The study and control of magnetic materials using ultrashort laser pulses is of great interest both from a fundamental point of view and for potential applications in the technology of information and communication. In this review the state of the art experimental and theoretical works in the field of Ultrafast Magnetization Dynamics is described, with a particular emphasis on the dynamics of nanostructures. The elementary physical mechanisms involving the spin dynamics when exciting magnetic nanostructures with femtosecond optical pulses are considered. The variety of experimental methods and theoretical approaches used to study the magnetic properties of the materials on a broad range of temporal and spatial scales are examined. The concepts of coherent photon-spin coupling, spin thermalization, precession dynamics and damping, magneto-acoustic interactions, are discussed. Some general trends and prospective works are also envisaged.
1 Introduction to ultrafast magnetization dynamics at the nanoscale
In modern magnetism, new experimental approaches utilize femtosecond laser pulses for studying the ultrafast spins dynamics in magnetic nanostructures. This research field is greatly motivated by the desire to develop new magnetic structures with performances surpassing the actual ones used in the technologies of recording and processing the information (typically one Terabit/cm2 with a switching time of a few nanoseconds). The general goal is therefore to investigate the dynamical properties of magnetic materials and nanostructures occurring in a time scale shorter than the switching time of magnetic devices, at a length scale below a few hundreds of nanometers. Towards that purpose, a generic experimental configuration consists in modifying the density of occupied states of charges and spins of a magnetic structure by absorption of an ultrashort light pulse [1-5]. A second laser pulse is then used as a probe to follow the relaxation of the spins either back to their initial ground state, or to a meta-stable excited state, or even to another magnetic ground state corresponding to the reversed magnetization defined by reference to the direction of an external static magnetic field . From a fundamental point of view, it is an ideal situation to explore the rich diversity of interaction mechanisms which are responsible for the modification of the magnetic order of nanostructures subject to an external perturbation in a temporal scale that extends from a few femtoseconds to a few nanoseconds. Such mechanisms include the spin orbit interaction , the exchange interaction , the structural anisotropy of the materials  and the spin-phonon interaction [10-13] among the most relevant but, in confined geometries, the dipolar interaction is important as well [14-16]. When studying ultrafast magnetic processes, in addition to the accurate temporal resolution provided by the duration of the optical pulses, the spectral characteristics of the laser pulses are also important. One can excite or probe the magnetic structures with a wide range of wavelengths to obtain specific information on the structure or electronic properties of the magnetic material studied. This is of great importance when specific information about the spins dynamics is desired like, for example, the understanding of the dynamical modification of the exchange band splitting , or about the effect of doping on the magnetization damping [18, 19], or else the importance of magnon dispersion on the motion of precession of the macro-spin in ferro and ferrimagnets [20-22]. Regarding this spectral aspect, several methods of investigation have been developed and used to probe the dynamics of spins, employing various laser techniques available either in ultrafast optics , Infrared  and TeraHertz spectroscopy [25-27] or more recently X-ray spectroscopy using synchrotron sources [7, 28] or table top laser systems . On the theoretical side, important concepts that play a role when considering the ultrafast spin dynamics can be confronted to experiments, like the conservation of angular momentum [5, 30] and the associated coherent spin-photon interaction [31, 32], the many-body exchange and correlation interactions [33-35], the relationship between fluctuation and dissipation  or else the shapes and anisotropies of the magnetic nanostructures [9, 37, 38].
Having a femtosecond temporal resolution is not sufficient in order to obtain information on the ultrafast dynamics of the charges and spins in structures having a reduced dimensionality. Ideally one would like to study individual magnetic nanostructures by performing experiments with simultaneously a femtosecond temporal resolution and a spatial resolution below a few hundreds of nanometers. The research activity in that area of “femtomagnetism” [39, 40] is still in its infancy owing to the relatively complex instrumentation required. Nevertheless, flourishing experimental methods are emerging which combine short laser pulses with the imaging techniques available in scanning optical microscopy [14, 41, 42], in electron microscopy  or else in near field scanning imaging . Importantly, an alternative way to study the effects of dimensionality on the dynamics of the spins and charges is to consider assemblies of nanostructures which are well controlled so that the inhomogeneous distribution of sizes plays a minor role. Such approach is much simpler from the point of view of the time resolved experiments but transposes the difficulties into the nanofabrication techniques. Still it is a very attractive approach due to the amazing progresses achieved during the past ten years using so-called “top-down” fabrication techniques employing the Molecular Beam Epitaxy of complex hybrid systems  associated with precise electron lithography or Focused Ion Beam patterning. Moreover, the progresses in self assembling of chemical species or nanoparticles, also provide assemblies of magnetic nanostructures having narrow distribution of sizes . Importantly, the size distribution does not necessarily warrant homogeneity as magnetic properties are also very sensitive to material structure and composition (stoichiometry in case of binary compounds or alloys for example) Let us emphasize that the study of assemblies of magnetic nanostructures is as important as the individual ones because the mutual interaction between the nano-systems may become a handicap for applications to high densities of information. This is particularly the case of magnetic systems due to the long range magnetic dipolar interaction. It is crucial to perform both types of investigations. We will therefore consider both aspects in what follows.
2 Ultrafast magnetic processes induced by laser pulses: from the initial spin-photon interaction to the onset of magnetic switching
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 . 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 . 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 .
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 CoPt3 thin film excited with low intensity 20 fs laser pulses, occurs within 50 fs . 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 . 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  or permalloy . 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 . 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 . The thermalization of the spins is identical in both cases .
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 Co70Fe30 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 . 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 CoPt3 and Ni thin films . 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 . Finally, the coherent control of the magnetization found a nice substantiation by switching GdFeCo thin films using circularly polarized femtosecond light pulses . 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 T2e 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 τe, 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 Tprec (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.
In the case of the laser induced spins dynamics, the change of the magnetization obviously results initially from optical transitions to excited spin states. These transitions can be considered in first approximation as dipolar optical transitions which, in the simple case of one excited electron in an atom, is given by the interaction Hamiltonian where is the potential vector of the laser field and the generalized momentum of the electrons. This generalized momentum contains the kinetic momentum , which is responsible for the optical transitions not involving the spins with interaction matrix elements between the quantum states and ; and an angular momentum due to the spin-orbit (S.O.)ion interaction between the spin vector and the field of the ions. This interaction (S.O.)ion = is deduced directly from the relativistic Quantum Electrodynamics, starting from the Dirac equation of the electron, after some simplifications . Another important interaction term is generally neglected and has to be considered. It is similar to the spin-orbit one and corresponds to the interaction of the spins with the potential associated to the laser field and not to the ions: (S.O.)laser = . These three interaction terms we name for simplification: kinetic momentum, (S.O.)ion and (S.O.)laser.
