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Symmetry-controlled temporal structure ofhigh-harmonic carrier fields from a bulk crystal

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Supplementary Information 1

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Symmetry-controlled time structure

of high-harmonic carrier fields from a solid

F. Langer1, M. Hohenleutner1, U. Huttner2,3, S. W. Koch2, M. Kira2,3, and R. Huber1

1Department of Physics, University of Regensburg, 93040 Regensburg, Germany

2Department of Physics, University of Marburg, 35032 Marburg, Germany

3Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor,

Michigan 48109, USA

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Supplementary Figure 1 | Multi-THz driving field and ultrashort gating pulse. a, The multi-THz waveform driving high-harmonic generation in gallium selenide is measured by electro-optic detection. The black curve is obtained after correction for the detector response. b, Spectral intensity of the 8-fs gating pulse used to resolve the temporal structure of crystal-generated high-harmonics (see Figure 3 of the main text). The intensity envelope of the pulses reconstructed from a SHG-FROG measurement is shown in the inset.

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Supplementary Figure 2 | High-harmonic generation in different crystal directions of gallium

selenide. Visible harmonic intensity (colour scale) as a function of the crystal angle ϕ. The photoluminescence signal is independent of the crystal orientation while the harmonic intensity shows a six-fold symmetry. Inset: For the orientation of the crystal axis of the sample, the intensity of the second harmonic components polarized parallel (red) and perpendicular (blue) to the fundamental field are measured. This experiment reveals that HH generation is most efficient in the

armchair directions ±e1, ±e2, and ±e3.

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Quantum many-body model of high-harmonic generation in GaSe in arbitrary directions. 8

A systematic analysis of high-harmonic generation (HHG) in solids requires a quantum-mechanical 9

descriptionS1 of the dynamical interplay between field-generated polarization and densitiesS2-S4 and 10

their accelerationS5 as well as relaxationS6. Jointly, these effects can induce quantum phenomena such 11

as quantum interferencesS7 or symmetry breaking from exclusively odd- to mixed even- and odd-order 12

harmonicsS5, which cannot be explained with classical recollision models, often applied in atomic 13

HHG. 14

As demonstrated in Refs. S5 and S7, excitations along a single excitation direction can effectively and 15

efficiently be described by one-dimensional excitations (1D-e) in momentum space, as long as the 16

excitations are polarization-direction (PD) conserving, i.e. they create light emission with identical 17

polarization properties as the excitation. For 1D-e, we can then systematically and efficiently include 18

the relevant many-body and light–matter interaction effects with the cluster-expansion approach 19

(Ref. S1). By using the same theory framework and material parameters, we already can quantitatively 20

explain measured spectra (Ref. S5) as well as the time-resolved emission properties (Ref. S7) 21

including the non-perturbative quantum interference (Ref. S7). Furthermore, an identical approach has 22

been applied to accurately describe atomic HHG on a microscopic level (Refs. S8 and S9). Therefore, 23

our theory framework is not only valid in the non-perturbative regime of strong excitations, typical for 24

HHG, but as well in the regime of perturbative nonlinear optics and conventional linear optical 25

excitations. 26

Since GaSe is a layered material with a hexagonal latticeS10, it has three equivalent in-plane bond 27

directions, 28

= √32 −12 , = −√32 −12 ,and = , in both real and reciprocal space. The unit vectors ej (j=1,2,3) are illustrated in Supplementary 29

Figure 3a as arrows pointing from a selenium atom (red sphere) to a gallium atom (dark sphere). Our 30

experiments in Fig. 1c and d confirm that GaSe has three principal, PD-conserving directions for 31

electronic motion, and identify those with the bond directions e1, e2 and e3. Only two of these three 32

vectors are linearly independent, e.g. the linearly independent vectors e1 and e2 define = −( + ). 33

These three directions define the ‘armchair directions’ in the honeycomb lattice of GaSe, whereas the 34

‘zigzag directions’ (e.g. e1-e2) line up with the vector connecting nearest atoms of the same species. 35

