Methods

# Attosecond control of solid-state high harmonic generation using
$\omega$-3$\omega$ fields

Adam Gindl,1 Pawan Suthar,1 František Trojánek,1 Petr Malý,1
Thibault J.-Y. Derrien,2 and Martin Kozák,1,∗

1Faculty of Mathematics and Physics, Charles University,
Ke Karlovu 3, 12116 Prague 2, Czech Republic
2HiLASE Centre, Institute of Physics, Academy of Sciences of
the Czech Republic, 25241 Dolní Břežany, Czech Republic

∗E-mail<EMAIL_ADDRESS>
###### Abstract

Controlling the electron dynamics in matter by individual oscillations of
light fields has recently led to the development of attosecond metrology. One
of the phenomena resulting from the nonlinear response of materials in the
strong-field interaction regime is coherent emission of high energy photons.
High harmonic spectra carry the fingerprints of sub-cycle electronic motion
and the energy structure of the studied system. Here we show that tailoring
the waveform of the driving light by using a coherent combination with its
third harmonic frequency allows to control the electron tunneling time within
each half-cycle of the fundamental wave with attosecond precision. We
introduce an experimental scheme in which we simultaneously monitor the
modulation of amplitude and emission delays of high harmonic radiation and the
excited electron population generated in crystalline silicon as a function of
the relative phase between the $\omega$-3$\omega$ fields. The results reveal
unambiguously the connection between the dynamics of electron tunneling and
high harmonic generation processes in solids.

## Main text

High harmonic spectroscopy has recently become an indispensable tool for
investigation of light driven coherent electron dynamics in atoms [1, 2, 3]
and solid-state systems [4, 5]. In a material illuminated by a nonresonant
light wave with high field amplitude, the regime of electron excitation
differs from classical optics in which the time-dependent potential generated
in a material by the applied field can be considered as a small perturbation
to the system. When the field amplitude of light increases such that the
probability of electron tunnelling to the excited state per half period of the
driving wave becomes larger than the perturbative transition probability, the
light-matter interaction takes place in the nonperturbative strong-field
regime [6], which represents the cornerstone of attosecond physics. This
regime is characterized by the dimensionless Keldysh parameter
$\gamma_{\text{K}}={\left(\omega\sqrt{m^{*}E_{\text{g}}}\right)}/{eF_{0}}<1$
[7], where $E_{\text{g}}$ is the band gap energy, $m^{*}$ is the reduced
effective mass of an electron-hole pair, $F_{0}$ and $\omega$ are the electric
field amplitude and frequency of the driving wave and $e$ is electron charge.

One of the phenomena resulting from the coherent nonlinear electron dynamics
in the strong-field regime is high harmonic generation (HHG) [8, 9]. When this
process occurs in solids, the photons with energies corresponding to higher
order multiples of the incident photon energy are generated as a consequence
of coherent nonlinear interband polarization and intraband electron dynamics
[10]. The interband HHG can be understood in the framework of a three-step
semi-classical model adopted from atomic physics [9]. The electron first
tunnels to the conduction band in a narrow time window close to each maximum
of the electric field of the driving wave. Subsequently, the coherent
electron-hole wavepacket is accelerated by the laser field in the crystal and
may eventually recombine while emitting high energy photon. The amplitude,
phase and polarization of the emitted high harmonic radiation are tightly
linked to the electric field waveform of the driving pulse and to the band
structure of the material. It has been shown that HHG can be controlled by
changing the polarization of the driving wave with respect to high symmetry
axes of the crystal [12, 13, 11], the carrier-envelope phase of ultrashort
pulses [14, 15, 16], by a coherent combination of phase controlled
$\omega$-2$\omega$ fields [17, 21, 18, 19, 20, 22, 23, 24] or by sub-cycle
control of electron excitation with respect to the THz driving field [25].

Although high harmonic spectroscopy in condensed matter has become a widely
used technique, there are only few experiments clearly separating the
individual processes contributing to the emission of the coherent high energy
photons [25, 16]. Up to now it was also not possible to measure the phase
shifts of the emitted harmonic radiation induced by tailoring the driving
waveform. The principle of coherent two-color optical control has initially
been developed to steer electron currents in semiconductors via quantum path
interference by the $\omega$-2$\omega$ field superposition [27, 26]. Since
then, the two-color coherent control has been demonstrated in many physical
systems, including the control of quantum wave function of electrons in atoms
[28], electron photoemission from metals [29, 30] or HHG in atoms and solids
[17, 18, 19, 20, 21, 22, 23].

In this report we demonstrate a scheme allowing to control the electron
tunneling time within each half-cycle of the driving wave with attosecond
precision using coherent superposition of an ultrashort infrared pulse with
its third harmonic frequency. By controlling the mutual phase between the
$\omega$-3$\omega$ fields, the driving waveform changes (see Fig. 1a) leading
to modulation of the amplitude and attosecond emission delays of high harmonic
radiation generated in a silicon crystal. To clearly separate the HHG process
from electron tunneling excitation we simultaneously monitor the excited
electron density as a function of the $\omega$-3$\omega$ phase by measuring
the transient reflectivity of the sample after the interaction with
$\omega$-3$\omega$ fields. We observe that even a small admixture of light at
the third harmonic frequency to the fundamental pulse (the ratio between light
intensities $I_{3\omega}/I_{\omega}\approx 10^{-4}-10^{-3}$) is sufficient to
reach high modulation visibility of both the high harmonic generation yield
and the excited electron population.

