Electric and magnetic terahertz nonlinearities resolved on the sub-cycle scale

Table-top sources of intense multi-terahertz (THz) pulses have opened the door to studies of extreme nonlinearities in the previously elusive mid- to far-infrared spectral regime. We discuss two concepts of fully coherent coupling of phase-locked THz pulses with condensed matter. The first approach demonstrates two-dimensional multi-THz spectroscopy of the semiconductor material InSb. By phase- and amplitude-sensitive detection of the nonlinear optical response, we are able to separate incoherent pump–probe signals from coherent four-wave mixing and reveal extremely non-perturbative nonlinearities. While this class of interactions is mediated by the electric field component of the THz pulse, the second approach is complementary, as it demonstrates that, alternatively, the magnetic THz field may be exploited to selectively control the spin degree of freedom in antiferromagnetic NiO.


Introduction
The incentive to study and control low-energy collective excitations in condensed matter on the femtosecond time scale has spurred on the development of a unique arsenal of sources and detectors in the terahertz (THz) spectral band [1,2]. Inherently phase-locked single-cycle THz pulses have become routinely available while ultrabroadband electro-optic sensors trace the transient field with respect to absolute phase and amplitude throughout the entire far-and mid-infrared range [3,4]. Low-energy dynamics have thus become amenable to observation on time scales shorter than a single oscillation period of light [5].
The crucial step from ultrafast linear to ultrafast nonlinear THz interactions has been heralded by the recent boost in energies of THz pulses beyond 1 µJ, whereby peak electric fields above 1 MV cm −1 have been reached [6][7][8][9][10]. When intense THz electromagnetic pulses interact with the many-body system of a solid, a variety of non-equilibrium scenarios may result. Recent experiments have explored intervalley scattering [11][12][13], interband tunneling [14], impact ionization [15,16], and the anisotropy of the effective mass [17] in semiconductors, anharmonicities of a soft phonon mode in ferroelectrics [18], as well as non-thermal suppression of electronic localization in strongly correlated systems [19,20]. Most studies have concentrated on the effects of THz excitation after the coherence between the external driving field and the induced polarization has decayed. On the other hand, first examples of coherent THz nonlinearities have been demonstrated with semiconductors [21][22][23][24] and rotational states of gas molecules [25,26]. These studies re-emphasize the importance of dynamics occurring on timescales shorter than the dephasing time of the electron system. Here we report on two novel classes of coherent nonlinearities. The recent breakthrough in the generation of ultra-intense multi-THz transients with peak electric fields up to 100 MV cm −1 [27,28] allows us to explore a regime of a light-matter interaction where the Rabi frequency becomes comparable to the oscillation frequency of the driving THz field. In this case, the nonlinear response reaches the limit of extremely non-perturbative optics [29]. Furthermore, the complementary magnetic component of THz pulses can strongly couple to a spin subsystem and coherently control the precession of a macroscopic magnetization [30,31]. Both aspects are covered in our discussion.
In the first part of this article, we report on the coherent nonlinear THz response of an interband polarization in the narrowband semiconductor InSb, studied by means of fieldresolved four-wave mixing (FWM) spectroscopy [32]. The carrier frequency of the THz driving field is set well below the interband resonance. In the limit of perturbative light-matter interactions, as are considered in classical nonlinear optics, the FWM process is fully governed by a third-order optical nonlinearity. However, this regime is realized only when the Rabi frequency is much smaller than the detuning of the driving frequency from a resonance. Our experiment demonstrates that this condition is violated for peak electric fields in excess of 3 MV cm −1 , as the FWM signal shows clear signatures of a non-perturbative response caused by coherent Rabi flopping.
The second part of this article focuses on an ultrafast control of collective spin precession in antiferromagnetic NiO at frequencies as high as 1 THz [33]. By using intense magnetic THz transients, we switch a coherent magnon on and off, with sub-cycle precision.