The spatial directions of atomic bonds define the preferred directions for electronic motion also in 36

momentum space, which can twist the polarization direction if the exciting field is not aligned with 37

these preferred directions. To analyse the direction dependence of HHG, we generalize the full 38

quantum analysis of Refs. S5, S7, and S8 to describe HHG for excitations along arbitrary directions 39

that are not PD-conserving. 40

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For this purpose, we evaluate the THz-field component 41 = ∙ , of the multi-THz driving field with respect to the three principal directions ej. Each of these 42

excitations induces an effectively 1D-e component in the momentum space. To fully include quantum 43

effects beyond a semiclassical description, the corresponding excitations must follow from the 44

dynamic interplay of the microscopic polarization , between bands λ and and electron (hole) 45

density , ( , ) in the conduction band eλ (valence band hλ) with crystal momentum ℏ . While the 46

vectorial projection of ETHz onto the bond direction relies on a real-space picture, our computations are 47

performed in reciprocal space and fully include the crystal periodicity. 48

As long as these principal directions are well separated, the 1D-e components should not mix 49

nonlinearly, despite each direction exhibiting strongly nonlinear HHG. The validity of this assumption 50

can be tested by adding the HHG emission resulting from these three effective 1D-e as linear 51

combinations of excitations (LCE) of the different j directions. Applying the LCE approach, we can 52

solve each ( , , , , , ) independently to construct the total polarization 53

Supplementary Figure 3 | Crystal symmetry of gallium selenide. a, Definition of the crystal

directions e1, e2, and e3 as well as the angle ϕ describing the direction of the THz polarization eTHz (blue arrow) with respect to the zigzag direction in between e1 and e2 (dashed line). The polarization direction of the high harmonic radiation EHH (purple arrow), emitted with respect to eTHz defines the

angle θ. The light yellow segment corresponds to the excitation angles ϕ used in the

experiments. b, Components |Ej/E0| of ETHz normalized by the peak field E0 along e1 (solid line), e2

(dashed line), and e3 (dotted).

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( ) = , = , , , , and current 54

( ) = , = , , , where the dipole-matrix element , the current-matrix element as well as the electronic band 55

structure are identical in all directions j. A similar approach is suggested in Ref. S11 to describe the 56

polarization dependence of high-harmonic generation in aligned molecules. The validity of the LCE 57

approximation is clearly justified based on the quantitative agreement between experiment and theory 58

in Fig. 2 and Supplementary Figures 5 and 6. The high-harmonic (HH) emission is then defined byS7 59

( ) = ( ) + ∂ ( ). The emitted HH intensity is proportional to = | ( )| after EHH is Fourier-transformed to the 60

frequency domain. For the strong THz excitations applied in the manuscript, the intensity IHH(ω) of the 61

higher harmonic orders is dominated by the polarization source P. 62

The LCE in direction j follows from the semiconductor Bloch equations (SBE)S1 which for a multi-63

band modelS5,S7 cast into the form 64

iℏ , = ℰ +i| | ( )∇ , − Ω , 1 − , − , + Γ ,+ Ω , , − Ω , , + Ω , , − Ω , , , iℏ , = ℰ +i| | ( )∇ , + Ω , , − , + Γ , + Ω , ,

− Ω , , + Ω , , − Ω , , ∗ , iℏ , = ℰ +i| | ( )∇ , + Ω , , − , + Γ , + Ω , ,

− Ω , , + Ω , , ∗ − Ω , , , ℏ , = −2Im Ω , , ∗ + Ω , , ∗ + | | ( )∇ , + Γ , ,

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ℏ , = −2Im Ω , , ∗ + Ω , , ∗ + | | ( )∇ , + Γ , , where ℰ ′is the transition energy between bands λ and , and Ω , = ( ) is the Rabi 65

energy. As demonstrated in Ref. S5, we need three hole bands h1, h2, and h3 and two electron bands e1 66

and e2 to accurately describe HHG in GaSe, using the standard values for and the bandstructure 67

from Ref. S5. The phonon interaction and Coulomb interaction between charge carriers induces the 68

scattering terms Γ , and Γ , that generate ultrafast scattering among electronic 69

excitationsS5,S6,S7,S12-S14. In HHG, these effects can be well approximated via constant dephasing 70 Γ , = − ℏ p , and a relaxation Γ , = − ℏ (f , − f , ) towards distributions with even parity. 71