In our experiments we use the fundamental pulses in the mid-infrared spectral
region with the photon energy of 0.62 eV (wavelength of 2000 nm) and the full-
width at half maximum duration of 35 fs. The third harmonic pulse (1.86 eV,
667 nm) is generated in a BBO crystal using type I phase-matching. Both pulses
propagate collinearly and are overlapped in space and time at the surface of a
silicon crystal (see the layout of the experimental setup in Fig. 1b, detailed
description of the setup can be found in Methods and in Extended data figure
1). Both the $\omega$ and 3$\omega$ pulses have the same orientation of linear
polarization along [100] crystallographic direction of silicon. The generated
harmonic radiation is collected in the reflection geometry to avoid
propagation effects in the sample. Due to the strong absorption of light at
photon energies above the direct band gap (3.4 eV), the collected harmonic
radiation is generated in a surface layer of the sample with the thickness of
only 5-100 nm depending on the photon energy. The relative phase difference
between the $\omega$ and 3$\omega$ fields is controlled by a pair of fused
silica wedges with a precision of 3 mrad corresponding to the time shift of
the two waves by 3 as. The long-term timing jitter of the setup is
characterized to be 2.8 as (RMS, see Methods and Extended data figure 2 for
details). We choose the combination of the fundamental frequency with its
third harmonics instead of more commonly used $\omega$-2$\omega$ combination
because it allows to control the time window of electron tunneling within each
half-cycle of the fundamental wave with better precision thanks to the shorter
period of the third harmonic field. Additionally, the $\omega$-3$\omega$
combination does not break the time symmetry of the waveform. As a
consequence, only the odd order harmonic frequencies are observed in the high
harmonic spectra generated in centrosymmetric materials. Simultaneously with
the modulation of high harmonic yield we monitor the excited carrier
population which is left in the sample after the interaction with
$\omega$-3$\omega$ pulses using transient reflectivity of an ultraviolet probe
pulse (3.62 eV, blue beam in Fig. 1b, details in Methods and Extended data
figure 3) incident on the sample with the delay time of 0.5 ps after the
$\omega$-3$\omega$ pulse combination. This allows us to clearly separate the
modulation of the integrated tunneling rate from the modulation of high
harmonic emission probability.

Coherent control of HHG in silicon by the two-color $\omega$-3$\omega$ field
is experimentally demonstrated in Fig. 1c, where we plot the measured HHG
spectra in the spectral region 2.8-10.5 eV as a function of the relative phase
shift $\varphi$ between the fundamental (peak electric field in silicon 1.61
$\pm$ 0.1 GV/m) and the third harmonic fields with the ratio between the peak
field amplitudes $r=F_{3\omega}/F_{\omega}=(3.9\pm 0.5)\%$. The results are
compared to numerical calculations using time dependent density functional
theory (TDDFT, details are described in Methods) shown in Fig. 1d obtained
with the fundamental electric field amplitude of 1.6 GV/m and the field ratio
of $r=4.3\%$. We observe a clear modulation of the spectra both in the
experimental and numerical data. The oscillations of individual harmonic
orders obtained by integrating the power emitted in the spectral window of 0.1
eV around each harmonic peak are shown in Fig. 1e both for the experimental
data (solid curves) and the numerical calculations (dashed curves). The data
are normalized and vertically translated for clarity. The HHG modulation is
compared to the modulation of the transient reflectivity of the sample $\Delta
R/R_{\text{0}}$ shown as a solid curve in the uppermost panel of Fig. 1e. Here
the dashed curve corresponds to the normalized population of electrons excited
to the conduction band resulting from the TDDFT calculations (see Methods for
details). These data are used to obtain absolute calibration of the mutual
phase between the $\omega$-3$\omega$ fields as the highest excited electron
population is expected for $\varphi=0$, for which the maxima of both waveforms
overlap (see Fig. 1a). We observe a good quantitative agreement between the
experiment and numerical calculations both for the phase shifts of the
oscillation maxima of individual harmonic orders, which are marked by squares
(experimental data) and stars (theory), and for the depth of modulation of the
high harmonic yield.

The phase of maximum generation yield differs for each harmonics as a
consequence of the dispersion of the propagating electron and hole in higher
energy bands of silicon, which tailors the interference between different
quantum paths contributing to the interband emission of high energy photons.
Remarkably, the maximum of the harmonic generation yield is reached for the
$\omega$-3$\omega$ mutual phase $\varphi\approx\pi/2$, which is shifted from
the phase corresponding to the highest population of excited carriers. This
observation suggests that the dominant mechanism of HHG driven by mid-infrared
light in silicon is the interband polarization [31] in contrast to the case of
HHG in wide band gap materials excited by near-infrared light, where the
intraband current was found to be dominating [16]. The phase of HHG modulation
maxima is only weakly dependent on the ratio between the amplitudes of
electric field of the 3$\omega$ and $\omega$ pulses $r$ (see Extended data
figure 4). The modulation visibilities defined as
$(S_{\text{max}}-S_{\text{min}})/(S_{\text{max}}+S_{\text{min}})$, where
$S_{\text{min}}$ and $S_{\text{max}}$ are the minimum and maximum HHG yields
at a particular harmonic frequency, are plotted in Fig. 1f as functions of
$r$. The visibilities reach high values even for very weak third harmonic
field only of about 1% of the fundamental field, which induces almost
imperceptible changes of the combined waveform. However, due to the strong
nonlinearity of the HHG process, this weak waveform modulation translates to a
strong modulation of the HHG signal.