Non-perturbative nonlinear response of a bulk semiconductor under off-resonant terahertz (THz) excitation
Semiconductors, due to our advanced theoretical understanding and experimental control of their material properties, provide a prime test bed for the investigation of light-matter coupling in quantum many-body systems. In particular, their interaction with extremely intense THz electromagnetic fields has attracted attention for potential applications in both modern optics and high-speed optoelectronics [11,12,14,16,17,34]. We utilize high-field multi-THz pulses to study the coherent response of bulk InSb by means of two-dimensional spectroscopy. Following the generic concept of a degenerate FWM experiment we address an interaction of two intensive THz pulses within a sample, as illustrated schematically in figure 1. The temporal position of pulse A is fixed while pulse B is delayed with respect to the former by a variable delay time τ .
The leading pulse (A or B, depending on the sign of τ ) can induce a change in the optical constants of the sample and, thus, affect the propagation of the subsequent pulse. This process gives rise to a nonlinear pump-probe signal which gradually varies as a function of τ . The pump-probe signal temporally coincides with the probe pulse and propagates in the same spatial direction. An alternative interaction pathway that is also possible proceeds as follows: a coherent polarization induced by the leading pulse can interfere with a trailing pulse, resulting in a periodic modulation of the optical constants. Part of the second pulse may then diffract from the emergent transient grating, generating a FWM signal. In the case of the non-collinear geometry shown in figure 1, the FWM signal propagates in a different direction than either pulses A or B, where the direction is defined by the conservation of the wavevectors involved in the nonlinear interaction. As we will demonstrate below, FWM and pump-probe signals may also be distinguished directly via the dependence of their respective phases. Complete amplitude and phase resolution of the nonlinear optical response will provide us with a sensitive tool to separate various nonlinearities and do so independently of their spatial propagation direction.

Field-sensitive two-dimensional spectroscopy
In order to test the validity of these considerations we explore the FWM response of a bulk sample of the semiconductor InSb. This material features a narrow direct bandgap of E g = 170 meV= 41.1 THz. A single crystalline window with a thickness of 30 µm is investigated at room temperature. The small energy gap of InSb results in a relatively large density of intrinsic charge carriers of n 0 = 2 × 10 16 cm −3 . High-field THz pulses are generated by a tabletop laser system using difference frequency mixing in a GaSe emitter [28]. The duration of the pulses amounts to 150 fs (full-width at half-maximum (FWHM)) and their carrier frequency is f 0 = 23 THz. Thus, the complete spectrum is located below the absorption edge of InSb, ensuring off-resonant driving of the polarization response. The THz radiation is tightly focused onto the surface of the semiconductor. The electric fields re-emitted by the oscillating polarization in the sample are collected by an off-axis parabolic mirror with large numerical aperture ensuring the angle-integrated detection. The field-resolved THz waveforms are measured electro-optically in a GaSe sensor. Further technical details of the experiment are summarized in [32].
The total field emitted from InSb, in the presence of both pulses A and B, is measured as a function of the electro-optic sampling delay time t and the relative pulse delay τ (see figure 2(a)). The peak electric fields at the focal spot, which has a diameter (FWHM) of 85 µm, amount to 2 MV cm −1 per pulse. This value corresponds to an intensity of ≈10.6 GW cm −2 . The nonlinear contribution is obtained by subtracting the transmitted fields of the individual pulses A and B from the total electric field. The resulting two-time-dependent nonlinear response is shown in figure 2 Even though these data contain all nonlinear optical processes in an angularly integrated way, the FWM signals may be sharply discriminated from the pump-probe signatures by their specific phase dependencies. Since the FWM process relies on the interference of both driving pulses, a FWM signal depends on the relative phase of both excitation fields and, therefore, the phase periodically varies as a function of τ . A Fourier transformation in two-dimensional frequency space allows us to separate FWM from pump-probe nonlinear processes. Figure 2(c) depicts the amplitude spectrum of the total nonlinear signal in frequency space. Different contributions to the nonlinear signal appear as distinct pairs of maxima conjugated by the inversion operation with respect to the origin [35]. For a qualitative understanding of the origin of the various maxima in frequency space, it is instructive to note that the Fourier transform (FT) of pulse A would be centered at the position k A = ( f 0 , 0) and its conjugate −k A . This fact is straightforward to understand since the electric field of pulse A, which is not shifted in time, does not oscillate as a function of τ , while it does oscillate with frequency f 0 along time t. As a consequence of this, nonlinear signals observed at k A represent changes in pulse A induced by pulse B. In other words, these signatures represent pump-probe signals with pulses B and A acting as pump and probe, respectively. As a shorthand we denote this interaction as PP-BA. The temporal evolution of this particular signal is obtained by the selective inverse FT of the signal at k A (see figure 2(d)). Due to causality, the signature PP-BA gradually vanishes for τ > 0 when the pulse B arrives after the probe pulse A.