The notion of coherence is easily deduced from what is known in ultrafast optics. A short laser pulse sets an electric polarization in the magnetic material which corresponds to dipoles oscillating at the laser frequency which can be resonant or not with the energy difference between the fundamental and excited energy levels. To simplify one can think of an assembly of Lorentz oscillators, except that they are strongly correlated in magnetic metals. The kinetic momentum is sufficient to describe the dynamics. The corresponding dephasing time T2e and lifetime T1 of these dipoles are easily described by a quantum two levels system. The ingredients to add now, in order to incorporate a change of the spin angular momentum, are the two (S.O.) interactions. The first one (S.O.)ion is the most important and explains most of the features of magneto-optics as derived a while ago by Argyres . In short, the spin angular momentum is modified and can be detected by measuring the polarization state corresponding to the particular transition , weighted by the population difference associated to the probability of occupation and of the two states. Correspondingly, this change of angular momentum induces a rotation and ellipticity in the polarization of the field of a propagating optical pulse that “sees” different complex refractive index when an external field is applied so that the spins are reversed with respect to the quantum axis (magnetic field direction). The second source for a change of the spin angular momentum is (S.O.)laser. It is exactly similar to the effect of the ion (S.O.)ion coupling term and gives rise to the same type of effects. It is this term which is at the origin of the coherent induced switching observed by Gamble et al. , using a high intensity pulsed electric field. It is also responsible for the perturbation of the nonlinear magneto-optical response during a single pulse propagation as observed in our group . Let us note that in this last experiment the usual (S.O.)ion is still the predominant one as the internal field induced by the laser remains relatively modest .
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 . 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 . 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, 33, 65, 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 , 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 T2e. 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 T2e 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 Co0.5Pd0.5 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 . 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 : 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 CoPt3 ferromagnetic films .
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 Epr and Epu (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 T2 and T1 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 . 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  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 EF 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 T1-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 .
Another way of considering the dynamics of the spins is to assume a time dependent temperature of the electrons in the Fermi distribution . This approach has been put forward and extended to the case of a spin temperature , different than the ones of the electronic charges and of the lattice . It corresponds to the three-temperature model which, in spite of being phenomenological, grasps essential aspects of the spins dynamics. It consists in assuming three different baths at temperatures Te, Ts, Tl with their corresponding specific heats Ce, Cs, Cl and coupling constant Ges, Gel, Gsl . In addition one can add a rate equation for the non-thermal electronic populations which “feeds” the electronic thermal baths at a rate which depends on the absorbed pump energy Em. A more sophisticated approach could be to consider the intraband relaxation of the charges to the Fermi level . The heat diffusion for each bath is also taken into account with the heat conductivities in the Laplacian of the temperature baths .
This model provides a simple explanation of the observation of a time dependent thermalization for the spins τs which increases non-monotonically when the density of absorbed laser energy Em increases. It is partially due to the divergence of the spin specific heat when the elevation of the spins temperature approaches the Curie point TC. Figure 7 (a) shows the variation of the change of spin temperature for two energy densities Em of the laser in the case of Nickel, using the three-temperatures model. The corresponding magnetization dynamics is represented in Fig. 7 (b) and the demagnetization times are also indicated. The slowing of the demagnetization is due to the temperature variation of the spin specific heat which diverges at TC as shown in the inset of Fig. 7 (a). Figure 7 (c) shows the three temperatures , and when the increase of spin temperature dynamically approaches the Curie point (Em = 10 mJcm−2). The inset displays the normalized non-equilibrium population together with the laser pulse assumed to be Gaussian with a temporal width of 50 fs. The asymmetry corresponds to the ad-hoc thermalization of the electrons and the delay between the pulse and is due to the temporal convolution between the two quantities. In Fig. 7 (d) the equilibrium time of the electron and spin temperatures to the one of the lattice is represented for different Em. They both increase monotonically with a slight difference due to the variation of Cs near TC. The importance of the thermal mechanism to explain the demagnetization of Ni films has been further confirmed recently . Importantly, the three temperature model can be replaced by more accurate thermodynamical approaches taking into account spatial and temporal fluctuations. Such approaches, difficult to implement in the bulk, will probably be very efficient to describe the ultrafast dynamics of nanostructures in a near future.
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 . 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 . 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 τel, considering a Hamiltonian of interaction that involves scattering events between electrons (creation: and annihilation: operators) and phonons (; ) with the interaction matrix element λep. Integrating over the occupied density of states of electrons at temperature Te and phonons at temperature Tl and, using the Fermi Golden rule, they obtain: where Ce and Cl are the specific heat of the electrons and phonons. DF is the constant density of electronic states at the Fermi level and Dp 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 Gel between the baths, the relaxation time is expressed in terms of the scattering rate λep 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 τsl 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 (λep) that plays the same role as the phenomenological constants Gel, and the spin-flip probability (a) that corresponds to Gsl. 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 . Let us remind that any model should also account for a thermalization time which depends on the density of laser excitation . Another source of debate regarding the Elliot-Yafet mechanism is the relationship between τsl and α which is in apparent contradiction with experiments performed by doping ferromagnetic transitions metals with impurities [19, 73]. 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 TC where other phenomena occur (divergence of Cs, local fluctuations of the spins density). More experimental results are obviously necessary to further clarify these points.
To close this section on thermal effects let us mention two other important aspects. We have stated above that one can transpose a hydrogen-like multi-level system to the case of a more realistic representation of a ferromagnetic metal by considering two bands shifted by the exchange energy. This approach, which can be named “rigid-band thermal model” does not take into account the dynamical change of the exchange interaction in the interaction Hamiltonian as the bands are supposed to be rigidly shifted. Experimentally, it is however known that the exchange splitting can be reduced as shown in the case of Gd . This is an opene theoretical question of great interest that will certainly find its justification in the coming years at the ab-initio level, considering for example simple dimmers or trimers of magnetic atoms or even further with full Time Dependent Density Functional Theory (TDDFT) calculations. It will necessitate considering the spatial modifications of the symmetric and antisymmetric spins wave functions, due to the laser excitation and giving rise to a dynamical . Conceptually, one can easily forecast that such approach will bring a missing piece in the overall puzzle of magnetization dynamics. In a way similar to the spin-orbit interactions (S.O.)ion and (S.O.)laser, which are present in both the coherent and thermal contributions in the magnetization dynamics as summarized above, it is very likely that one will find such coherent and population contributions also for the exchange terms.
