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1 Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations O. Schubert 1, M. Hohenleutner 1, F. Langer 1, B. Urbanek 1, C. Lange 1, U. Huttner 2, D. Golde 2, T. Meier 3, M. Kira 2, S. W. Koch 2 and R. Huber 1 1 Department of Physics, University of Regensburg, Regensburg, Germany 2 Department of Physics, University of Marburg, Marburg, Germany 3 Department of Physics, University of Paderborn, Paderborn, Germany NATURE PHOTONICS 1

2 1. Tuning HH spectra by variation of the driving frequency The fundamental THz fields are provided by a high-field source (see Methods Summary). This tabletop system allows us to readily tune the centre frequency while strictly maintaining absolute CEP stability of the THz waveforms. Figure S1 depicts a collection of HH spectra recorded while the driving frequency is swept between 27.3 THz and 31.8 THz. Figure S1 Frequency variation of HHG. Spectral intensity of high-order harmonics emitted from a single crystalline GaSe window (thickness: 220 µm) as a function of the frequency of the driving field (vertical scale). While the photoluminescence signal (PL) remains constant at 476 THz, the comb of high harmonic frequencies fans out with increasing fundamental frequency. The dotted white lines are a guide to the eye. 2. Phase coherence of HHG: Variation of time delay in f-2f interferometry In order to test the CEP stability of HH radiation we utilize f-2f spectral interferometry. As seen in Fig. 2c, the interference of the 12 th harmonic with the frequency-doubled wave of the 6 th harmonic gives rise to high-contrast spectral fringes. The fringe period is set by the temporal delay between the interacting waves. This delay may be sensitively tuned by inserting a dispersive glass (BK7) window of variable thickness L BK7 before the SH generation crystal (Fig. S2a). In Figs. S2b-d we find spectral periods of 4 THz, 3 THz and 2.5 THz for L BK7 = 0 mm, 5 mm, and 10 mm, respectively. The visibility of the interference fringes is only limited by the resolution of the spectrometer used and does not depend on the integration time, attesting to excellent phase coherence within the HH comb. Repeated measurements of the spectral interferogram over a time window of 10 minutes demonstrate the longterm stability of the spectral phase within the frequency comb (Fig. S2e). 2 NATURE PHOTONICS

3 SUPPLEMENTARY INFORMATION Figure S2 f-2f spectral interferometry within the THz-driven HH comb. a, Schematic setup: The HH radiation passes dispersive BK7 windows of variable total thickness L BK7, is focused into a BBO crystal (thickness L BBO = 3 mm) for phase-matched SHG, transmitted through a polarizer (P) and guided onto the entrance slit of a spectrometer (SPEC). b-d, Interference of the 12 th harmonic with the frequency-doubled 6 th harmonic for b, L BK7 = 10 mm, c, L BK7 = 5 mm, and d, L BK7 = 0 mm. e, Repeated measurements of the spectral interferogram over a time window of 10 minutes. NATURE PHOTONICS 3

4 3. Quantum model of ballistic charge transport To study the emission characteristics of GaSe, we perform detailed simulations based on a microscopic model. We start from the general semiconductor Hamiltonian S1 (1) with band index λ. The first term contains fermionic electrons within band. The light-matter interaction stimulates two different processes: The term with the dipole matrix element describes interband transitions where electrons are moved from one band to another due to the presence of an external electric field E(t). The term with k excites intraband currents where the electrons are accelerated by E(t). As shown in Ref. S2, the main effect of Coulomb interaction in the regime of extreme nonlinear optics can be modeled as dephasing among electronic excitations. Using the Heisenberg equation of motion S1, the operator dynamics can be evaluated, leading to the well known semiconductor Bloch equations (SBE) S3 which describe the time evolution of the microscopic polarizations and the electron and hole occupations, defined via (2) In the following, we analyze the excitation among conduction and valence bands and transform the equations into the electron-hole picture S1. Since GaSe has inversion symmetric bands with respect to the wave vector k, a two-band model with one conduction and one valence band cannot explain the occurrence of even harmonics in the emission spectrum. For this reason, we generalize the theory presented in Refs. 9 and 10 beyond a 2-band description. Note, that dynamic distortions of the band structure are included in this model via the field-induced transitions between all bands. The main approximation is in the number of bands we account for. In order to obtain a realistic description, we implement a 5-band model including two conduction bands (λ = e 1, e 2 ) and three valence bands (λ = h 1, h 2, h 3 ). The resulting SBEs become for the carrier occupations and (3) 4 NATURE PHOTONICS