Similarly, the spot located at k B = (+ f 0 , − f 0 ) corresponds to changes imprinted on pulse B. The temporal shift of pulse B leads to oscillations of its electric field along the τ -axis resulting in f τ = − f 0 . The negative sign of the frequency is caused by the positive phase shift in the field profile when the delay time t increases. The corresponding pump-probe signal PP-AB appears for τ > 0 when the time order of the pulses is reversed (see figure 2(e)).
Finally, the maximum at the position 2k A − k B = (+ f 0 , + f 0 ) and its conjugate correspond to a coherent FWM signal. As pointed out above, the FWM field also oscillates as a function of τ due to the phase-sensitive character of the interaction between the pulses A and B. However, in contrast to the PP-AB signal, the FWM signature appears at positive frequency f τ = f 0 . Therefore, its phase experiences a negative shift with increasing delay time t, as it is seen in the time-domain of the FWM signal shown in figure 2(f).
In contrast to the FWM signal, the pump-probe signals observed in our experiment are temporally extended far beyond the duration of the interacting THz pulses themselves. This observation indicates that the main contribution to the pump-probe response in InSb originates from a long-living electron-hole plasma produced by the intense THz field of the pump pulse. Possible mechanisms of free carrier generation by an excitation below the bandgap of InSb such as impact ionization [15], two-photon absorption (TPA) [36] or interband tunneling [37] have been studied previously. Here we concentrate on the FWM signal, which characterizes coherence induced in the sample by the phase-stable THz pulses.

Figures 3(a)-(c)
show the time-domain FWM obtained from measurements performed with peak electric fields of 2, 3.5 and 5.3 MV cm −1 per pulse, respectively. The envelope of the FWM signal qualitatively changes with increasing field strength and evolves from an oval shape at the lowest field (figure 3(a)) via an S-shape at 3.5 MV cm −1 (figure 3(b)) towards a split area at the highest excitation intensity (figure 3(c)). This splitting results in a minimum of the FWM signal along the diagonal of the time window (t = τ ) where the total excitation field reaches maximal values. In the following discussion we demonstrate that this behavior is an unambiguous signature of a non-perturbative polarization response in InSb.
In order to understand the character of the light-matter interaction in our experiment, we perform simulations of the FWM response assuming either a perturbative response due to a third-order nonlinearity χ (3) or a response of a quantum two-level system coherently driven by intense THz pulses. In both frameworks, the electric fields of the THz pulses are given by where z is the spatial coordinate along the beam axis, I (z, t) is the Gaussian intensity envelope with a FWHM duration of 100 fs, Z = √ µ 0 /ε 0 ε is the wave impedance, k and ω 0 = 2π f 0 are the wavevector and the carrier frequency, and (z, t) is the phase shift, which is initially set to zero (Fourier-limited pulses).