In this section we consider three important research directions which are prolific and very promising for future applications of Femtomagnetism at the nanoscale.
2.3.1 Precession dynamics: a short overview
Understanding the physics of the elementary switching time of a magnetic bit is of great importance for magnetic recording. It is the realm of micromagnetism which has been intensively considered during the past decade . From a formal point of view, the main task is to solve the Landau Lifschitz or Bloch equations in at least two dimensions, in a confined geometry. Three main ingredients are important towards that goal. First, one needs to describe correctly the effective field that contains the anisotropy (shape, magneto-crystalline, magneto-elastic), the static external field H0 that defines the quantum axis, as well as the initial conditions. Second, it is important to model the type of source used to drive the precession dynamics. One can use a radio frequency magnetic field which is well adapted for studying the ferromagnetic resonance (FMR)  in the frequency domain or a pulsed magnetic field that allows initiating a fast change of the angular momentum that drives the magnetization out of equilibrium [76-78]. The torque induced switching  or the magnetic domain manipulations and control using electronic currents  are also very fertile research directions leading to many applications in Spintronics as well as theoretical works . Another very efficient way to induce the switching is to use femtosecond laser pulses. This approach is more and more utilized mostly because of the temporal resolution which is the one given by optics (femtosecond time scale) with the possibility of following the dynamics up to the picsosecond and nanosecond temporal regimes. The third aspect of precessionnal dynamics is naturally the damping of the precession. It is important for obvious reasons of understanding the origin of the “frictions” that lead to the attenuation of the precession. It is a general topic of magnetism that was already attracting interest in the 1950 and which found a regain of interest in the context of spintronics. It is important to understand the origin of damping when characterizing new nanomaterials and nanostructures, trying to identify the underlying fundamental interaction mechanisms. With that respect, the concepts of magnetic fluctuations, already studied by Néel  and Brown  in the context of super-paramagnetism is finding new motivations with the possibility of modeling the spatio-temporal dynamics at the atomic scale , using either finite element methods  or a Monte Carlo approach [85, 86] to solve the Fokker-Planck equation describing the orientational distribution of the magnetization. An important related research topic is to evaluate the effects of inter-particle interactions [16, 87] as described further in Sections 'Assemblies of magnetic nanostructures' and 'Individual magnetic nanostructures'. As it is not possible to summarize here the tremendous amount of works developed in the field of precessional dynamics and switching, we simply focus on the laser induced precession.
2.3.2 Laser induced precession dynamics
The original approach in that direction was performed by G. Ju et al. . They reported about “an optically induced modulation in exchange biased ferromagnetic/antiferromagnetic thin bilayer films (NiFe/NiO) using laser pulses with a duration of 120 fs”. Their main result, showing the motion of precession observed in a pump-probe TR-MOKE configuration for a static magnetic field applied along either the easy-axis or hard-axis, is reproduced in Fig.8. Subsequently, this work was systematized by several groups and is sometimes named “Optically induced ferromagnetic resonance”  or “Magnetization precession induced by femtosecond laser pulses” as the approach does not only apply to ferromagnetic nanostructures. In our group, we have shown that one can retrieve the dynamics of the magnetization precession in the three dimensions of space with a femtosecond resolution performing measurements of the TR-MOKE polar, longitudinal and transverse signals. Figure 9 (a) shows the geometry used to retrieve the three components , , . This is ideal for studying the magnetization trajectory in real space and time, which is important for applications in Spintronics. In addition, the accurate temporal resolution allowed us to show that it is a powerful approach for studying the importance of the material anisotropy on the onset of precessional motion . Figure 9 (b) shows the case of two different Co thin films deposited on Al2O3 and MgO substrates giving rise respectively to a perpendicular or to an in plane easy axis magneto-crystalline anisotropy. The out of phase oscillations for these two samples are due to the fact that the laser heating reduces the anisotropy and gives an initial “kick” to the magnetization either towards the sample plane (Al2O3 substrate) or on the contrary towards the normal to the sample (MgO substrate). A detail view at short delays (not shown here) shows how the magnetization initially rotates in the polar/longitudinal plane. It allows observing the absolute phase and sign of rotation of the magnetization vector .
Let us summarize the main advantages of performing measurements of the precession of with femtosecond lasers as compared for example to radio frequency FMR.
First, it is a real time investigation of the precession and most importantly of the damping. The temporal resolution is limited by the pulse duration (20fs) which makes it possible to access the TeraHertz regime which is useful for studying very hard magnetic nanostructures.
Second, one can perform a real space analysis using the flexibility of magneto-optical configurations (polar, longitudinal and transverse).
Third, it is possible to perform a local probing of the magnetization: the spatial resolution is limited by the Abbe diffraction (laser spot diameter: ∼ 250 nm for the second harmonic of a Titanium Sapphire laser used with a 0.8 Numerical Aperture objective)
One can investigate nonlinear regimes of the precession of in particular when there is eventually a non-conservation of the modulus of the magnetization.
Let us develop more this last remark as it refers to an unusual situation when modeling the dynamics. First, it is important stressing that it is the variation of the effective field via a modification of the anisotropy which is responsible for the initial out of equilibrium situation after the laser pulsed excitation. This modification of the anisotropy however results from the reduction of the magnetization modulus because both the shape anisotropy and magneto-crystalline anisotropy , i = x, y, z. in the case of a uniaxial anisotropy axis with α the angle between the axis and the magnetization . Ka is an anisotropy factor which depends on the lattice temperature. Therefore, the effective field, which depends on the magnetization modulus, is not conserved. On the timescale of the precession dynamics (100 ps - few ns) in soft materials, electrons, spins and lattice are close to equilibrium. Therefore ultrafast processes, such as a non-equilibrium Fermi distribution, are not relevant any more, even though they play a crucial role in determining the initial conditions from which the magnetic system starts its motion of precession. The Landau Lifschitz equation can however be modified by the set of following equations:
Equations (3) and (4), together with the three temperature model described above (Te for the electrons, Ts for the spins and Tl for the lattice temperatures), allow calculating the three components of the precession of . The damping term “à la Gilbert”, which consists in adding an ohmic dissipation: in the effective field, is somehow interesting as the modulus of is not conserved during the recovery of the temperatures to the initial magnetic state (before the laser excitation). In particular this gives rise to new nonlinear effects in hard magnetic materials which have short periods of precession and fast damping, as the electrons, spins and lattice may not already be in equilibrium while the system starts its motion of precession. Another application of this non-conservative case of damping is that one can use the absorbed laser energy density Em to control the motion of precession, even in ferromagnets having a long relaxation. Figure 10 shows the precession of a cobalt film for two different laser densities Em. The period is longer for higher Em, as seen in the Fourier transform in the inset. We name this effect: “control of precession by laser intensity”.