5 SUPPLEMENTARY INFORMATION (4) for the microscopic polarizations. We solve this closed set of coupled differential equations numerically, for an initially unexcited system. A phenomenological dephasing is added to the polarization dynamics via the decay time T 2. The system is excited by E(t), which is modeled as a Gaussian THz pulse with a central frequency of 30.1 THz and a duration of fs. Since we want to include multiphoton processes, the non-rotating-wave contributions are fully accounted for. This yields a challenging propagation problem with t = 10 3 fs as time steps. The emission intensity of a coherently excited semiconductor contains both a polarization source P(t) and a current source J(t), (5) The current matrix element j λ (k) in J(t) is given by (6) where is the band structure of GaSe. Their Fourier transforms P(ω) and J(ω) provide the emission spectrum 9,S4 (7) NATURE PHOTONICS 5

6 We use tight-binding bands to model the valence bands and an improved model of the tight-binding bands according to Ref. 10 for the conduction band. The band structure parameters are taken from Ref. S5. According to Ref. S6 the inter-valence-band dipole-matrix element d h1h2 is an order of magnitude stronger than the transition probabilities d h1e and d h2e between valence and conduction band Carrier-envelope phase and Bloch oscillations Figure S3a shows the time evolution of the electron occupation in the conduction band for two different carrier-envelope phases (CEP) CEP. More specifically, we study the effect of a pure sign change by comparing CEP = 0 (blue, solid line) and CEP = π (red, dotted line) in Fig. S3a. Also the integrated occupation of the conduction band, n(t), is shown for CEP = 0 (shaded area) and CEP = π (red curve). It is remarkable that the occupation drastically increases if the electric field reaches negative values but hardly increases for positive field. This behavior holds for both CEP cases, regardless of CEP. This symmetry breaking is an inherent property of the 5-band model causing even harmonics to emerge in the HH spectrum and is the reason for the CEP dependence in Fig. 4. In a twoband model, the occupation increases symmetrically for both negative and positive fields (not shown here). Thus, neither even harmonics appear nor CEP control of Bloch oscillations is possible in the 2- band analysis. The carrier injection is also visible in the distribution of the conduction band electrons as clear spots at t = +/- 15 fs for CEP = 0 in S3b and at t = 0 fs for CEP = π in S3c. Figure S3 Control of Bloch oscillations via the THz-CEP. a, Driving THz field (left axis) with peak fields of 11 MV/cm for CEP = 0 (blue curve) and CEP = (red broken curve). The occupation of the lower conduction band (right axis) is shown as a function of time CEP = 0 (gray shaded and blue curve) and CEP = (red line. b, Distribution of conduction-band electrons (n e = f k e1 ), as a function of time for CEP = 0, and c, for CEP =. White lines trace the centre of the electron distribution for different delay times. 6 NATURE PHOTONICS

