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1 An effective magnetic field from optically driven phonons T. F. Nova 1 *, A. Cartella 1, A. Cantaluppi 1, M. Först 1, D. Bossini 2 #, R. V. Mikhaylovskiy 2, A.V. Kimel 2, R. Merlin 3 and A. Cavalleri 1, 4 * 1 Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany 2 Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands 3 Department of Physics, University of Michigan, Ann Arbor, Michigan , USA 4 University of Oxford, Clarendon Laboratory, Oxford OX1 3PU, UK *Corresponding authors # Present address: The University of Tokyo, Institute for Photon Science and Technology, Hongo, Bunkyo-ku, Tokyo , JAPAN NATURE PHYSICS 1

2 Supplementary material Supplementary S1 Experimental setup A schematic representation of the experimental setup described in the methods section of the main text. Figure S1 Experimental setup. Optical parametric amplifier (OPA), difference frequency generation (DFG), white light continuum (WLC). 2 NATURE PHYSICS

3 Supplementary S2 - Pump characterization The CEP stable pump pulses can be characterized by electro optical sampling (EOS). This technique allows for measuring the pulses electric field profile in the time domain (Figure S2), therefore allowing the retrieval of both the pulse duration and spectrum. The short gate pulses needed for the EOS were generated by a non-collinear OPA (NOPA), pumped with 200 µj of 800-nm light. After frequency-doubling of the pump, a broad portion of the WLC was amplified (ca. 60 nm FWHM bandwidth, centered at 900 nm) to about 2.5 µj pulses, which are compressed to about 21 fs using chirped mirrors. For shorter wavelengths where the gate pulses were too long to sample the electric field in time domain, the spectrum of our pump pulses was measured by a Michelson Fourier Transform Interferometer. Figure S2 EOS trace 22.7 THz, 97 fs pulse. NATURE PHYSICS 3

4 Supplementary S3 - Static birefringence measurement The in-plane sample orientation was determined by measuring its static birefringence at room temperature. The crystallographic axes can be identified by the orientations in which the birefringence is zero, i.e. the light polarization is not altered by the transmission through the sample. A half waveplate and a polarizer (P1) were used to define the incoming light polarization. The light transmitted by the sample was then passing through a second polarizer, and hit a detector. For each orientation of P1, the second polarizer (P2) was rotated to maximize the intensity on the detector. This procedure allowed for measuring the polarization rotation induced by the sample (Figure S3). Figure S3 Birefringence induced rotation of the probe beam polarization. 4 NATURE PHYSICS

5 Supplementary S4 - Polarization sensitive static reflectivity measurement The steady-state reflectivity of our sample was measured by means of Fourier Transform Infrared Spectroscopy (FTIR) with a Bruker Vertex 80v spectrometer. Our instrument allows for polarization resolved measurements. The spectrally resolved reflectivity, measured as a function of the polarization angle of the impinging light, is presented in Figure S4. The two crystallographic axes of the material could be easily identified from this measurement (dotted lines in Figure S4). Their orientation is in agreement with the static birefringence measurement mentioned above. These measurements were taken at 100 K. Figure S4 Spectrally resolved reflectivity measured as a function of the polarization direction of the impinging light. NATURE PHYSICS 5

6 Supplementary S5 Temperature dependence In the main text we reported the frequency of the observed magnetic excitations as a function of temperature (Figure 4c). Here, we show the corresponding time-domain dynamics for different temperatures in between (95 K) and below (85, 65, 10 K) the spin reorientation transition. Figure S5 Transient birefringence changes for different temperatures. The slow-varying component has been subtracted. For every temperature it is shown the complete oscillatory response (grey) upon vibrational excitation. The coloured lines represent a fit to the low frequency components associated with the magnetic response. (a) 95 K. The fit is composed of two exponentially decaying sinusoids of 0.75 THz and 0.15 THz frequency. (b) 85 K. One decaying sinusoid: THz. (c) 65 K THz. (d) 10 K THz 6 NATURE PHYSICS