Theory of perturbative FWM
The pulse propagation through the sample affected by χ (3) nonlinearities and free charge carriers can be described by the following system of equations [37]: Here β is the TPA coefficient and n is the excess density of free carriers generated via TPA. The coefficient α = σ FC (n 0 + n) describes the linear absorption due to the holes in the valence band, where σ FC = 5.6 × 10 −16 cm 2 is the interaction cross-section [37]. The TPA coefficient β is directly proportional to the imaginary part of the third-order nonlinear coefficient, Im χ (3) . In turn, the real part Re χ (3) is responsible for the Kerr effect, which leads to the phase modulation described by the second term on the right-hand side of equation (3). The first term in this equation takes into account the phase modulation caused by the plasma response of free carriers in the sample. The respective coefficients for InSb are A = 1.15 × 10 −24 m −4 [37] and B = 1.7 × 10 −19 m W [38]. Here the main contribution to the Kerr effect stems from the response of free electrons in the non-parabolic conduction band of InSb [38], which exceeds the nonlinear response of the bound valence-band electrons by more than an order of magnitude. The TPA coefficient β is set to be 0.3 cm MW in order to obtain the closest agreement with the experimental results. This value is somewhat smaller than the 2 cm MW −1 typically reported by studies that use picosecond and femtosecond pulses with peak intensities of several MW cm −2 [36,37]. The discrepancy between the two coefficients may indicate a saturation of the TPA process by Pauli blocking, as our experiment was performed using GW cm −2 peak intensities. Eventually, the TPA picture may break down due to the onset of a non-perturbative excitation regime.
Our simulations based on equations (2)-(4) assume plane wave propagation and, thus, neglect self-focusing effects which are assumed to be minor in the cases of the 30 µm-thick sample and the apertureless electro-optic detection utilized in our experiment. In order to analyze the FWM response produced by perturbative nonlinear effects we introduce them step by step in our simulations. Let us first consider only the Kerr effect related to the Re χ (3) nonlinearity and neglect all other contributions (B = 0, α = β = 0). The simulated time-domain FWM signals are shown in figures 3(d)-(f) for the same peak THz fields as those used in the experiment. The envelope of the FWM signature remains oval for all excitation intensities with a noticeable distortion of the constant phase lines due to a strong cross-phase modulation at the highest peak field of 5.3 MV cm −1 . Thus, the pure Kerr effect cannot explain the experimental observations, even though the amplitudes of the simulated FWM signals are comparable to the experimental values.
As a second step, we present the results of simulations that take into account solely the pulse attenuation by the TPA process (only equation (2) where β = 0 and α = 0). At the lowest field setting, TPA causes an oval-shaped FWM signal, as shown in figure 3(g). Higher driving fields lead to the characteristic S-shaped FWM signals shown in figures 3(h) and (i). The shape of these signals resembles the experimentally observed signatures for peak fields of 3.5 MV cm −1 ( figure 3(b)). However, TPA simulations for even higher driving fields (not shown) result in the same S-shaped FWM signatures and do not show any sign of the splitting observed in the experiment. Moreover, the amplitude of the simulated FWM signals saturates at a level of ≈50 kV cm −1 and stays far below the amplitudes observed in the experiment (see figures 3(h) and (i)). This fact can be qualitatively understood by keeping in mind that the TPA is proportional to the radiation intensity: during propagation the intensity of the pulses is mostly suppressed in the middle of the pulse, where the intensity is highest. Therefore, the intensity profile of the pulses develops a 'flat' top and the nonlinear signal saturates. Obviously, the TPA process cannot account for the field minimum observed in the center of the FWM signal ( figure 3(c)).
In fact, the switch-off analysis presented above is not completely physical. TPA has to be considered since 2hω 0 > E g , while the Kramers-Kronig relations enforce the simultaneous presence of Re χ (3) [39]. Moreover, the dominating contribution of the free carriers to the phase modulation and the energy absorption also must be taken into account. The results of such simulations are shown in figures 3(j)-(l). The shapes and amplitudes of the modeled FWM signals are similar to those shown in figures 3(g)-(i). Thus, we conclude that TPA dominates the simulated perturbative response. A strong nonlinear absorption makes the role of the Kerr effect almost negligible and the nonlinearity of the free carrier plasma leads to a broadening of the FWM signature along the time axis t and a partial compression along the τ -axis. In any case, our model demonstrates that neither the shape nor the amplitude of the experimental FWM signals are adequately described within the framework of perturbative nonlinear optics.