To qualitatively understand the observed phenomena we recall the semi-
classical model of HHG in atoms, in which we assume instantaneous electron
tunneling and classical propagation of a single electron in free space between
the tunneling and recombination events. In the adiabatic approximation, the
time-dependent potential generated by the oscillating electric field changes
slow enough to allow the quantum system to follow it. The instantaneous
tunneling rate of an electron to the conduction band can be approximated using
a Zener-like tunnelling formula [32]:

$W(t)\propto\left|E(t)\right|\exp\left(-\frac{\pi\sqrt{m^{*}E_{\text{g}}^{3}}}{2e\hbar\left|F(t)\right|}\right).$
(1)

Here $F(t)$ is the time dependent electric field in the material and $\hbar$
is reduced Planck’s constant. In Fig. 2a we plot the time evolution of the
coherent superposition of the fundamental and third harmonic fields
$F(t)=F_{\omega}\cos(\omega t)+F_{3\omega}\cos(3\omega t-\varphi)$ with the
amplitudes $F_{\omega}$=1.6 V/nm and $F_{3\omega}$=0.16 V/nm (ratio
$r=F_{3\omega}/F_{\omega}$=10%) for two values of the relative phase
difference $\varphi=1.24\pi$ (blue curve) and $\varphi=0.24\pi$ (red curve)
corresponding to weak and strong emission of high harmonic radiation,
respectively. The instantaneous tunneling rate describing the probability of
electron transition to the conduction band per unit time calculated using
equation (1) with parameters $E_{\text{g}}=$3.4 eV (1st direct band gap of
silicon) and $m^{*}=$0.45 $m_{0}$ (reduced mass corresponding to the lowest
electron and hole bands in $\Gamma$ point) is shown as blue and red shaded
areas. The classical trajectories of the electrons $x_{\text{e}}(t)$
calculated by solving the equation of motion $\ddot{x}_{\text{e}}=-eE/m$ with
the effective mass of the electron $m$=0.2 $m_{0}$ (effective mass of light
electron band in $\Gamma$-point) emitted in different times within one half-
cycle of the fundamental wave are shown as curves in the lower section of Fig.
2a. The color scale corresponds to the total energy of the recombining
electron and hole
$E_{\text{tot}}=1/2m\dot{x}_{\text{e}}^{2}+1/2m\dot{x}_{\text{h}}^{2}+E_{\text{g}}$
(we assume light holes with $m$=0.2 $m_{0}$ with trajectories
${x}_{\text{h}}(t)$) while the line thickness is proportional to the tunneling
probability of the electron corresponding to the particular trajectory. There
are two important effects which may contribute to the modulation of the HHG
yield. The first effect is a large difference between the maxima of the
instantaneous tunneling rates, which are related to the modulation of the
electric field amplitude of the combined $\omega$-3$\omega$ waveform. The
second effect contributing to the HHG modulation is the time shift $\delta t$
of the maximum of the tunneling window. The time shift has important
implications for the probability of the electron-hole recombination. When the
HHG is driven by a single frequency field, only the electrons generated during
the second half of each tunneling window can recombine and emit photons while
the ones created in the first half of the tunneling window propagate in the
laser field away from the original position. The time shift of the tunneling
window to later times with respect to the fundamental wave thus increases the
number of recombining electrons contributing to high harmonic emission while
the shift to earlier times leads to the opposite effect. This model suggests
that the maximum of the high harmonic yield is reached for different value of
$\varphi$ than the maximum of the excited carrier population, which agrees
with our experimental observations shown in Fig. 1e.

While this model captures most of the features observed in the experiments, it
cannot describe the complex electron dynamics in silicon involving the
intraband currents and interband transitions between multiple bands. To get a
deeper insight into the coherent highly nonlinear response of silicon we
analyse the nonlinear current obtained from TDDFT simulations. By applying
high pass Fourier filter to the calculated total time-dependent current with a
cut-off frequency of 4$\omega$ (only 5th and higher harmonics are taken into
account) we obtain the instantaneous current oscillating at high harmonic
frequencies. The radiation emitted by an accelerating charge is proportional
to the time derivative of the current, which gives us the time profile of the
radiated instantaneous high harmonic power. In Figs. 2b,c we plot the combined
$\omega$-3$\omega$ waveforms with the field ratio of $r$=5% for two different
values of the phase difference $\varphi$=1.5 (red curve in Fig. 2b) and
$\varphi$=0.5 (red curve in Fig. 2c) along with the calculated instantaneous
power of the emitted high harmonic radiation at photon energies above 2.8 eV
(blue shaded areas). The field of the fundamental pulse is shown for
comparison as dashed curves in Fig. 2b,c. The time shifts of the maxima of the
waveforms in these two cases of $\delta t=\varphi/(3\omega)=-135$ as (see the
inset of Fig. 2b) and $\delta t=135$ as (see the inset of Fig. 2c),
respectively, lead to dramatic changes of the emission probability even with a
very weak perturbation of the fundamental waveform by the third harmonic
field. We observe that the maxima and minima of the HHG yield correspond to
approximately the same values of excited electron populations, emphasizing the
role of the time shift of the electron tunneling window which is evidenced to
be the dominant mechanism causing the modulation of HHG in silicon.

When the time of electron excitation shifts with respect to the fundamental
wave, not only the amplitude but also the phase of the generated harmonic
radiation is expected to shift. To measure the phase delays of the emitted
high harmonic radiation we apply spectral interferometry by using a signal and
reference high harmonic fields [33, 34]. The reference field is generated by
an infrared pulse phase-locked to the $\omega$-3$\omega$ combination, which
arrives to the sample before the two-color waveform (layout of the
experimental setup is shown in Fig. 3a, the details of the experimental setup
are described in Methods and Extended data figure 5). The time separation of
the signal and reference fields of about 170 fs leads to spectral interference
in the harmonic spectra, example of which is shown in Fig. 3b for 5th to 9th
harmonic frequencies. By monitoring the shifts of the spectral interference
fringes we measure the relative shifts of the emission phase of harmonic
radiation as a function of the mutual phase between the $\omega$-3$\omega$
fields. We note that a real carrier population is generated in the sample
during the HHG by the reference pulse. While this can induce dephasing of
carriers during coherent HHG driven by $\omega$-3$\omega$ pulses which would
decrease the amplitude of the generated harmonics, it is not expected to
influence the phase of the emitted photons. In Fig. 3c we show the measured
high harmonic emission delays (field ratio $r$=5.5%) compared to the delays
obtained from the numerical TDDFT simulations ($r$=5%) in the time window
around the maxima of the harmonic emission shown in Fig. 3b. We observe that
close to the maximum of the emission yield, the emission delay scales
approximately linearly with the time delay between the two-color fields
$\delta t$ with different slopes for individual harmonic frequencies. The
emission delays of high harmonic photons relative to the fundamental wave thus
depend on the photon energy and cannot be understood only by considering the
time shift of the electron tunneling time. In contrast, the emission phase is
determined by the combination of the tunneling time and the quantum mechanical
phase acquired by the electron-hole wavepacket during its coherent dynamics
between tunneling and recombination, where multiple bands are involved in the
high harmonic generation process [35, 31]. The measured slopes $b$ of the
emission delays of 5th to 9th harmonics obtained by fitting the data shown in
Fig. 3c by a linear function $y=a+bx$ are plotted in Fig. 3d as a function of
the $\omega$-3$\omega$ field ratio $r$.

