulk and ∼1fs for surface plasmons). These elementary processes include intraband and interband transitions in the particular material considered, which often leads to a different spectral dependence of the optical and magneto-optical responses. On the same time scale quantum fluctuations of the magnetization are also important. They correspond to local changes of the magnetization, which result from the broad energy spectrum W associated to the spin distribution. Since the spin-density operator does not commute with the Hamiltonian, one expects local spin fluctuations with a typical time (for itinerant ferromagnets, the bandwidth of the spin density of states is of the order of 1 eV, corresponding therefore to fluctuations of ). On the time scale of 100 fs, the thermalization times τ_{e, s} of the charge (e) and spin (s) populations result from the redistribution in k-space of the electrons excited with a high energy above the Fermi level. The screened Coulomb interaction () and exchange interaction () are the main processes involved, leading to inelastic scattering of electrons and band filling (due to the Pauli exclusion) near the Fermi level. Note that both electrons and holes quasi-particles (particles dressed by a mean Coulomb interaction) are involved. The corresponding elementary scattering times to consider can be deduced from the time-energy Heisenberg inequality: , but the time scale of the overall process is integrated over the final density of states. Importantly, depending on the degree of electronic delocalization of the considered metal (itinerant versus localized ferromagnetism), one expects to conserve some local ordering with domains extending over a few nanometers. These domains disappear due to thermal fluctuations when the heating of the lattice occurs, mediated by the electron-phonon interaction (). The overall heating of the lattice is accomplished within a time scale of ∼ 1 ps and depends on the specific heat of the electrons and spins. Note that the electron-phonon interaction also contains a coherent contribution associated to the initial expansion of the lattice, heated by the laser pulse. When the material is thick enough, i.e. thicker than the absorption depth of light in the metal, a longitudinal acoustic pulse propagates and couples to the spins leading to interesting magneto-acoustic effects [59, 60]. On the time scale of a few picoseconds to a few nanoseconds the spins dynamics is determined by the precession of the magnetization with the two characteristic times *T*_{prec} (period of precession) and η the damping of the precession. Let us now formalize these temporal considerations, distinguishing between two classes of mechanisms: “thermal” and coherent.

#### 2.2 Coherent spin-photon processes and thermally induced ultrafast demagnetization

In this paragraph, we formalize the temporal considerations summarized above, shedding light on the important “thermal” and coherent mechanisms inducing a transient demagnetization.

2.2.1 Coherent spin-photon interaction.

The notion of coherence has two different meanings when considering the magnetization dynamics induced by laser pulses. The first one refers to the phase relationship between the laser pulse and the spin wave functions. It is a quantum effect and we name it “Coherent Spin-Photon dynamics”. The second meaning refers to the phase of the magnetization vector with respect to a sudden perturbation of the magnetic angular momentum (magnetic pulse or demagnetization) which induces a motion of precession. It is described with details further in paragraph 2.3. It is a classical effect which concerns the macro-spin associated to the magnetization vector. Let us first focus on the laser-related coherence and then on the “thermal” spins dynamics.

The coherent spin dynamics then simply results from these two interaction terms and gives rise to effects well known in ultrafast optics like coherent four-wave mixing (FWM) emission and pump-perturbed free induction decay. A detailed description of these effects has been made in our recent study of spin-photon interaction [32]. We propose here a new direction of research where all coherent processes known in optics like: self-induced transparency, four-wave mixing and conjugated mirrors, optical rectification, photon echoes, stimulated Raman scattering, coherent diffusion and coherent tomography, have their direct transposition to time resolved magneto-optics. One of these phenomena has recently been reported by our group, consisting of Magneto-Optical Four Wave Mixing generation in a Garnet thin film [64]. It is shown in Fig. 3, for the particular case of a two beam self-diffracted configuration. Let us emphasize that this aspect of the coherent photon-spin interaction is very promising for application to magnetic devices controlled by light. It does not involve a switching process but could be used in many configurations including “coherent integrated magneto-optics”.

2.2.2 Thermal effects: spin population dynamics versus three-temperatures model.

The preceding analysis based on the magneto-optical response involving the coherences can be simply pursued to the population dynamics. The three interaction terms (kinetic momentum, (S.O.)_{ion} and (S.O.)_{laser}) give rise to modifications of the population occupation numbers with spin flips that occur via the two (S.O.) coupling terms. This leads to a straightforward explanation of the ultrafast demagnetization: the change of angular momentum associated to the reduction of the total magnetization results from the spin-orbit coupling and is initiated by the laser. To give a simple idea of such effect, we show in Fig. 4 the dynamics of a hydrogen-like system (used only for the consistency of the level multiplicity). It shows the contribution of the projection of the time dependent orbital momentum and spin angular momentum to a magneto-optical Faraday signal obtained in a pump-probe femtosecond experiment, using 10 fs Gaussian pulses. The lifetime of the levels is assumed to be 150 fs.