Non-perturbative FWM in the two-level approximation
We now proceed by demonstrating that these deficiencies may be cured with numerical simulations of our FWM experiment using a simplified model of a two-level system, which represents the interband resonance in InSb. The pulse propagation is modeled by the Maxwell-Bloch equations [40]. Based on previous reports, we assume a dephasing time T 2 = 1 ps [41]. Since the timescales involved in our FWM experiment are much shorter than T 2 , the exact values of the relaxation time hardly affect the results of the simulations. For the same reasons we choose a depopulation time T 1 = 10 ps. The remaining two parameters of the model, i.e. the transition dipole moment µ 12 = 2.4 eÅ and the density of the two-level systems N = 2.9 × 10 20 cm −3 are adapted to provide best agreement with the experimental data. The simulated FWM signals are depicted in figures 3(m)-(o). Remarkably, these simulations reproduce all essential features observed in the experiment, such as the amplitudes of the FWM signals and, even more importantly, the characteristic splitting for the peak field of 5.3 MV cm −1 . This clearly demonstrates the non-perturbative character of the polarization response in InSb, in spite of the strong detuning of the excitation with respect to the interband resonance.
Better physical insight can be gained by examining the simulated coherent polarization dynamics of the interband resonances in InSb driven by pulses A and B at vanishing delay time τ . Our simulations assume an ensemble of 50 two-level systems with resonance frequencies distributed between 41 THz and 51 THz with an energy density proportional to h ω − E g , coarsely mimicking the continuum of interband transitions. Figure 4(a) shows the normalized temporal profiles of the driving THz pulses with peak fields of 2 MV cm −1 per pulse (4 MV cm −1 in total), the total polarization, and the frequency components of the polarization centered around the fundamental frequency f 0 , as obtained by Fourier filtering shown in figure 2(c). As one can see in figure 4(a), the total polarization P tot as well as its fundamental component P fund closely follow the driving electric field E THz . Thus, small deviations of P tot from E THz can be described within a perturbative approach. Correspondingly, the simulated FWM signal has an oval shape ( figure 3(m)). The situation changes when the peak driving fields reach 3.5 MV cm −1 per pulse. Figure 4(b) demonstrates that the total polarization oscillates at much higher frequencies after the driving field has assumed its maximum, i.e. at delay times t > 0. For such excitation conditions, the Rabi frequency R ∝ µ 12 E THz characterizing the strength of light-matter interactions becomes comparable to the detuning of the fundamental frequency from the lowest interband resonances. This leads to a non-perturbative response of the two-level systems driven into the regime of carrier-wave Rabi flopping [32,42]. The major contribution to P tot occurs at higher odd harmonics [43,44], whereas P fund starts to decrease (see figure 4(b)). At the highest peak driving fields of 5.3 MV cm −1 this effect becomes very pronounced and a distinct minimum develops in the center of P fund , as shown in figure 4(c). Notably, this effect is responsible for the splitting of the time-domain FWM signal observed in our experiment ( figure 3(c)). Thus, we establish an unequivocal manifestation of a non-perturbative response of InSb under an offresonant excitation by high-field THz pulses.

Spin control in antiferromagnets using magnetic THz transients
In the preceding section, the transient electric field E of intense THz pulses was employed to coherently drive matter into a non-equilibrium state. The magnetic field component B was ignored since it exerts a force (the Lorentz force) on an electron that is smaller by a factor of |v|/c than the electric contribution (the Coulomb force). Here, v is the electron velocity, and in atoms or molecules, |v|/c is of the order of the fine-structure constant of ≈1/137. Similar arguments indicate that the interaction energy between the B-field and its induced magnetic dipole is reduced by a factor of (v/c) 2 ∼ 10 −4 compared to the interaction energy between E and its induced electric dipole [45].
