Femtosecond optical transients are extensively used to study electronic coherence and population relaxation in condensed matter. In this work we show that it is possible to determine the spin coherence and population dynamics in magnetic materials using magneto-optical (MO) four-wave mixing transients. The methodology is applied to Bi-doped garnet films where an ultrashort coherent MO response is observed during the laser pulse and attributed to the spin–orbit interaction. In addition, long-lived spin populations () are shown to depend on the laser frequency detuning with respect to electronic resonance. Otherwise, our approach paves the way for realizing diffractive MO systems that can be controlled with external magnetic fields.
© 2017 Optical Society of America
As is well known, four-wave mixing (FWM) in molecular and semiconductor materials allows determining the relaxation times associated to the electronic coherent and population dynamics. Depending on the studied system, the coherent dephasing time () and the population lifetime () occur on a broad temporal range from tens of femtoseconds to nanoseconds [1–4]. Concerning the coherent dynamics of electronic spin states, many studies have been performed, for example, near the excitonic resonances of semiconductors . So far little is known about the coherence of spin states in correlated spin phases, even though it is important for understanding how to manipulate and control the magnetization of metals, oxides, or molecular magnets with optical beams. It requires development of adapted measurement methods as shown here.
Regarding spin dynamics, extensive studies of the ultrafast demagnetization in ferromagnets have been performed using femtosecond laser pulses [6–8]. A variety of approaches have been used, such as spin-resolved photo-emission , femtoslicing [10,11], table-top x-ray probing , terahertz resonant emission  or excitation, and ultrafast magneto-acoustics . In those experiments, the demagnetization results from a thermal process associated to the change of magnetization modulus and/or orientation, which can be described phenomenologically by a three temperature model [6,15] or by the time-dependent density functional theory [16,17]. The initial demagnetization is accompanied during the first few hundreds of femtoseconds by the thermalization of spins near the Fermi level [9,10,18,19].
Concerning coherent processes in ferri or ferromagnets, several mechanisms are important. First, the inverse Faraday effect induces a transient magnetic field, which has been used, for example, to control the precession of the magnetization in garnets [20,21]. The magnetic component of an intense terahertz field can also be used to coherently control antiferromagnetic spin waves via time-dependent Zeeman interaction . Second, the strong laser field may induce a magneto-electric anisotropy that modifies the magnetization dynamics. The underlying mechanism is the spin–orbit interaction. In ferromagnetic metals, it can be observed in femtosecond pump–probe magneto-optical (MO) Kerr or Faraday experiments . In bismuth-doped garnets, it is well known that a strong spin–orbit interaction occurs, which manifests by a large MO susceptibility far from the main resonances associated to the octahedral and tetrahedral sites . We therefore expect that this material is a good candidate for observing coherent effects associated to MO transient gratings. In this Letter, magnetic-field-dependent FWM signals are measured in Bi-doped garnet film. We use a three-beam configuration  so that the coherent and population components can be separately time resolved. The role of the spin–orbit interaction is investigated using a simple model consisting of an eight-level hydrogen-like system coupled to the laser field via a relativistic dipolar interaction, including the generalized electron momentum .
The MO-FWM experimental configuration is displayed in Fig. 1. Three linearly polarized frequency degenerate femtosecond pulses propagate in the directions , , and . The FWM signal is observed in the direction . is defined as the rotation of the FWM polarization for an angle of the external magnetic field with respect to the normal to the sample plane. is obtained with a polarization bridge consisting of a Wollaston prism and a synchronous detection scheme. can be measured either as a function of the temporal delay between pulses propagating along the directions and or as a function of the delay between pulses along directions and . The linearly polarized pulses with a duration of 50 fs are obtained from an amplified 10 kHz titanium:sapphire-based optical parametric amplifier laser system. In practice, the static magnetic field is set along the two complementary directions and , and the magnetic contribution to the FWM emission is obtained as
Importantly, in the case of a simple two-level system with dephasing time and lifetime , the two aforementioned signals would allow retrieving both and by varying the delays and , respectively. In a first set of experiments, designed to emphasize the coherent magnetic contribution, the central wavelength of the laser pulse is set off-resonance at 800 nm (1.55 eV) with the broadband energy levels of our sample as shown on the absorption spectrum displayed in the inset of Fig. 1. In a second set of experiments, designed to increase the ratio of the spin population component versus the coherence one, the central wavelength is tuned at 600 nm (2.07 eV) and 640 nm (1.94 eV) toward the resonant states associated to the higher energy levels at 2.38 and 2.87 eV in the tetrahedral and octahedral sites of our Bi-doped garnet film.
The sample is a 7 μm thick garnet deposited on a garnet substrate (0.5 mm thick) using liquid phase epitaxy . In the absorption spectrum of Fig. 1, the fringes correspond to the Fabry–Perot effect on the sample. It has a perpendicular magneto-crystalline anisotropy and a weak ferrimagnetic remanence at room temperature. Let us note that the off-resonance excitation is also tuned so that the lower energy transitions  (at 1.38 eV) are also avoided, even though they have a much weaker absorption cross section, as seen in the absorption spectrum in Fig. 1.