The observed smaller delays of higher order harmonics can be qualitatively
understood in the framework of the semi-classical electron dynamics shown in
Fig. 2a. The electrons propagating along trajectories corresponding to the
highest recombination energy are generated in a short time window within one
half-cycle of the fundamental wave. The recombination time of these high
energy electrons practically does not shift with the relative phase $\varphi$.
However, the electrons with lower recombination energies are generated in two
distinct time windows corresponding to short and long trajectories. Photons
within the intermediate energy range are thus emitted in two distinct times
within each half-cycle of the $\omega$ field (dark green curves in Fig. 2a)
with different amplitudes and emission phases. The change of the relative
phase shift $\varphi$ between the $\omega$-3$\omega$ field leads to strong
changes of the recombination probability of particular trajectories, which in
turn causes the delay oscillating between the values corresponding to the
short and long trajectories. The larger delays observed in the case of the 5th
harmonic frequency can be also influenced by the fact that its photon energy
is slightly below the direct band gap of undriven silicon. The intraband
current thus in this case may play more important role than for the higher
harmonics, where the dominance of the interband generation has been confirmed
by recent experiments [31].

The combination of coherent two-color control and phase-resolved detection
scheme with attosecond resolution introduced here brings opportunities for
gaining a deeper understanding of HHG in solids. In particular we study the
relation between the tunneling time, which is controlled within each half-
cycle of the driving wave by adding a weak perturbation at the third harmonic
frequency, and the amplitude and phase of the emitted high-energy photons. A
direct comparison between the modulation of high harmonic emission and the
modulation of the real carrier population induced by the $\omega$-3$\omega$
fields demonstrated here allows to separate the processes of electron
excitation from the nonlinear coherent dynamics leading to HHG. Similar
schemes can also be applied in other physical systems, in which the strong-
field nonlinear optical interactions induce coherent sub-cycle electron
dynamics [36]. Further, the $\omega$-3$\omega$ combination may find
applications in atomic HHG for enhancement of the emission yield or for
advanced gating techniques combining the coherent control with dynamic phase-
matching [37].

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## Funding:

Czech Science Foundation (project GA23-06369S), Charles University
(UNCE/SCI/010, SVV-2020-260590, PRIMUS/19/SCI/05, GAUK 349921). Funded by the
European Union (ERC, eWaveShaper, 101039339). Views and opinions expressed are
however those of the author(s) only and do not necessarily reflect those of
the European Union or the European Research Council Executive Agency. Neither
the European Union nor the granting authority can be held responsible for
them. T. J.-Y. D. was supported by the European Regional Development Fund, the
state budget of the Czech Republic (project BIATRI: CZ.02.1.01/0.0/0.0/15
003/0000445) and the Ministry of Education, Youth and Sports of the Czech
Republic through the e-INFRA CZ (ID:90140).

## Contribution

MK conceived and supervised the study. A.G., M.K. and P.S. performed the
experiments. A.G. and M.K. processed and interpreted the experimental data.
T.J.-Y.D. performed the numerical simulations. M.K. wrote manuscript with the
input from all the co-authors.

## Competing interests

Authors declare that they have no competing interests.

## Data availability

All data supporting the findings of this study are available from the
corresponding authors upon reasonable request.