We use the same dephasing and relaxation times T2 = 1.1 fs and τ = 6 fs as in Ref. S7, which agree 72

well with a fully microscopic HH computationS15. 73

We solve the SBEs for an initially unexcited system with a Gaussian multi-THz driving field . 74

Supplementary Figure 3a illustrates how the polarization direction of HH emission and THz field 75 = defines the angle ϕ (θ) with respect to the bond directions e1, e2, and e3 ( ). We 76

choose ϕ = 0° to align with the zigzag direction √ ( − ) in between the e1 and -e2 axes, such that 77

pointing along ϕ = −30° (ϕ = +30°) aligns with the armchair direction e1, (-e2). The various field 78

components then become 79 = cos( + 30°), = − cos( − 30°) ,and = − cos( + 90°). Supplementary Figure 3b shows how | / | (solid line), | / | (dashed line), and | / | (dotted 80

line) change as a function of ∈ [−30°, 30°],as in the experiments. The light yellow segment 81

corresponds to the excitation angles ϕ used. For this ϕ−range, the contributions of E1 and E2 are 82

always larger than E3. Therefore, the HH emission is dominated by its e1 and e2 components (i.e. Pj 83

and Jj), as demonstrated by Fig. 2a in the main text, because it scales strongly nonlinearly with Ej. 84

In order to analyse the polarization direction of the emitted HH radiation, we define the unit vector 85

pointing in direction α with respect to 86

= 1√3 cos( + ) ( − ) − sin( + )( + ), where − aligns with the reference line whereas + is perpendicular to it. The emission in 87

direction is then given by the projection 88 ( ) = | ( ) ∙ | .

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This is analogous to the experimental procedure where aligning a polarizer with angle α allows only 89

components parallel to eα to pass. The maximum of ( ) with respect to α then defines the 90

polarization direction θ at frequency ω. A straightforward calculation yields maximal HH emission in 91

direction 92

=12 arctan 2√3 −Re[( Σ)∗Δ ] + Re[( , )∗Δ ]3|Δ | − | Σ| − | , | + 2Re[( , )∗ Σ] − ,where we follow the branch of maximal ( ). The quantities Σ = ( , + , ), and Δ =93 ( , − , ) enhance even ( Σ) and odd (ΔE) orders of , ( ) = −i ( ) + ( ), 94

respectively, as explained in the main text. For an excitation along = 0°, contains only even 95

harmonic orders whereas ΔE contains exclusively odd harmonic orders and EHH,3 vanishes. 96

We observe that the = 0° excitation separates the even and odd HHs into perpendicular directions. In 97

this situation, the THz field ETHz = E e1 – E e2 points in the middle of the e1 and e2 bonds while its 98

components have opposite signs along e1 vs. e2 and no component along the e3 direction, following the 99

definitions of Fig. 2 in the main text and Supplementary Figure 3. We have demonstrated in Ref. S7 100

that a HH response , = + along a single bond direction ej produces even (odd) 101

harmonics ( ) that have an even (odd) parity with respect to the driving field. Since the e1 102

and e2 bonds are excited with sign-reversed fields, we find that a response , = + in e1 103

is connected with = − in e2 due to the parity difference between even and odd 104

harmonics. In case the excitation into directions e1 and e2 couple only linearly, the full vectorial HH 105

emission, = , + , = ( + ) + ( − ), separates even and odd 106

harmonics emitted along and perpendicular to the driving field, respectively. The corresponding 107

vectorial algebra is illustrated in Figs. 2c, d, and e. 108

The perfect directional separation of even and odd harmonics is also explained by the conventional 109

symmetry arguments stating that inversion symmetric systems can emit only odd harmonics. Since the 110

GaSe crystal is inversion symmetric only in the zigzag direction (e1-e2), it is clear that excitation and 111

subsequent emission into that direction supports only odd harmonics. In this special case, the broken 112

overall inversion symmetry manifests as even harmonics in the perpendicular emission direction, i.e. 113