7 SUPPLEMENTARY INFORMATION 3.2. Role of Bloch oscillations in high-harmonic generation The HHG originates from multiple processes where both polarization and current effects are nontrivially intertwined as discussed above. Therefore, it is not a priori clear how strongly the Bloch oscillations contribute to the HHG. We perform next a careful switch-off analysis in order to analyze this question. Since Bloch oscillations result from electrons crossing from one side of the Brillouin zone edge to another, the Bloch oscillations can formally be switched off from the HHG by preventing this process without modifying the remaining polarization and current dynamics. One cannot naturally implement this to actual experiments, while we can do this in our computations, as explained below. More specifically, we may artificially extend all the bands beyond the Brillouin zone as flat bands while setting the dipoles to zero in this artificially extended region. In this situation, those excitations that are accelerated to the edge of the Brillouin zone continue to the flat, extended, regions without contributing to the Bloch-oscillation related HHG. By comparing the HHG of full versus switch-off computation, we may directly deduce the relative strength of the Bloch oscillation in the HHG. Figure S4 compares the computed HHG spectrum with (solid lines) and without (dashed lines) the Bloch oscillation contributions for two carrier-envelope phases CEP = 0 (red curves) and CEP = (blue curves). We observe that without the Bloch oscillations the CEP has very little effect on the HHG spectrum. At the same time, the Bloch oscillations significantly increase the HHG peaks and yield a strong CEP dependence. For example, the 15 th harmonic is changed by almost 90% by the Bloch oscillations, while this peak remains small and CEP-independent when the Bloch oscillations are excluded. Therefore, we conclude that the Bloch oscillations both dominate the high frequency components of the HHG and induce their CEP sensitivity. At the same time, the lower frequencies are less influenced by them. This observation confirms that the CEP control analyzed in Fig. 4 indeed originates from the CEP sensitivity of the Bloch oscillations. Figure S4 Role of Bloch oscillations in the HHG spectrum. The spectrum of full computations (solid lines) is compared with an artificial computation (dashed lines) where the Bloch oscillations are artificially switched off, for two carrier-envelope phases CEP = 0 (red curves) and CEP = (blue curves). The thin vertical lines indicate the expected positions of the harmonics (order number indicated next to the line) NATURE PHOTONICS 7

8 4. Bloch oscillations and HHG in different crystal directions During a full period of a Bloch oscillation, the crystal electrons traverse the entire Brillouin zone. The coherent radiation emitted by the oscillating wave packet encodes important information regarding the electronic band structure in the direction of the electric field. Since the band structure of crystalline solids is usually not isotropic, coherent Bloch emission depends critically on the direction of acceleration with respect to the crystal axes. In Fig. S5, we record HH spectra from GaSe while we continuously rotate the crystal by an angle about the surface normal (see inset of Fig. 1b), thus driving frequency-modulated Bloch oscillations in various in-plane directions. In this measurement we employ a GaSe crystal with a thickness of only 40 µm under perpendicular incidence to suppress phase-matching effects. The HH intensity follows a 60 periodicity, in line with the intrinsic crystal symmetry. Intensity maxima are observed at angles = 0, 60 and 120, whereas the signal is strongly reduced at = 30 and 90. These angles can be correlated with the absolute orientation of the Brillouin zone by monitoring the angular dependence of the SH intensity in parallel with the HH spectra. Considering the 62m point group of GaSe, the symmetry of the (2) tensor is known and the angle can be unambiguously linked to the high-symmetry directions in reciprocal space S7. According to this assignment, HHG is most efficient when the THz driving field is oriented along the -K direction, while it is strongly suppressed along the -M direction. In contrast, the interband excitation does not strongly vary as a function of since the intensity of the PL peak at 476 THz is independent of the crystal orientation. This observation confirms that the highest harmonic orders are dominated by intraband currents, which sense details of the electronic band structure far into the Brillouin zone (see Fig. S5c). Pronounced band mixing and scattering into higher bands might suppress coherent transport. In order to utilize Bloch oscillations as a means to quantitatively map out the electronic structure, a refined theoretical model will be indispensible. An extension of our fully quantum mechanical theory to various symmetry directions is underway. 8 NATURE PHOTONICS

9 SUPPLEMENTARY INFORMATION Figure S5 Harmonics generation as a function of crystal orientation. a, Angular dependence of the horizontally polarized second harmonic signal on the polar angle (see inset of Fig. 1b) measured with a calibrated pyroelectric detector. From the symmetry of the (2) tensor, the orientation of the GaSe crystal becomes evident. This second harmonic signal is maximized along the -K direction S7 (see sketch of the hexagonal Brillouin zone). b, Dependence of HH spectral intensity on polar angle (colour coded) for a GaSe crystal of thickness 40 µm at perpendicular incidence. Photoluminescence (PL) from interband recombination leads to a strong signal at a frequency of 476 THz. The PL intensity is strictly constant for all polar angles. In contrast, the HH intensity depends strongly on, showing a 60 symmetry. c, Band structure of GaSe reproduced from Ref. S5. NATURE PHOTONICS 9