Figure 1: Fig. 1. Modulation of high harmonic generation yield by coherent
superposition of $\omega$-3$\omega$ fields. (a) Time evolution of electric
field of a combined $\omega$-3$\omega$ waveform for different values of the
mutual phase difference $\varphi$ within one period of the fundamental wave.
(b) Layout of the experimental setup. (c) Measured and (d) calculated
dependence of high harmonic spectra generated in silicon on the mutual phase
$\omega$-3$\omega$ fields $\varphi$. The field ratios are $r=(3.9\pm 0.5)\%$
in (c) and $r=4.3\%$ in (d). (e) Experimental (full curves) and numerically
calculated (dashed curves) modulation of individual harmonic orders (panels
labelled 5-15) and the transient reflectivity signal of the probe pulse (top
panel). The black points label the values of the mutual phase corresponding to
the maxima of the measured (squares) and theoretical (stars) modulation for
individual harmonics. The HHG data were obtained by integrating the spectral
window with the width of 0.1 eV around the maxima of the respective harmonic
orders. (f) Measured visibility of modulation of the harmonic generation yield
for 5th-15th harmonics as a function of the ratio of $\omega$-3$\omega$ field
amplitudes $r$. Figure 2: Fig. 2. Time-dependence of high harmonic emission
driven by $\omega$-3$\omega$ fields. (a) Semi-classical model showing the time
evolution of the electric field of the combined $\omega$-3$\omega$ waveform
for two values of the mutual phase difference $\varphi=1.24\pi$ (blue curve)
and $\varphi=0.24\pi$ (red curve). Instantaneous electron tunneling rate
calculated using equation (1) is shown as blue and red shaded peaks. The
classical trajectories of electrons excited in different times are shown in
the lower part of the figure with the color scale corresponding to the energy
in the moment of recombination and the line thickness proportional to the
tunneling probability associated with the particular trajectory. (b) The
instantaneous power of high harmonic emission calculated using time-dependent
density functional theory (blue shaded area) for the relative phase
$\varphi$=1.5 $\pi$ between the $\omega$-3$\omega$ fields and for the field
ratio of $r$=4.3%. The time evolution of the electric field of the combined
waveform (red solid curve) is compared with the electric field of the
fundamental pulse (black dashed curve). Inset: The peak of the waveform shifts
by $\delta t$=-135 as. (c) The same as in (b) with the phase difference of
$\varphi$=0.5 $\pi$ leading to the positive time shift of the field maximum of
$\delta t$=135 as. Figure 3: Fig. 3. Attosecond emission delays of high
harmonic radiation induced by modulating the relative phase of
$\omega$-3$\omega$ fields generated in a silicon crystal. (a) Layout of the
experimental setup for spectral interferometry of high harmonic fields. (b)
Spectral interference of the 5th, 7th and 9th harmonics measured as a function
of the relative phase between $\omega$-3$\omega$ pulses $\varphi$. (c)
Measured (solid curves, $r$=5.5%) and calculated (dashed curves, $r$=5%)
attosecond emission delays obtained from the data shown in (b) as a function
of the time delay of the 3$\omega$ field with respect to the $\omega$ field.
The shaded areas show the standard deviation of the delays resulting from the
least squared fits of the data using equation (2). (d) Slopes of the measured
emission delays of the 5th (black squares), 7th (red circles) and 9th (blue
triangles) harmonics as a function of the field ratio $r$ obtained by fitting
the data shown in (c) using linear function in the time interval of 400 as
around the modulation maxima. Errorbars correspond to the standard deviation
of the slope obtained by least squares fits.

## Methods

### Experimental setup

The experiments demonstrating coherent two-color control of high harmonic
generation in silicon are performed using pulses generated in a noncollinear
optical parameteric amplifier with subsequent difference frequency generation
(NOPA-DFG, setup is described in detail in [1]) and the pulses at its third
harmonic frequency. The NOPA-DFG setup is pumped by solid-state femtosecond
laser system Pharos SP2-6W (Light Conversion) with ytterbium-doped active
medium. The NOPA-DFG output pulses have central wavelength of 2000 nm and
pulse duration at the sample of 35 fs (FWHM of instantaneous power). The
coherent $\omega$-3$\omega$ pulse combination is generated in the setup, which
is shown in Extended data figure 1. The beam passes through a half wave plate
to control the direction of linear polarization of the fundamental pulse.
Subsequently, the beam is focused using an off-axis parabolic mirror with a
focal distance of 15 cm to a BBO crystal, where the pulses at third harmonic
frequency 3$\omega$ are generated. We use phase-matching angle of
$\theta$=27.2° for direct third harmonic generation ($oooe$-type phase-
matching) leading to vertically polarized 3$\omega$ pulse (polarization along
the direction of projection of the optical axis of the BBO crystal in the
plane perpendicular to the light propagation). In the first experiment, the
half wave plate is adjusted such that the fundamental polarization is
horizontal. The pulse thus propagates as an ordinary ray and we obtain only
one replica of the fundamental pulse after the BBO. When the half wave plate
is rotated to 22.5°, the polarization incident to BBO crystal has both the
horizontal and vertical components with the same amplitude. As a consequence
of the birefringence of the BBO crystal we obtain two orthogonally polarized
pulses, which are delayed due to different group indexes of the ordinary and
the extraordinary rays. After the BBO crystal, the two collinear beams at the
fundamental and the third harmonic frequencies are collimated by a second off-
axis parabolic mirror with focal length of 15 cm.

The relative phase between the $\omega$ and 3$\omega$ pulses $\varphi$ is
controlled using a pair of fused silica wedges with an apex angle of
$\alpha$=4°. One of the wedges is placed on a translation stage. The phase
shift of the third harmonic field with respect to the fundamental pulse
resulting from the propagation in the wedges over a distance $\Delta L$ can be
expressed as $\varphi=3\omega\Delta L\left(n_{3\omega}-n_{\omega}\right)/c$ ,
where $c$ is speed of light in vacuum and $n_{\omega}$=1.4381 and
$n_{3\omega}$=1.4561 are the refractive indexes of fused silica at frequencies
$\omega$ and $3\omega$, respectively. One period of the high harmonic yield
modulation corresponds to the shift of the relative $\omega$-$3\omega$ phase
$\varphi$ by 2$\pi$ which gives $\Delta L=$37 $\mu$m. The calculated shift of
the wedge needed to change the thickness of the material by $\Delta L$ is
$\Delta x=\Delta L/\tan(\alpha)=0.59$ mm, which matches well the
experimentally determined value of 0.574$\pm$0.005 mm. While the refractive
indexes of fused silica at frequencies $\omega$ and 3$\omega$ differ strongly,
the difference between group velocities of the two pulses in the wedges are
much smaller (group indexes are $n_{g,\omega}$=1.4673 and
$n_{g,3\omega}$=1.4732) leading to a small relative shift of the pulse
envelopes when changing the relative phase by several periods of the $3\omega$
field. To compensate the total group delay which the $\omega$ and 3$\omega$
pulses obtain between the first BBO crystal and the sample we let the beam
pass through a second BBO crystal with the optical axis cut under the angle
$\theta$=50°. Due to the negative uniaxial birefringence of BBO, the ordinary
ray at frequency $\omega$ propagates slower in the crystal than the
extraordinary ray at 3$\omega$. The beam is finally focused on the sample
surface by an off-axis (90°) silver coated parabolic mirror with focal length
of $f$=35 mm. The spot sizes of the two beams on the sample are
$w_{\omega}=15$ ${\mu}$m and $w_{3\omega}=8$ ${\mu}$m. The dependence of
nonperturbative HHG yield in silicon on the incident light intensity of the
fundamental pulses was previously measured to be cubic [31]. When considering
Gaussian profile of the fundamental beam, the harmonics are produced from the
area with a radius decreased by a factor of $w_{HHG}=w_{\omega}/\sqrt{3}=8.7$
${\mu}$m, which is well matched to the size of the third harmonic beam. To
adjust the polarizations of all the pulses to be linear in the same direction
we put a thin polarizer (Thorlabs, LPNIRA) between the final focusing
parabolic mirror and the sample.