On the other hand, magnetic fields represent the most direct handle to control the quantummechanical spin S of an electron. As S is associated with a magnetic dipole moment it couples to B through the Zeeman Hamiltonian γ S · B where γ denotes the gyromagnetic constant. As a consequence, a perturbing magnetic field exerts a torque γ S × B, thereby pulling the spin out of its equilibrium direction, set by an effective internal magnetic field B int ( figure 5(a)). Note that the weak Zeeman torque is resonantly enhanced when the driving frequency equals γ B int . In ferromagnets, this so-called Larmor frequency does not exceed 10 GHz, in contrast to antiferromagnets in which the perturbing field destroys the perfectly antiparallel spin configuration ( figure 5(a)). Thus, the exchange interaction becomes operative and induces a strong repulsive torque, thereby pushing the resonance frequency of long-wavelength spin waves into the THz range. These high frequencies make antiferromagnets, which represent the majority of magnetically ordered solids in nature, particularly interesting for high-speed spin manipulation. So far, direct Zeeman-type control of magnons has been demonstrated in ferromagnets on time scales of ∼100 ps [46][47][48], yet it has been inhibited in antiferromagnets due to a lack of sufficiently fast and intense magnets. Instead, optical pulses have been used to generate effective transient magnetic fields either through heating or stimulated Raman processes [49]. Ultrafast electron heating leads to a rapid change in the internal field B int , yet the associated cooling dynamics limit the speed and thus the compatibility of electron heating with coherent control of THz resonances. The non-thermal Raman approach invokes the electric field of an off-resonant laser pulse but requires materials with massive spin-orbit coupling. In addition, the underlying microscopic processes of both optical approaches are not yet fully understood [49] and lack the simplicity and universality of the generic Zeeman-type control.
Here, we exploit the recent development of sources of high-field phase-locked electromagnetic transients with amplitudes of ∼1 MV cm −1 and ∼0.5 T that cover the frequency range below 10 THz [6,[8][9][10]17]. Using such pulses, we switch a coherent 1-THz magnon oscillation in antiferromagnetic NiO on and off on a single-cycle timescale.

Experimental details
A schematic of our experiment is shown in figure 5(b): an intense THz magnetic field transient incident on the sample system is expected to perturb the electron spins via direct Zeeman interaction. The induced magnetization M(t) is probed by a near-infrared laser pulse as a function of the delay time t after the THz pump. Here, we take advantage of the Faraday effect: the projection of the magnetization onto the propagation direction e k of the probe pulse causes a transient circular birefringence which rotates the probe polarization by an angle [49] θ F (t) = V d e k · M .
(5) Here, V and d are the magneto-optic Verdet constant and the sample thickness, respectively, whereas . denotes averaging over the probed sample volume. We choose the textbook antiferromagnet NiO as a test system for this study. The high Néel temperature T N = 523 K allows us to conduct all experiments under ambient conditions. Figure 5(c) shows the quasi-cubic crystal structure. Below T N , the Ni 2+ spins are ordered ferromagnetically within {111} planes where they point along the 112 -axis [50]. Adjacent planes are oppositely magnetized due to antiferromagnetic coupling. Each of the four equivalent 111 stacking directions defines a twin (T) domain which is subdivided into spin-rotation (S) domains as the spins can point in three equivalent in-plane directions. Birefringence induced by a slight lattice contraction along the stacking axes allows us to map the domain structure with a polarization-sensitive microscope as seen in figure 5(d). For our free-standing NiO window (thickness d = 45 µm), we find dimensions of the T domains of the order of 100 µm. The S domains are not resolved since they are less than 1 µm wide [50]. Inelastic neutron and light scattering as well as infrared absorption have identified long-wavelength antiferromagnetic magnon modes in NiO at ≈ 0.2 and 1 THz [51].