2.3.3 Ultrafast Magneto-acoustics
This is a new research direction which has been been recently developed by two groups working with magnetic semiconductors  and with ferromagnetic metals . It consists in propagating acoustic waves using femtosecond laser pulses in a metallic film and to use the strain deformation associated to the acoustic pulse to induce a precession of the magnetization, either in another ferromagnetic layer or in the metal itself which can be ferromagnetic. The propagation of picosecond acoustic pulses, induced by femtosecond optical pulses, is a well known field in ultrafast optics [88-90] and it has been extensively studied in a variety of metallic structures, with a strong interest during the past five years for application to nanostructures. In the experiment of A.V. Scherbakov et al.  a single semiconducting GaMnAs layer doped with 5% Mn is grown on a semi-insulating (001) GaAs substrate. The acoustic pulses are generated by excitation with 200 fs optical pulses of a 100 nm thick Al layer deposited on the front face of the sample. The Kerr rotation angle of the ferromagnetic semiconducting structure is observed on the rear face of the sample at 1.6 K using the field of a supraconducting magnet. In our experiment  we have used a 200 nm thick nickel film to directly inject the acoustic wave and probe its effect on the precession of the magnetization on the rear part of the sample at room temperature. Figure 11 shows the precession from the rear side of the sample in the magneto-optical Ker ellipticity. We have developed a model that allows explaining the magnetization dynamics (precession and damping) in terms of the perpendicular anisotropy induced by the longitudinal acoustic strain. The torque provided by this strain induces the nonequilibrium dynamics of the magnetization which starts its motion of precession .
2.3.4 Switching dynamics induced by laser pulses
We have focused on describing processes which either lead to a demagnetization or to a motion of precession and damping of . These effects can be used for devices where only a transient change of the magnetization is necessary. We have seen that it is particularly interesting in the case of coherent diffractive magneto-optical structures, or for modulating devices with a laser control of the precession for example. For purposes of storage it is naturally necessary to switch the magnetic nanostructures. This is a subject of intense research and an important breakthrough occurred when it was shown that one can control the switching in a film of GdFeCo using circularly polarized light . Recently, the same group have used time-resolved single-shot pump-probe microscopy to investigate the mechanism of the all-optical magnetization reversal . They have shown that the reversal does not involve precession but results from a strongly non-equilibrium state. Further details of the all-optical switching are described in ref. .
3 Ultrafast magnetization dynamics in nanostructures
3.1 Spatial scale and mechanisms
Experimental works on the ultrafast magnetization dynamics in nanostructures remain rare because one has to combine the temporal and spatial resolutions, which necessitate different strategies depending on the spatial scale of the studied phenomena. In what follows we focus on objects with sizes from a few nanometers to about . Some important physical mechanisms for magnetic interactions can be classified as follows:
Diffusive electronic and spin processes occur on the micron scale. Typically, a spin can lose its polarization after multi-scattering events with phonons. The corresponding diffusion time is for electrons with a Fermi velocity . Ballistic electrons generated by femtosecond laser pulses are even faster. Therefore diffusive processes participate to the thermalization of charges and spins (∼100 fs as mentioned in Section 'Ultrafast magnetic processes induced by laser pulses: from the initial spin-photon interaction to the onset of magnetic switching'). Obviously it is pertinent to study the diffusion on samples having thicknesses of a few tens of nanometers. For example one can perform pump-probe spectroscopy by pumping on one side of the sample and probing on the other side. The only limiting factor then is the penetration depth of light which is of the order of 20 nm in metals (naturally it varies a little depending on the particular material investigated.
The size of magnetic domains depends on the materials (soft or hard magnetic layers). Typically it lies in a range from 100 nm to 10 microns for bulk ferromagnets. Such domains structures can therefore be spatially resolved with conventional confocal optical microscopy. Magnetic monodomains in confined geometries are however smaller, typically less than 30 nm for ferromagnetic nanoparticles.
Superparamagnetism in nanoparticles, corresponding to fluctuations of the direction of the magnetization (random switching), occurs when the anisotropy energy barrier is smaller than the thermal energy: , where V is the volume of the nanoparticles and Kan the anisotropy. For cobalt nanoparticles superparamagnetism occurs at room temperature typically below ∼7 nm, assuming an anisotropy . Naturally, the concept of suparparamagnetism depends on the timescale of the probing instrument. Here we assume that the magnetic state of the nanoparticles is investigated with a femtosecond time resolution which, on the scale of the superparamagnetic fluctuations, is similar to a δ-function probing. In that sense, the dynamics refers here, as well as in Section 'Diluted Cobalt nanoparticles', to the trajectory during a single reversal of the magnetic vector.
The width of a Bloch wall (Bw) is given by where J is the exchange interaction coupling constant ( for 2 spins distant of ). Typically .
The range of the spin density oscillations for conduction charges close to a magnetic impurity (Friedel oscillations) is another important length scale in magnetism. It is given approximately by the inverse of the Fermi wave vector . Let us precise that spin density oscillations correspond to a static representation. When spatial and ultrafast temporal aspects intervene simultaneously, one should consider the propagation of the spin excitations. In the case of molecular systems, it is intimately related to the concept of wave packet dynamics with a possible transfer of the spins. In the case of continuous systems (thin films or large nanostructures) long range correlations and hopping from site to site, including thermal fluctuations, are necessary concepts to consider. Such concepts of spin wave packets are particularly important in molecular magnets as briefly described further in Section 'Molecular Magnets'
The magnetic dipolar interaction induces collective effects between close neighbor nanoparticles when they are separated typically by a few nanometers.
Spin scattering at surfaces has been suggested to be responsible for the increase of the Gilbert damping α when decreasing the size of the particles. This mechanism becomes important for nanoparticles with diameters of less than ∼5nm [93-95].
One should also distinguish between different types of nanoparticles depending on their size, their structure and their organization
The various processes are sketched in Fig. 12. The wide spatial range in which they extend necessitates different experimental strategies to resolve the dynamics of the nanostructures.