(e1+e2). 114

Supplementary Figure 4b shows computed HH intensity envelopes of the parallel polarization 115

component of EHH(t) as a function of time for ∈ [−30°, 30°] on the same timescale as ETHz (black 116

solid line in Supplementary Figure 4a). Similar to the measurement in Fig. 3 of the main text, we find 117

that for an excitation along the bond directions e1 ( = −30°) and −e2 ( = 30°), the HH pulse train 118

consists of bursts emitted exclusively at positive (e1, solid curve) and negative (−e2, dashed curve) 119

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crests of ETHz (Supplementary Figure 4c) because of the non-perturbative quantum interferenceS7. This 120

quantum interference manifests in the emission spectrum as the appearance of even and odd harmonic 121

orders with comparable strength as explained in Ref. S8. It is created by the interference of two 122

excitation paths: one direct one from valence- to conduction band, and another one connecting an 123

intermediate valence band with roughly equal transition amplitude. This is qualitatively different from 124

other quantum interference processes, e.g. leading to a Cooper minimum in the spectrum of atomic 125

HHG in Ref. S16. Exciting along = 0° (Supplementary Figure 4d, solid curve), i.e. in the middle of 126

the e1 and −e2 direction, every field crest of ETHz = E e1 – E e2 has a component in e1 as well as in e2 127

direction, as explained above. However, the two components have opposite signs, such that one of 128

both directions is always experiencing a positive field crest, effectively creating HH emission at every 129

field cycle of ETHz. For intermediate angles, the contour in Supplementary Figure 4b shows the 130

transition from a unipolar emission at = ±30°, to an emission at every field crest at = 0°. 131

Supplementary Figure 4 | Computed high-harmonic pulse trains generated in different crystal 132 directions. a, Driving waveform ETHz applied in the computations of IHH. b-d, Computed HH intensity 133 envelopes containing only the parallel polarized components of EHH(t) as a function of time t and 134

crystal angle ϕ for ∈ [−30°, 30°] as contour (b) and for crystal angles ϕ = 30° (dashed line), 135

ϕ = −30° (solid line) (c) and ϕ = 0° (d). 136

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Role of non-perturbative excitations 137

The drastic polarization and frequency-comb selectivity of HHG analysed in the main text stems from 138

a non-perturbative quantum interference. It results from a superposition of different excitation 139

pathways from the valence band to the conduction band. In detail, sufficiently strong fields open two 140

excitation paths (a direct one between valence and conduction band, and another one also employing 141

an intermediate valence band) with roughly equal transition amplitudes – even when the related 142

transition energies differ significantly (Ref. S7). In crystal directions with broken inversion symmetry 143

(i.e. the Ga-Se bond direction, ϕ = ±30°), this quantum interference produces the unipolar emission 144

seen in Figs 3c and 4b, which underlies the observed polarization dependence and frequency-comb 145

properties. In particular, when the fields are strong enough, the interfering transition amplitudes 146

remain similar for a broad range of THz field strengths due to non-perturbative excitationsS7. 147

Therefore, the behaviour of the polarization depends only weakly on the applied electric field strength, 148

as long as the THz amplitude is kept in or close to the non-perturbative regime as confirmed by our 149

measurements and simulations. 150

The transition from the perturbative to the non-perturbative regime becomes evident from the scaling 151

of the HH intensity with the applied THz field strength, as demonstrated in Ref S5. Supplementary 152

Figure 5 shows the measured (a) and computed (b) intensities of the 8th to 11th harmonic orders as 153

functions of ETHz. Initially the intensity increases very rapidly and polynomially over several orders of 154

magnitude with ETHz, as expected for perturbative nonlinearities, in theory and experiment. 155

Approaching the non-perturbative regime at field strengths above ~25 MV/cm, a saturation-like 156

behaviour sets in and the HH intensities grow at a much lower power of the field strength. 157

Nevertheless, there is no abrupt threshold but a rather smooth crossover between these two regimes. 158

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Supplementary Figure 5 | Field scaling of harmonic orders. a, Experimentally determined harmonic intensity IHH as a function of the applied electric field ETHz for biasing in armchair direction. Orders 8 to 11 are shown (see legend); the noise level is indicated by the dotted line. b, Computed intensity IHH of the 8th to 11th harmonic order as a function of ETHz.