Figure 2(a) displays the FWM rotation signals at 1.55 eV for and 180°. The difference (MO-FWM) between these two signals is shown in the inset. This signal corresponds to the coherent contribution of the MO response. Since in the present case the frequency of the laser is detuned from the resonances associated to the crystal field split energy levels , the coherence time is shorter than the duration of the laser. Figure 2(b) shows the magnetic contribution up to at 1.55 eV (closed circles). This signal corresponds to the spin population dynamics associated to the weak remaining absorption at 1.55 eV. An exponential fit to the relaxation leads to a lifetime of . During the first 50 fs, when the three pulses are overlapping in time, the coherent contribution dominates again as the three pulses interact coherently. The coordinate scale is interrupted for convenience to better show the weak population contribution for . In Fig. 2(b) we also display the signals for laser photon energies 2.07 and 1.94 eV. The ratio between population contribution and coherence increases when approaching the energy levels (transitions ).
We attribute the large MO-FWM emission displayed in the inset of Fig. 2(a) () to the coherent coupling between the photons and the spins. This is sustained by the magnetic field dependence of the FWM emission. The microscopic origin of this coherent coupling can be modeled by a dipolar interaction between the laser field and the electrons that are coupled to the spins via the spin–orbit interaction. Such an approach has been proposed to describe MO Faraday experiments performed in ferromagnetic metallic thin films . In the present case of a ferrimagnetic garnet, it is better suited to consider a multilevel system since the description of the MO response in the visible spectrum is usually accounted for by considering the discrete transitions associated to the sites.
We adopt a Hamiltonian similar to the one previously used to describe the general relativistic electrodynamic interaction of a laser with discrete levels . The corresponding eight levels considered in this model are inspired from a hydrogen-like system with transitions to better identify the cases of strong () and weak () spin–orbit couplings due to the magnetic field. We consider the following interaction Hamiltonian, which takes into account the photon–spin interaction, including spin–flip processes via the coupling of the laser field with the orbital and spin angular momenta:
Using the density matrix formalism , we have solved the Liouville equation in the weak field basis, with the appropriate temporal sequence of pulses which, for a given orientation of the external magnetic field (), gives rise to the FWM rotation signal in the direction . These signals are then computed for opposite directions ( and ) of the static field , leading to the MO-FWM signal . Importantly, the third-order perturbation theory requires consideration of the time ordering of the fields to which corresponds either the coherent contribution when , for a fixed and , where is the pulse duration and the dephasing of the spin states, or the population contribution when is fixed and and . Figure 3 shows the coherent and population contributions of the three-pulse MO-FWM, using the above hydrogen-like system. The parameters used are at a photon energy , detuned 18% off the main transition, for each of the three pulses. , , and . is assumed to be shorter than to simulate the experimental result of Fig. 2(a). Also, for simplification, a single population lifetime is considered and assumed to be equal to the experimental case of Fig. 2(b). The modeling of the FWM signals using the interaction Hamiltonian (Eq. 2) show that they are dependent on the magnetization direction, and that it is possible to distinguish between the coherence and population dynamics. The origin of this behavior is due to the spin–orbit coupling term appearing in the generalized electron momentum . It is, therefore, a manifestation of the relativistic electro-dynamics coupling between the laser field and the spins.
In conclusion, the coherent emission from Bi-doped garnet film obtained in a MO-FWM configuration is shown to strongly depend on the orientation of an external magnetic field. The three-beam MO configuration adopted allows investigating the spin coherence and population dynamics in ferro- and ferrimagnetic systems. In the case of Bi-doped garnets excited in their transparency region at 800 nm (with a slight overlap with the transitions), the strong spin–orbit interaction leads to large MO signals. Using femtosecond pulses in a three beam geometry, in the presence of a weak magnetic field, we are able to distinguish between the dephasing and population dynamics of spin states. The dephasing of the spins states () occurs in a time scale much shorter than the relaxation of the populations (1.35 ps). By tuning the laser wavelength toward the transitions of the tetrahedral and octahedral sites, we are able to modify the ratio of the coherent versus population components in the MO-FWM emission. Our observations are well described by a model including the spin–orbit Hamiltonian applied to a simple eight-level hydrogen-like system, with phenomenological dephasing time and lifetime of the spins states. Apart from understanding the initial coherent photon–spin interaction in ultrafast magnetism, the present work shows the high potential of coherent magneto-optics for realizing femtosecond diffractive optical devices controlled by magnetic fields.
European Research Council (ERC) (ERC-2009-AdG-20090325); Agence Nationale de la Recherche (ANR) (ANR-10-EQPX-52).
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