*Analysis of an interesting and productive controversy*. Let us now clarify one important dynamical aspect of the orbital momentum and spin angular momentum . There has been recently a debate regarding the validity of modeling the time dependent magneto-optical response to obtain information about the spins dynamics [13, 65, 33, 66]. There are three main aspects in the controversy raised by P. M. Oppeneer et al. First, they mention that the probe pulse should be treated correctly in a pump-probe TR-MOKE modeling. They are right. By essence, it is a third order nonlinear process which involves two pump and one probe laser fields delayed with respect to one another. Let us note that generally speaking all orders of the fields are involved which contribute differently depending on the particular experimental geometry of the laser beams (pump-probe non-degenerate or degenerate, four-wave mixing for example). Recently [32], we have shown that the time ordering of the probe and pump fields is essential when considering coherent and population effects. It is the case for the “pump-perturbed free induction decay” (PP-FID) signal which can be viewed as a situation where the time ordered sequence of the fields creates a coherent magneto-optical grating in the magnetic material, established by the probe and the pump pulses. This grating exists only for a delay less than the dephasing time *T*_{2e}. The pump pulse is then self-diffracted on this grating and contributes to the coherent magneto-optical contribution. Note that this time ordering is different than for the polarization coherent signal where the pump arrives first. In the case of the M.O. response due to the population dynamics, which exists only when the probe pulse arrives after the pump and after the *T*_{2e} dephasing time, the probe does not play a role else than reading the populations, at second order in the pump field. In that sense G. Zhang et al. are right since their purpose was not about the fine details of the coherent magneto-optical response, which must consider properly the time ordering in the sequence of probe and pump pulses. This first point illustrates very well why, in some previous articles, some researchers have sometimes been confused between the role of the TR-MOKE signal and the actual spins dynamics.

The second aspect raised in the controversy concerns the role of the spin orbit coupling. Here, it is clear, as discussed above, that the two parts of the interaction Hamiltonian (S.O.)_{ion} and (S.O.)_{laser} play a role via the coupling of the spins to the fields of the ions and of the laser due to the dipolar interaction with the generalized momentum. This effect does induce spin-flips. Is it the only source of such spin flips? Certainly not and the reader is invited to examine further our discussion below, concerning the missing piece of the puzzle in femtomagnetism due to the dynamical exchange interaction. It is important to emphasize that, by now, the spin orbit has already been shown to play an important role. In our experiment performed in a ferromagnetic 15-nm-thick Co_{0.5}Pd_{0.5} film with perpendicular anisotropy at the femtoslicing beam line in Bessy, we could measure the time evolution of the z-projections, along the quantification axis set by the external magnetic field, of the orbital and spin angular momenta [7]. The result is unambiguous: both quantities vary with a slight delay between the two, preceding . Therefore, the dynamics of the spin flips after the laser perturbation of the orbital momentum is demonstrated experimentally. It answers a question mark put forward a while ago [67]: are the orbital and spin momenta both varying during the demagnetization? The answer is yes and our interpretation of the TR-MOKE experiment in terms of a demagnetization (with spin flips) was correct. Figure 5 shows the temporal variation of both quantities. Note that, in the present case, it is the (S.O.)_{ion} which affects the population dynamics and the coherent response is not involved (the coherence between the visible pump pulse and the X-ray probe is lost). Importantly the total momentum is not conserved as both and decrease. It shows that the dynamical magneto-crystalline anisotropy plays a major role in the dynamics. Also, one has to keep in mind that, due to the coherent spin photon interaction, the laser carries away some momentum by acquiring a very significant amount of polarization rotation and ellipticity as demonstrated in Ni and CoPt_{3} ferromagnetic films [31].

The third aspect of the controversy concerns the questions to know if, strictly speaking, the TR-MOKE fully represents or not the spins dynamics. This is an important point concerning the magnetization dynamics, with unfortunately some technical subtleties of the theoretical response that we want to popularize here. Rigorously, the pump-probe M.O. signals are represented by terms proportional to in the time ordered integrations of the density matrix, where *E*_{pr} and *E*_{pu} (respectively ) correspond to the probe and pump (resp. complex conjugate) fields. Therefore, to obtain information on the dynamical variation of the and angular momenta, one has to consider the population dynamics contained in the second order nonlinear contributions and of these quantum observables, because the first and third order populations of the density matrix are respectively constant and zero. Note that these two second order terms then lead to the third order polarization which allows deducing the TR-MOKE signals. In Fig. 6 we have represented the variation of both second order nonlinear components, as calculated with our hydrogen-like model. Clearly the spin populations are affected by the (S.O.) terms and are the quantities responsible for the TR-MOKE signals. The assumption of G. Zhang et al., even though not fully rigorous in the sense just developed above, is therefore correct. But P. W. Oppeneer et al. are also correct in the sense that, for particular materials, the temporal shape of the observable and might not be exactly the same as the ones of , from which the rotation and ellipticity signals and are deduced. It is not the case in our model because we use simple multiplicity levels with *T*_{2} and *T*_{1} transverse and longitudinal relaxation times. The experiment performed by the group of M. Murnane, which we already briefly discussed above, also shows that TR-MOKE and spins dynamics gives rise to the same dynamical behaviors in nickel [56]. In conclusion, the theoretical and experimental considerations related in this section hopefully elucidate the interesting scientific controversy.