High harmonic radiation is collimated by a UV fused silica lens with focal
distance of 100 mm. The spectra are detected using a grating spectrograph
(Andor, Shamrock 163) with a grating containing 600 lines/mm blazed for 300
nm. The spectrometer is equipped with a cooled CCD camera (Andor iDUS 420).
Prior to entering the spectrometer, the light is spectrally dispersed in the
vertical direction by a pair of fused silica prisms. This allows to filter out
most of the radiation at wavelengths above 500 nm before the high harmonics
enter the spectrometer. Because the dispersion is applied in the vertical
direction, all the harmonics are transmitted through the entrance vertical
slit of the spectrometer. The harmonics 11th-15th have photon energies higher
than 6 eV and are strongly absorbed in air. The modulation of the high
harmonic orders is thus measured in vacuum using a vacuum ultraviolet
spectrometer (EasyLight, H+P Spectroscopy) combined with a microchannel plate
detector.

Due to the inline setup of the nonlinear interferometer, in which both beams
propagate through the same beam paths, the drifts of the relative phase
difference $\varphi$ between the $\omega$ and 3$\omega$ pulses are negligible
and the time jitter of the two fields is only few attoseconds over the
measurement time of one hour (see the measured modulation of the 5th harmonic
frequency over the time period of 60 minutes shown in Extended data figure 2).

The carrier population excited in the sample by the coherent superposition of
$\omega$-3$\omega$ fields is monitored by measuring the transient reflectivity
of the sample. We use an independent probe pulse with the wavelength of 343 nm
and photon energy of 3.62 eV (third harmonics of the Pharos laser output at
1030 nm), which is incident on the sample with a controlled time delay with
respect to the coherent $\omega$-3$\omega$ pulse combination. The high photon
energy of the probe is selected due its short penetration depth into silicon
of only few nanometers, which prevents accumulation of the signal coming from
different depths which would be modified by the fact that the phase velocities
of the pulses at $\omega$ and 3$\omega$ frequencies in silicon strongly
differ. The second reason for selecting a short probe wavelength is the
possibility to focus the probe beam to a smaller spot size than the $\omega$
and 3$\omega$ beams, which allows to probe the region of the sample with
approximately homogeneous spatial distribution of the excited carriers. In
this experiment, the pump $\omega$-3$\omega$ combination propagates through an
optical chopper, which modulates the beam. After being reflected from the
sample, the power of the probe beam at 3.62 eV is detected by a silicon
photodiode. The electronic signal is measured using a lock-in amplifier
(SR830, Stanford Research Systems) at the frequency of the optical chopper.
The measured transient reflectivity dynamics is shown in Extended data figure
3 as a function of the time delay between the $\omega$-3$\omega$ pulses which
excite the carriers and the ultraviolet probe pulse. The excited carrier
population is linearly proportional to the transient reflectivity change in a
short time after the excitation. To observe the oscillations of the carrier
population as a function of the relative phase between the $\omega$-3$\omega$
fields $\varphi$ we set the time delay of the probe to 0.5 ps.

### Data processing

The measured high harmonic spectra are not compensated for the absolute
spectral efficiency of the detection setup. The absolute calibration is not
required because we are only interested in the relative changes of the HHG
yield of individual harmonic frequencies as a function of experimental
parameters (mutual phase difference between $\omega$-3$\omega$ fields, ratio
of the field amplitudes $r$). The generation yield at each harmonic frequency
is obtained by integrating the spectral window with the width of 0.1 eV around
the spectral peak of the particular harmonics.

The spectral interferometry data shown in Fig. 3b are processed by fitting the
peaks of the individual harmonics in each spectrum corresponding to specific
$\omega$-3$\omega$ phase differences $\varphi$ using a function:

${f(\hbar\omega)=A_{1}\text{exp}\left[-\frac{(\hbar\omega-\hbar\omega_{HHG})^{2}}{2\Delta\omega_{HHG}^{2}}\right]+A_{2}\text{exp}\left[-\frac{(\hbar\omega-\hbar\omega_{HHG})^{2}}{2\Delta\omega_{HHG}^{2}}\right]\text{sin}\left(\omega\tau+\Delta\varphi_{HHG}\right)+A_{3}}$
(2)

Here $\omega_{HHG}$ and $\Delta\omega_{HHG}$ are the central frequency and the
spectral width of the particular harmonic peak. Fitting parameters are the
delay between the two harmonic pulses $\tau$, their relative phase shift
$\Delta\varphi_{HHG}$ and the coefficients $A_{1}$, $A_{2}$ and $A_{3}$ which
correspond to the amplitude of the slowly varying Gaussian envelope ($A_{1}$),
the amplitude of the spectral interference ($A_{2}$) and the constant
background ($A_{3}$). The data are fitted in two steps. First the time delay
$\tau$ is determined from the data with the strongest spectral interference
shown in Extended data figure 5. In the second step, $\tau$ is kept constant
and the relative phase shift of harmonic fields $\Delta\varphi_{HHG}$ is
obtained for different values of the relative phase between the driving
$\omega$-3$\omega$ fields. The relative emission delays shown in Fig. 3c are
calculated from the relative phase shifts of the individual harmonic peaks as
$\delta t_{HHG}=\Delta\varphi_{HHG}/\omega_{HHG}$, where $\omega_{HHG}$ is the
central frequency of the particular harmonics.