10 5. Phase-matching of second harmonic and high-harmonic generation In GaSe SHG of the THz wave ( THz /2 = 30 THz) is possible at an external phase-matching angle of = 41. While higher-order harmonics beyond the 5 th order cannot be perfectly phase-matched, the coherence length increases with the external incidence angle, for all harmonic orders. This also manifests itself in an increased HH intensity. In Fig. S6 we show HH spectra emitted by a GaSe crystal (L = 220 µm) for three different angles = 63, = 69 and = 75, respectively. The relative intensities of the various harmonic orders remain largely unchanged whereas the global intensity varies as a function of. The local structure of the harmonics displays modulations reminiscent of the CEP scan presented in Fig. 4. These details result from the fact that as a side-effect of adjusting one may also slightly vary the Gouy phase of the THz driving field. Figure S6 HH intensity for different angles of incidence. HH spectra generated in GaSe (L = 220 µm) for the angles = 75 (blue), = 69 (red) and = 63 (black). For each angle we optimized the focal position, leading to slight variations in the local shape due to the modified Gouyphase. The spectral intensities of neighbouring curves are offset by a factor of 100 for clarity. 10 NATURE PHOTONICS

11 SUPPLEMENTARY INFORMATION 6. Scaling of harmonic orders in the visible regime The HH intensity depends on the driving THz field in a non-trivial fashion, deviating from the scaling law expected for perturbative non-linear optics. This fact has been pointed out in Fig. 2b for the case of the 13 th harmonic. Here we complement this discussion with more experimental data for other harmonic orders. In our experiment the peak electric field is varied using a combination of two wire-grid THz polarizers. We record spectra of HH emission from a GaSe crystal (thickness: 220 µm) for different settings of the THz peak field using a monochromator and a silicon CCD. Subsequently the intensities for the harmonic orders are extracted and plotted as a function of the external peak field E a (Fig. S7). All harmonic orders significantly deviate from a power law for large fields. For fields substantially lower than 30 MV/cm, the intensity of all harmonic orders n approaches the scaling law ~E 2n, as expected within the limit of perturbative non-linear optics. Figure S7 Scaling of harmonic intensity with THz peak field. a-h, Intensity of the 11 th harmonic to the 14 th and 18 th to 21 st harmonic as recorded with a monochromator and a Si CCD. For comparison the respective scaling law ~E 2n is shown as a dashed line in each panel. The dotted line represents a linear dependence on the external field. NATURE PHOTONICS 11

12 7. Semi-classical model High-harmonic generation in gases is frequently explained in the so called three-step model, where an electron is excited into the continuum via a tunnel process (step #1), accelerated by the external field (step #2) and recollided with the ion core after half a period of the driving field (step #3). S8 A similar model has been employed to interpret high-order sideband generation via recollision of electron-hole pairs in quantum well heterostructures at low temperatures. 21 Such excitonic effects are not expected to be dominant for our experiment, where a bulk solid is strongly biased at room temperature. We might find some analogy with the three-step model if we think of the interband polarization as an electron wave packet, which is generated via interband tunneling (step #1) and gets accelerated (step #2) in the field. The process of HHG in solids is qualitatively different, however: While gases and excitons may be treated as a quasi-classical two-particle system, the quantum mechanical wave nature of the electrons becomes dominant in a periodic crystal lattice; Bragg reflections at the lattice have to be taken into account and render a quasi-classical treatment in real space difficult. For a more intuitive picture of the electron dynamics (steps #2 and #3) we switch to reciprocal space and limit the discussion to intraband effects within one band. The change of the crystal momentum ħk with time is equal to the force: S9 dk ee (8) dt Therefore a pre-existing electron of charge e is accelerated by the constant external field E, causing the wave vector k to increase linearly with time. For simplicity we assume a constant electron population of the conduction band. Scattering events may be neglected on the relevant timescales. The temporal evolution of the electron wave packet in k-space is obtained via integration of Bloch s acceleration theorem (Eq. 8): t e k( t) E( ) d, where we model the electric field E(t) as a sum of two fields oscillating at frequencies THz and 2 THz, respectively, with Gaussian envelopes (Fig. S8a). Assuming an electric field transient as depicted in Fig. S8a, polarized along the -K direction of the crystal, the electron wave packet oscillates in k- space up to a time of -80 fs, when it first hits the Brillouin zone boundary (BZB) and is Braggreflected (Fig. S8b). With increasing fields the electron packet is driven more strongly through k-space and performs more than one Bloch oscillation for the strongest fields at the center of the THz transient. 12 NATURE PHOTONICS