What is missing so far? Firstly, the above considerations do not include correlations of the many body interacting electrons like the quantum exchange and the Coulomb screening, corresponding to a real ferromagnetic metallic material. Secondly, the motion of the ions (or the phonon bath) is not included either. From a theoretical point of view, one of the advantages of the nanostructures is the reduced dimensionality. Electron correlations and ionic motion are then easier to implement than in the bulk, particularly for small clusters where the confinement or the shell structure [68] become important. Let us consider simple descriptions that allow understanding the underlying Physics.

Regarding the electrons and spins correlations, qualitatively one can extend the multi-level hydrogen-like system mentioned above to a quasi-continuum of states distributed in two bands, corresponding to the majority and minority spins, shifted by the exchange splitting. The Fermi level *E*_{F} is degenerate with these bands. From the point of view of the magneto-optical response the exchange interaction plays then a role similar to a Zeeman splitting in a discrete multi-level system under an external field . In that case it is simply taken into account in the unperturbed Hamiltonian as which defines the initial conditions (under the applied magnetic field) and does not affect the dynamics. Similarly, for the exchange-split two-band model of the quasi-continuum, the dynamics of the populations occurs only via the interaction Hamiltonian, assuming that the majority and minority bands are already split. Two approaches can then be considered. The first one is formal and the populations are modified by the diagonal part of the density matrix, like in the case of discrete states. It is then simply a computation task to solve the density matrix of the quasi-continuum of states where the relaxation is given either by a *T*_{1}-like model or considering the coupling with an external bath (phonons). Importantly, the angular spin momentum does not necessarily need to be modified by the external bath to obtain the demagnetization, as the initial interaction with the laser and S.O. couplings modify the angular momentum. This type of approach can also be performed within the Time Dependent Spin Density Functional Theory [69].

*Ultrafast demagnetization explained by spin flips: Elliot-Yafet mechanism*. We have shown above that the extended spin-orbit interaction, involving (S.O.)_{ion} and (S.O.)_{laser}, modifies both the coherent response and the population dynamics. In a different spirit, the phenomenological three-temperatures model successfully accounts for many observations related to the demagnetization, including high temperature magnetization behavior (close to the Curie point). Another model has been proposed to explain the change of magnetization and the corresponding dynamics of the spin populations [10]. It is based on elementary spin flip processes due to scattering of the charges and spins with phonons, similar to the Elliot-Yafet scattering of conduction electrons on impurities [72]. Let us briefly summarize this model. First, considering only the charges, the authors derive an expression of the equilibrium time between the electrons and the lattice τ_{el}, considering a Hamiltonian of interaction that involves scattering events between electrons (creation: and annihilation: operators) and phonons (; ) with the interaction matrix element λ_{ep}. Integrating over the occupied density of states of electrons at temperature *T*_{e} and phonons at temperature *T*_{l} and, using the Fermi Golden rule, they obtain: where *C*_{e} and *C*_{l} are the specific heat of the electrons and phonons. *D*_{F} is the constant density of electronic states at the Fermi level and *D*_{p} the number of phonons considered as harmonic oscillators. This approach is equivalent to the two-temperatures model where, instead of the ad-hoc electron-phonon coupling *G*_{el} between the baths, the relaxation time is expressed in terms of the scattering rate λ_{ep} between the populations of electrons and phonons. Second, the authors incorporate the interaction of the spins with electrons and phonons, assuming spin flip processes where *a* is a probability of spin-flip and (, ) the spin operators. Considering the approach to equilibrium of the spins the authors obtain an expression of the spin equilibrium time . Using the Fermi's Golden rule, and considering the different possibilities of absorption and emission of phonons as well as spin flips, the authors obtain: , where is the Gilbert damping. The calculations are made far from the Curie point, assuming that . In short, the equilibrium time of the spins τ_{sl} with the lattice is inversely proportional to the Gilbert damping α. This model, is in agreement with the three temperatures approach, except that it considers the population dynamics which allows giving expressions for the scattering probability (λ_{ep}) that plays the same role as the phenomenological constants *G*_{el}, and the spin-flip probability (*a*) that corresponds to *G*_{sl}. The model does not consider the non-thermal populations of charges and spins which play an important role during the first ∼200 fs when the thermalization of the spins occurs as well as the demagnetization [52]. Let us remind that any model should also account for a thermalization time which depends on the density of laser excitation [31]. Another source of debate regarding the Elliot-Yafet mechanism is the relationship between τ_{sl} and α which is in apparent contradiction with experiments performed by doping ferromagnetic transitions metals with impurities [73, 19]. In those experiments, the variation of the doping concentration is another way of varying the Gilbert damping α, which does not affect the magnetization relaxation time as predicted by the Elliot-Yafet model. Finally, the model may not apply when approaching the Curie temperature *T*_{C} where other phenomena occur (divergence of *C*_{s}, local fluctuations of the spins density). More experimental results are obviously necessary to further clarify these points.