For structures with sizes larger than 100 nm, time resolved magneto-optical confocal microscopy is an ideal technique. In Fig. 13, we have represented such microscope developed for that purpose . It is a collinear pump-probe set-up, with pump pulses at 800 nm (150 fs pulse duration) and probe pulses at 400 nm (120 fs). The overall spatial resolution is of 300 nm, corresponding to the spatial cross-correlation of the two beams. The beams are focused onto the magnetic sample with an objective with a Numerical Aperture NA = 0.6. A dichroic plate collects the reflected probe beam which is focused onto a 20 m aperture. Its polarization state is analyzed with a polarization bridge to obtain the time dependent magneto-optical Kerr rotation or ellipticity as a function of the delay between the pump and probe pulses. The contrast ratio of the probe polarization measured without magnetic sample on a reference plate is .
For structures with sizes less than 100 nm, high spatial resolutions can be achieved with Scanning Probe Microscopy techniques , X-ray photoemission electron microscopy  or Lorentz Transmission Electron Microscopy , but the temporal resolution is generally in the picosecond regime. For example with “Time Resolved Photoemission Electron Microscopy (TR-PEEM)”, after the sample is perturbed by a visible pulse, an X-ray probe pulse excites the photo-electrons of the particular magnetic element of interest. The spatial resolution is typically of ∼10 nm and the temporal one is of a few tens of picoseconds depending on the source of X-rays (different functioning mode of synchrotron sources for example). For sizes close to 50–100 nm, confocal scanning microscopy might be interesting but did not receive attention yet in magneto-optics, while it has been employed for studying the dynamics of surface plasmons in individual noble metal nanoparticles . Alternatively the femtosecond dynamics of the surface plasmons in arrays of holes made in noble metal films have also been studied . Both types of approaches have not yet been transposed to ferromagnetic structures. Near field magneto-optical microscopy is difficult to employ in metallic nanostructures having small M.O. rotations because the fiber and its metallic aperture tend to depolarize the probe beam, reducing the contrast ratio Rc defined above. For these structures TR-PEEM is more adequate.
For nanostructures or mechanisms to investigate below 20 nm the spatial resolution is even more difficult to achieve. Two techniques seem promising. The first one is “time resolved magnetic X-ray diffraction experiment” where X-ray circular magnetic dichroism (XMCD) is performed to probe the magnetic chemical elements of the material together with visible femtosecond pulses to excite the sample. Experiments are currently being performed with such configuration using X-ray pulses produced by SXR beamline  of the Linac Coherent Light Source (LCLS)  at Stanford, USA. The magnetic information is obtained thanks to the XMCD configuration. The spatial resolution is obtained via the analysis of the diffraction pattern generated by the spatial “non-homogeneity” of the sample resulting from the specific process to analyze (fluctuations of short range magnetic domains for example). The resolution in Fourier space of the diffraction pattern is obtained with appropriate Charge Couple Device detectors. A temporal resolution of ∼200 fs is given by the jittering between the bunch of electrons and the laser pulse. The spatial resolution is related to the Fourier analysis of the diffraction pattern and can reach in principle ∼5 nm. It requires some interpretation as the diffraction pattern results mainly from different scales (100 nm: sample non-homogeneities due to the growth process; 10 nm: spin density fluctuations).
Another attractive technique which should give even better spatial resolution consists of a new generation of electronic microscope, like the one developed in A. Zewail's group  at Caltech, USA for detecting electronic processes. It is certainly an interesting approach to implement for detecting spins density fluctuations. We are currently building such facility in our research institute . Our intention is to perform Lorentz force electron microscopy with spins resolution at the spatial scale of 5 nm together and with a temporal resolution of a few hundreds of femtoseconds. These last two techniques, TR-XMCD-Diffraction and TR-MET, although promising ultimate techniques to gain one order of magnitude in the spatial resolution, are still in their infancy. The results presented in the next two sections are in the range of 100 nm, using more conventional techniques. We describe some experimental studies in the field of ultrafast magnetism at the nanoscale, distinguishing between measurements of assemblies and individual nanostructures.
3.2 Assemblies of magnetic nanostructures
The ultrafast magnetic studies of assemblies of nanoparticles bring information both on the nanoparticles themselves when they are sufficiently diluted and on the inter-particle interactions when they are closely packed (inter-particle distance less than 5 nm), which is an important situation for high density recording.
3.2.1 Diluted Cobalt nanoparticles
The first study of the kind has been performed with femtosecond TR-Faraday on cobalt nanoparticles dispersed in a SiO2 matrix or Al2O3 , which corresponds to the regime of diluted nanoparticles concentration. The diameter of the nanoparticles is either 4 ± 1 nm or 10 ± 1 nm. In the larger size nanoparticles the ordering at room temperature is close to superparamagnetic (low energy barrier) but they are large enough to observe in real time the gyroscopic behavior predicted by L. Néel  and W.F. Brown  who derived an expression for the prefactor time τ0 in the expression of the fluctuation time τ of the macrospin given by the Arrhenius law: where K, V and T are respectively the anisotropy constant, the volume and the temperature of the nanoparticles. The important result of the study is the fast damping of the magnetization precession. Surprisingly, the measured precession period and the damping time of the magnetization precession have a value close to the calculated Brown's prefactor τ0 obtained from the Langevin analysis of the Landau-Lifschitz equation in the presence of a fluctuating field:
where Ms is the magnetization at saturation and K the anisotropy constant. In this expression, we isolate two contributions: one which corresponds to a coherent behavior of the magnetization and the other one which describes the fluctuations. The coherent term contains the parameters η and γ0 of the gyroscopic motion and is given by: . The term related to the fluctuations contains the ratio between the anisotropy energy barrier and the thermal energy and is given by: . Using the known material parameters (Co) the calculation gives a prefactor τ0 = . As the three times, Tprec, η and τ0 are of the same order, it means that the precession is damped before the magnetization is reversed (τ0 is the minimum time for such reversal). Therefore, this study questions the existence of a fully coherent pathway during the statistical superparamagnetic fluctuations between the two directions of the magnetization with respect to the external magnetic field.