### Numerical calculations using time dependent density functional theory
(TDDFT)

#### Ground state preparation

Silicon (100) crystal is prepared using a primitive cell made of two silicon
atoms along with non-orthogonal periodic boundary conditions as implemented in
_Octopus_ [2]. Inter-atomic distance is chosen equal to the experimental value
($a=5.431$ angstroms) [3] The inter-atomic potential is described using the
method of norm-conserving _ab-initio_ pseudo-potentials [4]. The real-space is
meshed regularly using a step of $\delta x=0.23$ angstroms, and the momentum
space is meshed uniformly using a grid of $k=40\times 40\times 40$, repeated 4
times centered at points X and L.

The ground state is prepared using the Tran-Blaha functional (TB09) [5]. The
convergence of the ground state was verified by comparison with available
literature of more precise methods [6]. The resulting direct band gap
$E_{\Gamma}$ of silicon is found to be $E_{\text{g}}$=3.04 eV.

#### Time-dependent simulations and control of convergence

The time-evolved Kohn-Sham (KS) states are prepared by solving the KS equation
expressed in the velocity gauge and in Hartree atomic units:

$\displaystyle\left[\left(-\frac{i\hbar}{2m_{e}}\boldsymbol{\boldsymbol{\nabla}_{r}}+\frac{\left|e\right|}{c}\boldsymbol{A}\left(t\right)\right)^{2}+\hat{v}_{\text{ion}}\left(\boldsymbol{r}\right)+\hat{v}_{\text{H}}\left[n\left(\boldsymbol{r},t\right)\right]\left(\boldsymbol{r}\right)+\right.$
(3)
$\displaystyle\left.+\hat{v}_{\text{xc}}\left[n\left(\boldsymbol{r},t\right)\right]\left(\boldsymbol{r}\right)\right]\times\psi_{n,\boldsymbol{k}}\left(\boldsymbol{r},t\right)=i\hbar\frac{\partial}{\partial
t}\psi_{n,\boldsymbol{k}}\left(\boldsymbol{r},t\right).$

Here the vector potential describing the laser pulse $A(t)$ is introduced
using the dipolar approximation in the minimal coupling formulation. In Eq.
(3), $i$ is the pure imaginary number, $\hbar$ is the reduced Planck constant,
$m_{e}$ is the electron mass at rest, $\boldsymbol{\boldsymbol{\nabla}_{r}}$
is the real-space gradient operator, $\left|e\right|$ is the elementary charge
of the electron and $c$ is the light velocity. In the Coulomb gauge, the
vector potential $\boldsymbol{A}\left(t\right)$ in the dipolar approximation
is related to the electric field $\boldsymbol{F}\left(t\right)$ via the
relation

$\boldsymbol{A}\left(t\right)=-c\int_{-\infty}^{t}\boldsymbol{F}\left(t^{\prime}\right)\,dt^{\prime}.$
(4)

In Eq. (3), $\hat{v}_{\text{ion}}\left(\boldsymbol{r}\right)$ represents the
potential of the atomic lattice. Note that the non-local term originating from
the pseudo-potential is not detailed for simplicity. $\hat{v}_{\text{H}}$
denotes the Hartree potential, corrected by the exchange-correlation potential
noted $\hat{v}_{\text{xc}}$. The incident wavelengths is normalized to the
band-gap energy obtained using the TB09 functional. Orientation of the
linearly-polarized electric field is set along the $(100)$ direction of the
cubic crystal. From the time-evolved Kohn-Sham orbitals
$\psi_{n,\boldsymbol{k}}\left(\boldsymbol{r},t\right)$, the time-dependent
electrical current $\boldsymbol{J}\left(t\right)$ is computed as indicated in
Ref. [7].

The electric field of laser pulses at frequencies $\omega$ and 3$\omega$,
respectively, which is used in the numerical simulations can be written as:

$\vec{F}(t)=\vec{F}_{\omega}\times\text{sin}\left(\omega
t+\varphi_{0,\omega}\right)\times
f\left(t\right)+\vec{F}_{3\omega}\times\text{sin}\left(3\omega
t+\varphi_{0,3\omega}-\varphi\right)\times f\left(t\right),$ (5)

where:

$f\left(t\right)=\text{$\sin^{4}$$\left(\frac{\pi}{2}\frac{t-t_{0}-\tau_{\text{corr}}}{\tau_{\text{corr}}}\right)$}\\\
\times
H\left(t-t_{0}+\tau_{\text{corr}}\right)\left[1-H\left(t-t_{0}-\tau_{\text{corr}}\right)\right].$
(6)

Here the correction of the FWHM for $\sin^{4}$ envelope can be expressed as:

$\tau_{\text{corr}}=\frac{4\tau_{\text{exp}}}{\pi}\arcsin\left(2^{-1/4}\right).$
(7)

$\tau_{\text{exp}}$=35 fs is the experimental pulse duration and $H(x)$ is
Heaviside function of argument $x$. Note that 4th-power sinus is commonly
employed to decrease the spurious oscillations originating from the Fourier
transformation upon computation of high-harmonic spectrum [8, 31].