A high-field THz transient is generated by optical rectification of 5-mJ, 800-nm, 100-fs laser pulses in a large-area ZnTe emitter and focused onto a single T domain (white circle in figure 5(d)). The microscopic geometry depicted in figure 5(c) precisely corresponds to the experimental situation for one of the three possible S domains. Figure 6(a) shows the time trace B(t) of the incident magnetic pulse in its polarization plane as recorded by electro-optic sampling [17]. This single-cycle transient reaches a magnetic peak field of 0.13 T and covers a broad spectral range from 0.1 to 3 THz, fully overlapping the magnon resonance at 1 THz ( figure 6(c)). An 8-fs pulse with a central photon energy of 1 eV serves as our probe to map out the pump-induced Faraday rotation θ F as a function of time t. Figure 6(b) displays the experimental results: a harmonic oscillation with a period of 1 ps sets in within a single cycle, reaches its maximum amplitude at t = 3 ps, and decays exponentially with a time constant of 39 ps, thereafter. Coherent oscillations exceeding the noise floor may be resolved even after several hundreds of picoseconds (not shown). Figure 6(c) shows that the FT of the Faraday transient consists of a narrow peak at 1.0 THz, clearly identifying the signal as the magnetization signature of a coherent oscillation of a long-wavelength antiferromagnetic magnon in NiO. The spectrum of the Faraday signal also features a barely discernible faint peak at about 0.2 THz (figure 6(c)) that can be assigned to a lower-frequency long-wavelength magnon [51]. Our data do not show indications of broadband excitations such as Joule heating. Free-carrier and lattice absorption are also negligible because NiO is an insulator, and all optical phonon resonances are located above 12 THz, far beyond the frequencies considered here.

Single-pulse excitation
Most remarkably, the observed spin dynamics are driven by the magnetic field B only. Using a symmetry argument, the electric field E may safely be ruled out as an excitation channel: since the Faraday signal is proportional to the pump field [33], an electric dipole interaction with magnons (interaction energy ∝ M · E) would represent a linear magnetoelectric effect. In centrosymmetric materials like NiO such processes are forbidden.

Microscopic description
In order to gain further insight into the spin motion and its driving force, we also performed microscopic simulations of the spin dynamics. As the driving THz transient has a wavelength that is large compared to the NiO lattice constant, spin motion is uniform within each 111 lattice plane. Thus, the magnetic state of NiO is adequately characterized by the spins S 1 and S 2 of adjacent 111 Ni 2+ planes, and the spin Hamiltonian of [51] simplifies to Here, the first term describes the exchange interaction, and the exchange energy J is negative as the planes are antiferromagnetically coupled. The second term captures intrinsic anisotropic effects such as spin-spin magnetic-dipole coupling and the influence of the crystal potential on the spins mediated by the spin-orbit interaction. This anisotropy ensures that the spins are preferentially oriented in-plane along the 112 direction. The last term describes the Zeeman interaction between the spins and the driving magnetic field. Considering S 1 and S 2 in equation (6) as classical observables, we obtain Landau-Lifshitz-Gilbert-type equations of spin motion, where B eff j = B(t) − J S 3− j /γ + (D x S j x , D y S j y , 0) t /γ is an effective magnetic field acting on spin j. Apart from the Gilbert damping constant α, which is adjusted to (2.1 ± 0.1) × 10 −4 to best replicate the measurement results, all parameters of equation (6) have been determined previously [51]. In particular, the driving field B is determined by taking the measured incident transient B(t) (figure 6(a)) and accounting for multiple reflections at the NiO crystal surfaces.
By numerically solving equation (7) for S 1 and S 2 , we obtain spin dynamics that are dominated by a precession with a period of 1 ps ( figure 7(a)). S 1y exhibits an additional weak component with a period of about 5 ps, corresponding to the lower-frequency long-wavelength magnon of NiO at a frequency of ≈0.2 THz [51]. An image of the three-dimensional spin trajectory is displayed in figure 7(c): the Zeeman torque pulls the spins along the x-axis, thereby triggering a precessional motion about the equilibrium (z) direction. While the x component is antiparallel for both spins, the anisotropy field causes a smaller y component that leads to a non-vanishing macroscopic transient magnetization component M y .