In another study, we measured the dynamics of CoPt nanoparticles . Two different phases of the nanoparticles have been investigated. First, the nanoparticles are core-shells with the core made of cobalt. The blocking temperature is 66 K and therefore the ultrafast dynamics of the magnetization, measured by magneto-optical TR-Faraday, displays a monotonous relaxation without precession as shown in Fig. 14(a). After a thermal annealing at 700 K the Pt diffuses inside the core and forms a CoPt crystalline phase which is observed by transmission electron microscopy. Figure 15 shows the gradual evolution between the Co-core/Pt-shell and CoPt phases on the same individual nanoparticle as a function of temperature. The new CoPt crystalline nanoparticles have a blocking temperature of 347 K, allowing to observe the motion of precession of the magnetization at 300 K, as shown in Fig. 14(b). This study establishes clearly the fact that in the superparamagnetic regime the motion of precession is strongly damped and confirms the results obtained with the diluted Co nanoparticles. Let us emphasize that, in these studies, the dynamics of the magnetization vector in the ferromagnetic nanoparticles corresponds to a slight excursion out of the direction of equilibrium. After a few nanoseconds the magnetization comes back to its initial fundamental state. Ideally one would like to observe the full pathway of the magnetization trajectory in a situation where superparamagnetic jumps occur. Such experiments should also be performed with individual nanoparticles which is a difficult experimental challenge not yet achieved.
3.2.3 Breathing modes of nanoparticles assembled in supra-crystalline order
Supra-crystalline ordering of nanoparticles can be obtained either in two or three dimensions. The ordering is obtained by self assembling of inter-digitated nanoparticles dressed by organic ligands as obtained for example with cobalt nanoparticles . Such supra-crystalline assemblies display collective vibration motion when they are excited with femtosecond laser pulses. In section two we have mentioned that thin metallic films present coherent vibrations associated to the displacement of atoms in the lattice. The period is typically in the range of a few picoseconds. These vibrations, which are excited by the hot electrons and quickly damped, also exist in noble metal nanoparticles [104, 105] and in semiconductor quantum dots . They correspond to breathing modes of the individual nanoparticles with spherical shapes. In the supra-crystal assemblies, the whole nanostructure can also vibrate at a lower frequency. We have shown the existence of such coherent vibrational dynamics, due to the supra-crystalline order, both with Co  and CoPt nanoparticles . The time resolved reflectivity of 3D long-range ordered supra-crystals of cobalt nanoparticles, excited with 120 fs optical pulses at 800 nm and probed with 150 fs pulses at 400 nm, is shown in Fig. 16 (a). Large oscillations with a period of 130 ps are present. Note that the 3D long-range assemblies are constituted of well-ordered pavements of . The measurements being performed at the sub-micron scale, using a confocal microscope, guarantees that single pavements are probed. As soon as a small disorder is introduced in the 3D structures, by performing a thermal heating, the vibrations are suppressed as seen in Fig. 16 (a). In the case of the Co-core/Pt-shell nanoparticles, the same behavior occurs as shown in Fig. 16 (b) before and after thermal treatment. The period of the collective oscillations is of 146 ps. Importantly the heating process does not suppress the coherent vibrational dynamics of the individual nanoparticles as seen in Fig. 16 (c) where a close-up of the dynamics (differential reflectivity up to 4 ps) is shown after the thermal process. The period of the individual oscillations slightly increases from 1.95 ps to 2.15 ps and is less contrasted after the thermal process because a crystalline CoPt phase starts forming with different vibration properties.
3.2.4 Modeling of the magnetization dynamics in closely packed CoPt nanoparticles
To model the magnetization dynamics in the CoPt nanoparticles we have numerically integrated the Fokker-Planck equation that governs the dynamics of the magnetic moment associated with the nanoparticles. The goal is to estimate the dipolar interaction between the nanoparticles, which we performed using a mean-field approximation . Starting with the individual nanoparticles, the probability distribution of the magnetic moment is given by:
Where V is the volume of the particle, α is the Gilbert damping constant, γ0 is the gyromagnetic factor, and Ms is the magnetization at saturation. We have computed the relaxation time τ as a function of the density barrier , where . Figure 17 (a) shows τ in the case of isolated nanoparticles (black circles) which is in agreement with Brown's expression (dashed line) at low temperatures, with a 10% accuracy. At higher temperatures, the numerical solution departs from Brown's expression and the computed relaxation time tends to zero. These results are consistent with those obtained using different numerical techniques such as Monte Carlo or Langevin dynamics  and validate our approach which is then used for calculating inter-particles effects. The effect of the dipolar interaction on one particle exerted by all the others is expressed as a self-consistent magnetic field. We consider a system of interacting nanoparticles distributed over a spatial lattice. At sufficiently low temperature (or high energy barrier), the magnetic moments fluctuate in the vicinity of the positive and negative directions of the z axis. In this case, the self-consistent field is also aligned along the z direction and can be written as , where S is a constant that depends on the geometry of the lattice, is the probability of occupation of the sites, d is the center-to-center inter-particle distance, and is the average z component of the total magnetic moment of the system:
The numerical results for interacting particles are shown in Fig. 17 (b), for the case with and three values of the interparticle distance d. The Gilbert damping constant is taken to be , a high value which accounts for the large damping observed in experiments on small Co nanoparticles . The relaxation time decreases with decreasing interparticle distance, in agreement with previous experimental measurements . Beyond a certain distance ( nm), the dipolar interaction becomes negligible and the result for the corresponding isolated nanoparticles is retrieved. In contrast, the relaxation time decreases with increasing site occupation probability p. Therefore, the magnetization reversal process is accelerated for short inter-particle distances and large concentrations.
The comparison of the theoretical results with experimental measurements shows that the inter-particle dipolar interaction plays a significant role. We have performed TR-MOKE measurements using 150 fs laser pulses at 400 nm for the pump and 800 nm for the probe. The core Co nanoparticle has a 2.4 nm radius and is surrounded by a 1.8 nm Pt shell. The inter-particles distance (Co-Co) is ∼8.6 nm and the particle concentration is large enough so that in the simulations we can safely assume p = 1. Using the three temperature model, we compute the elevation of lattice temperature which we assume to be constant during the relaxation time. For parameters relevant to the experiment it yields an equilibrium temperature around K. This value of the temperature was used in the Fokker-Planck simulations of the magnetization dynamics. Figure 17 (c) shows the computed relaxation time together with the measured damping as a function of the external static magnetic field. The relaxation time calculated with inter-particle dipolar magnetic interaction gives a better adequacy with the experimental results than the case of isolated nanoparticles.
3.3 Individual magnetic nanostructures
The TR-MOKE confocal microscope described in Fig. 13 is useful for studying individual magnetic nano-structures with a size larger than ∼100 nm. Alternatively it can be employed for writing and reading patterns on thin films. In the following we summarize some important steps performed in that field.