7 Supplementary S6 - Estimate of total magnetization change Before every pump-probe measurement, we also measured the probe polarization rotation due to the reversal of the sample ferromagnetic moment (i.e. the static Faraday rotation), achieved by flipping the external magnetic field. This was done in the same experimental conditions (position on the sample, light intensity on the diodes, etc.) and with the same setup used for measuring the corresponding pump-probe trace. Furthermore, while the probe is transmitted by the sample, and therefore the probed volume is constant, the penetration depth of the pump (and therefore the pumped volume) is strongly wavelength dependent, due to the vicinity of a resonance. To take all these effects into account, for each data point, the pump induced change of magnetization was computed as the ratio: measured pump induced rotation MM penetration depth of the pump MM = 100 static Faraday rotation thickness of the sample The penetration depth of the pump was retrieved from the FTIR measurements shown above (Figure S4) by fitting the reflectivity using the Variational Dielectric Function method implemented in the software ReFit ( NATURE PHYSICS 7

8 Supplementary S7-1D-FDTD solution of Maxwell s equations The polarization inside ErFe0 3 is simulated by solving Maxwell s equations in space and time. This is done by means of the one-dimensional finite difference time domain method (1D-FDTD) 1. The space and time discretization is done using the Yee grid 2, which is suitable for the simulation of Maxwell equations in absence of free charges, since it intrinsically satisfies the divergence equations EE = 0 and HH = 0. The curl equations are explicitly implemented inside the 1D-FDTD loop. The optical properties of ErFe0 3 are inserted in the simulations through the constitutive equation DD = εε 0 εε rr EE. According to the Lorentz model εε rr is expressed as a series of damped harmonic oscillators corresponding to the IR active phonons, plus an εε taking into account the permittivity due to high energy excitations: εε rr = εε + kk 2 ωω pp,kk 2 ωω 0,kk ωω 2 + iiiiγ kk The values of the plasma frequency ωω pp, the TO frequency ωω 0 and the damping Γ for each of the kk phonons are extracted from a Lorentz fit to the static reflectivity measurements, while εε is extracted from the literature (Ref. 18 of the main text). The constitutive equation for the magnetic field is BB = μμ 0 HH. In the FDTD simulation, perfectly absorbing boundary conditions are implemented. The lattice polarization inside the sample at any time and space can be calculated from PP = εε 0 kk 2 ωω pp,kk 2 ωω 0,kk ωω 2 + iiiiγ kk Since the simulations are 1D, in order to simulate the effect of the pump pulse polarized at 45 with respect to the crystallographic axes, each simulation was performed twice, to calculate the polarization along a and b independently. 8 NATURE PHYSICS EE

9 Supplementary S8 - Estimate of the effective magnetic field As mentioned above, our simulations are one dimensional. To estimate the effective field value let us first consider a single axis (a). We start from the measured reflectivity, which is fitted to yield the dielectric constant (ε(ω)). With this information we simulate the time and space dependent polarization along this axis triggered by the pump pulse. It is known 3 that the polarization can be expressed as a function of the atomic displacements through the formula: where is the electronic charge, is the unit cell size, is the Born effective charge and goes over the 20 atoms of the unit cell. Given the lighter mass of O compared to Er and Fe, we assume that the most significant displacement is due to the oxygen atoms. This is a reasonable approximation especially for the higher energy phonons as those in Fig. 1c. Furthermore, we consider that the oxygen atoms involved equally contribute to the total motion, i.e. they exhibit the same effective charge. We can therefore express the polarization as that is a direct proportionality between polarization and atomic displacement bar a constant. If we now repeat the calculation for the b axis we can obtain the total polarization: NATURE PHYSICS 9

10 From this expression it is clear that the formula is effectively an estimate of the time dependent angular momentum ( ) acquired by the ions upon vibrational excitation. This expression is then included in the Landau-Lifshitz equation (solved following the approach developed in Ref. 25 of the main text) as the source term. The effective field which has the dimensions of an angular momentum, can be estimated from the calculated values by comparing the simulated magnetic dynamics with the measured one, without requiring the knowledge of any of the proportionality constants. Supplementary references 1 Allen Taflove, Computational Electrodynamics: The Finite-Difference Time- Domain Method, Artech House, (2005). 2 Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Transactions on Antennas, 14 (1966). 3 Zhao, X. and Vanderbilt, D. Phonons and lattice dielectric properties of zirconia. Phys. Rev. B 65, (2002) 10 NATURE PHYSICS