#### Normalization of pulse frequency to band gap energy

Density functional theory gives for most of the materials systematically lower
values of the band gap compared the experimental values. For the HHG process,
an important parameter is the ratio between the driving photon energy and the
minimum band gap. As shown in [31], the driving frequency can be normalized to
the reduced band-gap provided by DFT [6] to fulfill the relation
${\hbar\omega_{\text{TB09}}}/{E_{\text{g},\text{TB09}}}={\hbar\omega_{\text{exp}}}/{E_{\text{g},\text{exp}}}$.
In the calculations, the wavelengths of $\lambda_{\omega,\text{TB09}}$=2237 nm
and $\lambda_{3\omega,\text{TB09}}$=745.66 nm are used instead of the
experimental values 2000 nm and 666.67 nm, respectively. The time axis in the
calculation results is normalized back to obtain harmonic spectra with
frequencies corresponding to experimental data.

#### Calculation of harmonic spectra

Harmonic spectra are calculated using Larmor’s formula and Fourier transform
of the total current obtained by the TDDFT simulations. The Fourier transform
is applied to a temporal window that decays in $\sin^{4}$ at the end of pulse
to avoid generating spurious oscillations in the spectra as a consequence of
no dephasing present in the TDDFT calculations (similarly to Ref. [31]).

#### Calculation of the excited electron density

The number of electrons excited from the valence band to the conduction bands
is computed using the method described in Ref. [9]. Note that transient values
of electron density are subjected to gauge dependencies [10]. Therefore,
reported values of quantity of excited electrons are captured after the laser
pulse, i.e., in absence of fields, a situation where gauge is not problematic
[10, 11].

## References

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  * [2] Tancogne-Dejean, N. et al. Octopus, a computational framework for exploring light-driven phenomena and quantum dynamics in extended and finite systems. J. Chem. Phys.. 152 124119 (2020).
  * [3] Okada, Y. & Tokumaru, Y. Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K. Journal Of Applied Physics. 56, 314-320 (1984).
  * [4] Troullier, N. & Martins, J. L. Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B. 43, 1993-2006 (1991).
  * [5] Tran, F. & Blaha, P. Accurate Band Gaps of Semiconductors and Insulators with a Semilocal Exchange-Correlation Potential. Phys. Rev. Lett.. 102, 226401 (2009).
  * [6] Lherbier, A. et al. Band widths and gaps from the Tran-Blaha functional: Comparison with many-body perturbation theory. Phys. Rev. B. 87 075121 (2013).
  * [7] Wachter, G. et al. Ab Initio Simulation of Electrical Currents Induced by Ultrafast Laser Excitation of Dielectric Materials. Physical Review Letters. 113, 087401 (2014).
  * [8] Sato, S. A., Hirori, H., Sanari, Y., Kanemitsu, Y. & Rubio, A. High-order harmonic generation in graphene: Nonlinear coupling of intraband and interband transitions. Phys. Rev. B. 103, L041408 (2021).
  * [9] Derrien, T. J.-Y. et al. Photoionization and transient Wannier-Stark ladder in silicon: First principle simulations versus Keldysh theory. Phys. Rev. B. 104 L241201 (2021).
  * [10] Ernotte, G., Hammond, T.J. & Taucer, M. A gauge-invariant formulation of interband and intraband currents in solids. Phys. Rev. B. 98 235202 (2018).
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Figure 4: Extended data figure 1. Experimental setup for generation of
coherent superposition of $\omega$-3$\omega$ pulses. (a) The setup for the
measurements of amplitude modulation of high harmonic generation. The half
wave plate (HWP) is rotated to 0° (fundamental polarization remains
horizontal). The pulse is focused by off-axis parabolic mirror to the first
beta-barrium borate (BBO) crystal phase-matched for direct third harmonic
generation. After the crystal, the beam is collimated and propagates through
the fused silica wedges, where the relative $\omega$-3$\omega$ phase is
adjusted by translating one of the wedges. The group delay between the pulses
is compensated in the second birefringent BBO crystal. The beam is focused on
the sample by an off-axis parabolic mirror (all mirrors are displayed as
lenses for simplicity). Both pulses propagate through a thin polarizer rotated
under 45° to generate linear polarizations with their axis being parallel to
each other on the sample surface. (b) The setup for the measurements of
attosecond phase delays of high harmonic radiation. The components are the
same as in (a) with the half wave plate rotated by 22.5° leading to
polarization rotation of the fundamental beam by 45°. As a consequence, two
temporally separated fundamental pulses leave the first BBO crystal and co-
propagate to the sample, where the latter pulse is temporally overlapped with
the 3$\omega$ pulse and the former one serves as a reference pulse for the
spectral interferometry measurements.

Figure 5: Extended data figure 2. Long-term stability of $\omega$-3$\omega$
interferometer. (a) Measurements of the modulation of 5th harmonic generation
as a function of the relative phase difference between $\omega$-3$\omega$
fields $\varphi$ repeated over 60 minutes (color scale) along with the phase
of the maxima obtained from fitting the data with harmonic function (black
curve). (b) Timing jitter and long-term drift calculated from the measured
data shown in (a). The RMS timing fluctuations of the $\omega$-3$\omega$
fields measured over 60 minutes are 2.8 as.

Figure 6: Extended data figure 3. Transient reflectivity dynamics in silicon.
The black curve shows the dynamics of the transient reflectivity change
induced by the pump consisting of the $\omega$-3$\omega$ combination probed by
the ultraviolet pulses with the wavelength of 343 nm. The amplitude of the
signal is linearly proportional to the excited carrier population.

Figure 7: Extended data figure 4. Modulation of high harmonic generation yield
in silicon by $\omega$-3$\omega$ fields for different field ratios $r$. Curves
show the measured data of the HHG yield of the 5th harmonics (uppermost
panel), 7th harmonics (central panel) and 9th harmonics (bottom panel) as a
function of the relative phase between $\omega$-3$\omega$ fields $\varphi$.
The phase corresponding to the maxima of modulation for each value of $r$ are
marked by black squares.

Figure 8: Extended data figure 5. Spectral interferometry of HHG in silicon.
The measured data (black curve) of the spectral interference of the 5th, 7th
and 9th harmonics are fitted using equation (2) (red dotted curves).