To arrive at the Faraday rotation, we use equation (5) and average M ∝ S 1 + S 2 over all S domains. Figure 6(b) demonstrates a striking agreement between experiment and theory, both with respect to the shape of the transient and its absolute magnitude. This result shows that the magnetic field of the incident THz pulse triggers a spin precession corresponding to the high-frequency long-wavelength magnon at 1 THz through the Zeeman interaction.

Coherent THz control of magnons by two-pulse excitation
The long lifetime of the spin precession (figure 6(b)) allows for it to be coherently controlled over quite a large time window. For this purpose, we excite the NiO with different sequences of two pulses. When the second THz pulse is applied precisely six precession cycles after the first, the amplitude of the induced magnetization doubles ( figure 6(d)). This amplification arises because the torque of the second pulse is in phase with the spin precession induced by the first pulse. The amplification effect is reversed if the second pulse arrives 6.5 precession cycles after the first (figure 6(e)). In this case, the torque of the second pulse is entirely out of phase with the spin precession and switches the dynamics off. We note that our simulations again provide curves in excellent agreement with the experimental data.

Future directions: magnon probes and spin flips
We have successfully demonstrated the coherent control of ultrafast spin precession in antiferromagnetic NiO using the Zeeman torque of an intense THz transient. Such a control scheme is minimally invasive (as it leaves the non-spin degrees of freedom of the NiO in their ground state) and universal (as it is mediated by the Zeeman interaction). Coherent magnons may now be used as probes for ultrafast interactions of electron spin with orbital motion and lattice modes in essentially all THz-transparent matter and at all relevant frequencies.
We emphasize that Zeeman-driven spin motion is expected to be widely scalable, and our simulations based on equation (7) indicate a novel regime of dramatic THz nonlinearities beyond the perturbative regime. In contrast to the weak-field regime (figure 7(a)), THz peak amplitudes of 11 T lead to oscillations of S 1x significantly slower than the high-frequency magnon at 1 THz, and the evolution of S 1y is dominated by the low-frequency magnon at ≈0.2 THz ( figure 7(b)). Note that both signals exhibit a modulation depth of nearly 100 %. Most remarkably, the change of S 1z from 1 to nearly −1 demonstrates the induction of a spin flip, which also becomes apparent in an image of the three-dimensional spin trajectory ( figure 7(d)). It is not obvious whether in such an extreme excitation regime the parameters of the spin Hamiltonian of equation (6) can still be assumed to be constant. However, recent modeling based on an extended Hamiltonian has demonstrated similar magnetization dynamics [52]. In any case, future studies of high-field-driven THz spin dynamics will enter an entirely new territory of many-body physics.

Conclusions and perspectives
Intense electric and magnetic multi-THz fields have become versatile tools to selectively and coherently control electronic or spin degrees of freedom of condensed matter in a previously elusive spectral range, and in the sub-optical-cycle temporal regime. Our study of multi-THz two-dimensional spectroscopy confirms the onset of a non-perturbative response under even off-resonant excitation of interband transitions in InSb. Peak electric fields above 5 MV cm −1 lead to carrier-wave Rabi flopping between the valence and conduction bands. These results underpin the large potential of novel strong-field multi-THz technology for high harmonics generation [53] and coherent quantum control of low-energy excitations [22,54]. The unique capability of field-sensitive multi-dimensional spectroscopy to separate purely coherent and incoherent nonlinearities may open a new chapter in the investigation of collective low-energy excitations including inter-molecular vibrations in hydrogen-bonded liquids [55] and Cooper pair breaking transitions in superconductors [56]. On the other hand, all-magnetic spin control opens the door to the new concept of THz electron spin resonance. Coherent magnons in systems as diverse as ferro-and antiferromagnets or high-temperature superconductors may, for instance, be employed as unique probes of ultrafast interactions with lattice and charge degrees of freedom. Furthermore, our experiments may inspire the development of ultrafast next-generation memory devices, spin-based quantum computation and spintronics. This is only the beginning of a new era of sub-cycle multi-THz coherent control of condensed matter.