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In the main text, we concentrate on the new crystallographic control of the polarization and CEP of the 160

HH pulse train in this non-perturbative regime, due to several pragmatic reasons. Since the strength of 161

HHG naturally grows monotonically as a function of the excitation’s field strength ETHz, the strongest 162

signals and the best signal-to-noise ratios are reached at high fields, which obviously makes non-163

perturbative measurements “easier”. 164

Additionally, the balancing of excitation paths by quantum interference in the non-perturbative regime 165

tends to equalize the height of the non-perturbative even and odd peaks, as also suggested by the 166

saturation of HH intensities observed in Ref. S5. In contrast to this, the perturbative regime creates 167

higher-order peaks that tend to grow from the tails of low-order peaks. In other words, non-168

perturbative excitations produce clearer individual peaks for each order than perturbative excitation. 169

Pragmatically, this means that the polarization direction of a given HH peak is easier to resolve and 170

interpret for non-perturbative than for perturbative excitations. Therefore, the non-perturbative regime 171

(roughly above 25 MV/cm) is the simplest operational range for the studied effects. 172

Crystal angle dependence of HH peaks 173

Figures 1b and 2a show that the HH intensity is reduced when the crystal angle ϕ is detuned from the 174

bond directions (ϕ = ±30°). This behaviour is caused by the effectively reduced field strength because 175

only the projection onto ϕ = ±30° contributes to HHG. To quantitatively present the ϕ−dependence of 176

the false-colour radial plots of Fig. 1b and Supplementary Figure 2, the measured and computed 177

intensity IHH is shown in Supplementary Figures 6a and 6b as a function of ϕ for the 9th to the 13th 178

harmonic. We observe that all orders exhibit qualitatively a similar ϕ−dependence and our quantum 179

mechanical model convincingly captures the quantitative behaviour of all involved orders. For 180

example, the 9th order has the weakest and the 12th order has the strongest ϕ−dependence. We have 181

chosen to present these extreme cases in Fig. 2a to illustrate the typical range of behaviours. 182

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Prospects of polarization- and CEP-shaping of high-harmonic waveforms 184

It is extremely exciting that quantum effects yield a binary control of the periodicity of macroscopic 185

light pulses (see Fig. 4) as well as a continuous control of the polarization direction as a function of the 186

crystal direction (see Fig. 2 and Supplementary Fig. 7). Crystal symmetries in conjunction with strong-187

field quantum interference offer a higher level of polarization and CEP control in HHG as compared to 188

atomic gasesS17 that is also easily accessible. In the experiments presented in the manuscript, we 189

demonstrate the following scenarios of CEP and polarization settings: 190

(i) For a crystal orientation of ϕ = 0°, a HH pulse train with repetition rate 2νTHz, a pulse-to-pulse 191

CEP slip of ΔϕCEO = π and a polarization parallel to ETHz (see Fig. 4c and e) is generated. This 192

situation is achievable in a comparable manner with atomic gasesS18 and other crystalsS19,S20. 193

(ii) Another HH pulse train with repetition rate νTHz and ΔϕCEO = 0 emerges for ϕ = ±30° with a 194

polarization parallel to ETHz (see Fig. 4b and d). A qualitatively similar scenario can be achieved 195

in atomic gases only with two-color pumpingS21. 196

(iii) For a crystal orientation of ϕ = 0°, also a pulse train with a repetition rate of 2νTHz, ΔϕCEO = 0 197

and a polarization perpendicular to the driving field (see Figs. 4c and e) can be generated, which 198

has not been achieved in atomic gases, so far, to the best of our knowledge. 199

(iv) For intermediate angles 0° < ϕ < 30°, HH pulses with oblique polarization can be generated, 200

which also has not been achieved in atomic gases. 201

Supplementary Figure 6 | Crystal-angle dependence of HH intensity. a, Experimentally

determined harmonic intensity IHH as a function of the crystal angle ϕ for harmonic orders 9 to 13. b, Corresponding computed IHH.

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Supplementary Figure 7 | Polarization of high-harmonics in dependence on the crystal angle.

Frequency-resolved high-harmonic intensity IHH as a function of the crystal angle ϕ. Black arrowsindicate the polarization of harmonic orders 9 to 13 relative to the THz driving field, whereby the THzpulse features a horizontal polarization.

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