3.3.1 Magnetization dynamics of individual ferromagnetic discs
We have studied the magnetization dynamics of individual discs with sizes larger than 200 nm [109, 110]. For example, Fig. 18 shows the dynamics of CoPt3 discs. The magnetic hysteresis of an individual disc is shown in Fig. 18 (a). The images displayed in Fig. 18 (b) correspond to two adjacent dots made by e-beam lithography. After saturation of both discs, with an external field +H ∼ 4 kOe, one of them has been switched with a pump pulse under a smaller negative field −H ∼ 1 kOe, showing the capacity to address individual discs. The TR-MOKE signal of one disc (Fig. 18 (c)), excited with 3.8 mJcm−2 pump pulses, displays the characteristic features of the demagnetization at short delays followed by a recovery of the initial state on the picosecond time scale with two characteristic times. τsl = 2.27 ps corresponds to the spin-lattice relaxation during the heating of the disc and τdiff = 530 ps (not shown in Fig. 18) corresponds to an efficient thermal diffusion to the Al2O3 substrate. In this case the dynamics is measured under an external applied magnetic field (corresponding to the saturation field) which explains the fast return to the equilibrium. In addition, high thermal conductivity of the substrate starts to play a non-negligible role on the timescale of a few tens of picoseconds. The spin-lattice relaxation time varies linearly as a function of the pump energy density as shown in Fig. 18 (d). The dynamics of the dot can also be performed by scanning images as a function of the pump-probe delay. It allows retrieving the transverse thermal expansion on the disc as shown in Fig. 19.
The preceding results have been obtained with a 5 kHz repetition rate laser. Li et al. have developed a high repetition rate version of the microscope functioning at 11MHz  They obtain a spatial resolution of 210 nm and temporal resolution of 230 fs using a high numerical aperture objective (NA = 0.95). The high repetition rate allows performing a double modulation scheme (photoelastic modulation and mechanical chopping) which considerably improves the signal to noise ratio and allows obtaining simultaneously the Kerr rotation and ellipticity. For such high repetition rates a drawback can be a permanent heating of the nano-discs which may prevent a return to the magnetic ground state between each laser pulse. Nevertheless, the authors could demonstrate high contrast imaging of the magnetization dynamics of patterned Co/Pt and Fe/Gd multilayer samples.
Two main effects are important when reducing the size of magnetic discs: the increase of the shape and magneto-crystalline anisotropy on each individual disc and the dipolar magnetic interaction between adjacent discs in case of closely packed discs. This last situation is naturally important for high density recording. Systematic studies have been performed to show the importance of both effects [14, 15, 111, 112].
3.3.2 Anisotropy effects on the magnetization dynamics of individual discs
A. Barman et al.  studied the dynamics of magnetization of individual Ni disks with sizes varying from 5 m to 125 nm. They observe a precession of the magnetization with a period that decreases with the diameter of the nano-discs. The effect appears for sizes below 1 m and becomes prominent below ∼300 nm. In addition, on large discs the frequency of the precession increases as expected on thin films. But for smaller discs with a size below 1 m the variation of the frequency with the external field presents a reduction which is consistent with the appearance of a surface anisotropy that competes with the demagnetizing field. For sizes below 250 nm, the magnetization of the nanostructures undergoes a transition from an in-plane to an out-of-plane direction. In addition, they become fully single-domain below 125 nm. Another work has shown the importance of the anisotropy. S. Lepadatu et al.  have compared the ultrafast magnetization dynamics of a Fe film to an array of four 50 nm square nanodots separated by 50 nm elaborated in the same iron film. After demagnetization, the magnetization relaxation is monotonous in the case of the film whereas a precession occurs in the case of the nanodots. The precession frequency increases with the applied field amplitude, as expected. However, the effective magnetization calculated from the Kittel model is reduced as compared to saturation magnetization. The authors attribute this result to a large interface perpendicular anisotropy between the substrate and the Fe film.
3.3.3 Interaction between nanodiscs
A. Barman et al.  studied the precession dynamics in the Ni discs as a function of their spatial density, showing an increase of the precession damping, which is attributed to inter-discs dipolar interaction.
Recently, B. Rana et al. studied first the collective dynamics of square arrays of permalloy dots with a width of 200 nm as a function of the areal density of the arrays . They have shown that for very high areal density the magnetization dynamics displays a single frequency precession corresponding to a collective behaviour of all elements in the array. Non uniform collective modes of the array show up when decreasing the areal density. At very low areal density, the elements become uncoupled and center and edge modes of the isolated elements dominate. In a second work, the authors studied 50 nm permalloy dots with several inter-particle distance from 50 to 200 nm, down to the single nanodot regime . For single nanodots the precession dynamics reveals one dominant resonant edge mode with slightly higher damping than that of the unpatterned thin film. With the increase in areal density of the array both the precession frequency and damping increase significantly due to the increase in magnetostatic interactions between the nanodots. A mode splitting and a sudden jump in the apparent damping are observed at an edge-to-edge separation of 50 nm. A study of Ni ellipsoids with axis dimensions of 320 nm/130 nm also concludes that inter-particle effects are present on the long time scale (GHz regime), which is attributed to the nanostructing induced anisotropy .
Overall these studies clearly set the characteristic spatial scales that influence the magnetization dynamics. The shape anisotropy starts playing a significant role on the frequency of the precession when the size of the individual discs or dots is below ∼200 nm. The magneto-static interaction between the structures becomes important when their separation is less than ∼50 nm.
3.3.4 Magneto-Optical Pump-Probe Imaging and laser induced nanostructures
The time resolved magneto-optic confocal microscope can be used efficiently for writing patterns on a thin film as well as reading this pattern afterwards. For writing, only the pump pulses are used with a sufficient energy density (typically 5–10 mJcm−2 in ferromagnetic films) to demagnetize and switch locally the magnetization under a weak reversed static magnetic field. Then, to read the written dot we perform the TR-MOKE measurement using a weaker pump intensity (typically 0.5 mJcm−2). Naturally the sample is scanned in 2D to obtain the image. The advantage of reading using the TR-MOKE is the very high contrast of the images due to the fact that it is differential, thanks to the modulated pump beam. We name this technique MOPPI for Magneto-Optical Pump-Probe Imaging.