13 SUPPLEMENTARY INFORMATION Figure S8 Semiclassical model of intraband transport. a, Terahertz fundamental wave superimposed with second harmonic. b, For an external field of E a = 44 MV/cm the position in k-space oscillates harmonically for times up to -80 fs. Afterwards electrons are Bragg reflected at the Brillouin zone boundary (BZB) and enter the Brillouin zone from the opposite side. The position k(t) in reciprocal space may be translated into a trajectory in real space using the well known relation for the group velocity v g (t) of the electron packet: 1 d v g (t) dk where (k) denotes the energy dispersion of the lowest conduction band. Assuming constant conduction band population (n e = const.), the time derivative of the intraband current reads: dj d en evg enea, g dt dt where e denotes the charge of an electron and a g (t) the group acceleration. High-harmonic spectra are calculated by Fourier transforming a g (t). For our calculations we fit the first conduction band for GaSe from Ref. S5 with a sum of 4 cosine functions: ( k) a cos( b k c. i k( t) i i i) The intraband dynamics depicted in Fig. S8 leads to emission at frequencies throughout the visible, strongly modulated at the carrier frequency of 30 THz. For higher fields the overall intensity of the high-harmonic spectra increases and spectral components at higher frequencies appear. We may define a cut-off frequency co as the highest frequency where the high-harmonic intensity still exceeds a typical noise floor I t. The cut-off frequency co follows a linear trend, which is plotted in the inset as a function of THz peak field. We follow a similar route in the experiment (inset of Fig. S9). The extracted co for a typical value of I t is in good agreement with the numerical estimate. NATURE PHOTONICS 13

14 Figure S9 Field-dependence of cut-off frequency. Simulated high-harmonic spectra for THz peak fields of E a = 24 MV/cm (blue), 33 MV/cm (green) and 44 MV/cm (red). The dashed line indicates the sensitivity level used to determine the cut-off frequency co. Inset: Cut-off frequency co as a function of peak field (red line). Black dots correspond to experimental results. The shaded area indicates where strong interband PL in the experiment prevents a faithful extraction of the cut-off. 14 NATURE PHOTONICS

15 SUPPLEMENTARY INFORMATION References S1. Kira, M. & Koch, S. W. Semiconductor Quantum Optics (Cambridge University Press, Cambridge, 2012). S2. Golde, D., Meier, T. & Koch, S. W. Microscopic analysis of extreme nonlinear optics in semiconductor nanostructures. J. Opt. Soc. Am. B 23, (2006). S3. Haug, H. & Koch, S. W. Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 2009). S4. Kira, M. & Koch, S. W. Many-body correlations and excitonic effects in semiconductor spectroscopy, Progress in Quantum Electronics 30, (2006). S5. Schlüter, M. et al. Optical properties of GaSe and GaSxSe1-x mixed crystals. Phys. Rev. B 13, (1976). S6. Segura, A., Bouvier, J., Andrés, M. V., Manjón, F. J. & Muñoz, V. Strong optical nonlinearities in gallium and indium selenides related to inter-valence-band transitions induced by light pulses. Phys. Rev. B 56, (1997). S7. Catalano, I. M., Cingolani, A., Minafra, A. & Paorici, C. Second harmonic generation in layered compounds. Optics Communications 24, (1978) S8. Corkum, P., Plasma perspective on Strong-Field Multiphoton Ionization. Phys. Rev. Lett. 71, (1993) S9. Krieger, J. and Iafrate, G., Time evolution of Bloch electrons in a homogeneous electric field. Phys. Rev. B 33, (1986) NATURE PHOTONICS 15