Figure 20 shows several dots written on CoPt3 thin films (15 nm thick), with perpendicular anisotropy, deposited either on a Al2O3 (top two figures) or glass substrate (lower two figures). The sample on Al2O3 corresponds to an epitaxial growth, while on glass it is polycrystalline. The dots have been written for two laser energy densities: 4 mJcm−2 (left two figures) and 8 mJcm−2 (right two figures). To better see the detailed structure of the dots, we used a Magnetic Force Microscopy (MFM) imaging. The dots on the Al2O3 substrate display a labyrinth structure for both intensities, with a mean diameter that increases for the higher intensity. This structure corresponds to self organized domains in epitaxial magnetic structures. The labyrinth-shape of the domains is mostly determined by the boundaries of the dot and governed by the magneto-crystalline and shape anisotropy. In the case of the glass substrate, the shapes of the switched dots are quite different and strongly depend on the laser density of energy. At low energy density (lower left corner) a single dot is written with a diameter d = 200 nm slightly smaller than the theoretical Abbe diffraction of the probe beam (250 nm). For the highest energy density, the dot splits into a multi-domain structure made of several smaller dots. These two behaviors (single or multi-domains) show that the structural anisotropy dominates due to the high disorder present in the polycrystalline phase of CoPt3/glass. These studies show that the MOPPI technique is a powerful technique for exploring the effect of the material parameters on laser induced nanostructures. Note that the patterning can be more sophisticated than the simple dots reported here, like for example magneto-optical diffractive arrays .
4 Conclusion: Ultrafast magnetization dynamics in reduced dimensionality
In this concluding section we give some trends that might be of interest for new researches in systems which have intrinsic effects related to dimensionality without necessarily requiring the design of structures made with sophisticated nano-fabrication techniques.
4.1 Ultrafast magnetization dynamics in reduced dimensionality systems
The magnetization dynamics at the nanoscale does not necessarily involve nanostructured dots or nanoparticles. The physics of the interactions in reduced dimensionality is important as well and can be studied in “macroscopic” structures displaying 2D magnetic behavior. This is the case in systems where fluctuations of the magnetization occur. For example, it is known since the fundamental work of Mermin and Wagner , that in some systems the magnetic ordering does not occur at any temperature because the long range order cannot establish. This is the case because the density of spin waves diverges in reduced dimensionality favoring spatial fluctuations of the magnetization. Such situations are common in magnetic insulators where the ordering is essentially mediated by the exchange interaction. Therefore studying this class of materials with ultrashort laser pulses is an ideal situation for investigating the dynamics of the exchange interaction.
Let us mention an interesting ab-initio approach of the ultrafast magnetization dynamics in molecular systems [117, 118]. The authors consider spin-orbit induced spin flips in linear molecules like NiOCo or NiMgCo. Using an ab-initio approach, they show that the bridging oxygen or Manganese atoms play an important role on the spin flip process. They also find that the ultrashort laser pulses can favor the spin transfer along the chain and suggest coherent spin manipulation to realize spin-logic devices.
4.2 Vortex dynamics
The existence of magnetic vortices in confined structures is well known . They are stable structures which form perpendicular to the surface of the micrometer-size discs or squares to overcome the energy cost due to the shape anisotropy which tends to maintain the magnetization in the plane. The dynamics of these structures has been investigated experimentally with time resolved MOKE, with a pump perturbation which is not optical but consists of a magnetic field pulse of ∼150 ps duration  and also with time resolved X-ray imaging . The dynamics of the vortex shows that two modes are present in permalloy discs with diameters ranging from 0.5 to 2 m. One of them corresponds to a precession around the internal field and the other one corresponds to a gyroscopic motion of the entire vortex. The switching dynamics of vortices has also been investigated theoretically, performing 3D micromagnetic simulations of the Landau-Lifshitz-Gilbert equation . The authors have shown that during the switching process the core reversal occurs after a vortex-antivortex pair is created and vanishes while the vortex with opposite polarity forms. That sequence has been obtained with picosecond magnetic pump pulses which is the source for the change of angular moment.
4.3 Molecular Magnets
The reduced dimensionality is inherent to molecular magnets which constitute another interesting class of materials. In particular, in metal organic molecular compounds the local exchange interaction on each molecule together with their strong anisotropy favor the existence of well defined “low spin” and “high spin” energy levels. Then, a ferromagnetic order occurs when the temperature is sufficiently low so that the fundamental state can be the “high spin” configuration. Interesting quantum effects can then be explored like the tunneling between different configurations that can be monitored in the presence of an external magnetic field. The tunneling is a slow process but there are other mechanisms that could be used for ultrafast manipulation of molecular magnets. For example, the vibronic spectrum is very rich in the organic “cage” surrounding the metal ions and it can be easily excited with femtosecond laser pulses. Several advantages can be forecast. The spectral selectivity of the laser allows exciting the vibronic modes resonantly, including in the fundamental state using IR ultrashort pulses. The dynamics associated to the molecular motion is particularly interesting when a coherent superposition of states is excited, leading to the propagation of a wave-packet on the potential of the molecules  which can be as short as 23 fs in conjugated systems  with strong electronic correlations and their associated structural transitions in the excited state . Importantly, the magnetization dynamics can be strongly modified by the coupling of the spins to these transient wave-packets especially in systems having a large spin-orbit interaction. In addition, these wave packets can be coherently controlled using the variety of laser techniques available in the coherent control of atomic and molecular systems. One could therefore have other ways of inducing ultrafast magnetic changes utilizing these concepts. The rich variety of molecular magnets is certainly of great importance as to future developments in that field. Note that, to our knowledge, no experimental report of ultrafast magnetism in molecular magnets has been made so far, probably because of the restrictive experimental conditions (low temperature, high magnetic field, stability of the molecules under illumination). Dimensionality effects can be explored not only at the molecular level but also between magnetic planes in ordered structures of such molecular magnets. For example ferromagnetism or antiferromagnetic coupling between adjacent planes of molecular magnets can be monitored by varying the spacing between the molecular planes . Interestingly, the variety of bottom-up elaboration techniques for the self-assembly of single molecule magnets  on surfaces opens a wide range of possibilities both for fundamental research and applications in ultrafast magnetism.
J.-Y. Bigot thanks the European Research Council for financial support via the ERC Grant “ATOMAG” ERC-2009-AdG-20090325♯247452. Some of the works reported here have been supported by the Centre National de la Recherche Scientifique (CNRS), the University of Strasbourg and the Agence Nationale de la Recherche (ANR). The authors deeply thank their colleagues of IPCMS who helped performing some of the experiments reported here: L. Andrade, M. Albrecht, J. Arabski, M. Barthelemy, E. Beaurepaire, C. Boeglin, O. Ersen, V. Halté, H. Kesserwan, J. Kim, A. Laraoui, G. Manfredi, S. Moldovan, H. Vonesch.