# Multiband dynamics of extended harmonic generation in solids under
ultraviolet injection

Yue Lang1, Zhaoyang Peng1, Zengxiu Zhao1111Corresponding author:
<EMAIL_ADDRESS>1 Department of Physics, National University of
Defense Technology, Changsha, P. R. China

###### Abstract

Using one-dimensional semiconductor Bloch equations, we investigate the
multiband dynamics of electrons in a cutoff extension scheme employing an
infrared pulse with additional UV injection. An extended three-step model is
firstly validated to play a dominant role in emitting harmonics in the second
plateau. Surprisingly, further analysis employing the acceleration theorem
shows that though harmonics in both the primary and secondary present positive
and negative chirps, the positive (negative) chirp in the first region is
related to the so-called short (long) trajectory, while that in the second
region is emitted through ‘general’ trajectory, where electrons tunnelling
earlier and recombining earlier contribute significantly. The novel
characteristics deepen the understanding of high harmonic generation in solids
and may have great significance in attosecond science and reconstruction of
band dispersion beyond the band edge.

##  1 Introduction

Highly nonlinear interaction of multicycle laser pulse with gaseous targets
enables the generation of high harmonics (HHG) and lays the foundation of
attosecond science[1]. The process of up-converting the fundamental frequency
can be well described with the semiclassical three-step model including
tunneling, acceleration and recombination[2, 3] and has been extensively
studied during the past three decades[4, 5, 6, 7, 8, 9]. Due to the rich
information carried by HHG from the microcosm, it has been widely applied to
the detection of structures and control of ultrafast dynamics with
unprecedented spacial and temporal resolutions[10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20]. The cutoff in gases is theoretically and experimentally given by
$\hbar\omega_{\mathrm{cutoff}}\approx I_{p}+3.17U_{p}\propto I\lambda^{2}$,
illustrating the possibility of extending the energy of harmonic photons well
into extreme ultraviolet and even “water window” region employing longer
wavelengths[21, 22, 23, 24].

HHG from solids has also been scrutinized theoretically during the same
period[25, 26, 27, 28], partially due to its promise of intenser and higher
flux of emission with much denser electrons. However, the experimental
observation was reported only recently in Zinc Oxide[29], benefiting from
longer wavelength of the incident pulse to circumvent radiation damage in
condensed materials. Mechanism of solid HHG still remains under intense debate
and is partially shed light on in pump-probe experiments[30, 31, 32], where
the rescattering model is found to predict satisfyingly the observations. The
high efficiency of HHG in solids replenishes the development of table-top
attosecond sources[33] and enables petahertz-bandwidth signal manipulation
through ultrafast control of current injection[34, 35]. Also, deeper insights
into the mechanism of HHG in solids hold the promises of reconstruction of
band structures[36], potential field imaging[37], detection of topological
transition[38], etc. Different from the counterpart in gases, the solid-state
cutoff is observed to scale linearly with the electric field[29] and has been
demonstrated to depend on the band dispersion[39], while direct relevance with
driving wavelength is still under debate[40]. As a result, potential
applications call for deeper understanding of solid HHG and new methods to
extend the cutoff.

Various schemes have been explored intensively to increase the cutoff and
intensity of harmonics in solids, such as the employment of solid Ar (or
Kr)[33], the consideration of pre-acceleration[41], properly-doped
semiconductor[42, 43], etc. On the other hand, different compositions of
incident pulses, such as two color or even three color, have been explored
where the relative intensity, delay, carrier-envelope phase (CEP), and pulse
shape can be adjusted to achieve better characteristics of resulting harmonic
spectra[44, 45, 46, 47, 48]. In solids, further extend of harmonics to higher
energy region requires intenser incident pulse to drive direct transitions
from the highest valence band to higher-lying conduction bands. Due to the
large gaps, however, the incident pulse should be intense enough to achieve
significant population, which can possibly cause irreversible damage to
materials. An appearing bypass is coincidence of infrared (IR) and properly-
delayed ultraviolet pulses, hereinafter referred to as IR+UV scheme. We
recently have proposed a model to extend the cut-off energy of harmonic
generated from solids by injecting both the IR and UV pulses [49]. In this
study, we further verify the extended three-step model and discuss the effects
of multiple bands. In particular, we examine the difference in the interband
dynamics of the two plateaus. The paper is organized as follows: The
theoretical model and approaches are given in the methods section; Then, we
present the extended three-step model and focus on the multiband dynamics in
the extended plateau; Finally, we summarize the findings in the conclusion
part.

##  2 Methods

The dynamics of electrons in laser solid interaction can be given by the
semiconductor Bloch equations (SBE)[50, 51, 52]:

$\displaystyle i\frac{\partial}{\partial
t}{\rho}_{mn}^{\mathbf{k}}\left(t\right)=$
$\displaystyle\left[E_{m}^{\mathbf{k^{\prime}}}-E_{n}^{\mathbf{k^{\prime}}}-\frac{i\left(1-\delta_{mn}\right)}{T_{2}}\right]\rho_{mn}^{\mathbf{k}}\left(t\right)$
(1)
$\displaystyle+\mathbf{F}\left(t\right)\cdot\sum_{l}{\left[\mathbf{d}_{ml}^{\mathbf{k^{\prime}}}\rho_{ln}^{\mathbf{k}}\left(t\right)-\mathbf{d}_{ln}^{\mathbf{k^{\prime}}}\rho_{ml}^{\mathbf{k}}\left(t\right)\right]},$
(2)

where $\mathbf{k}$ is the crystal momentum, $m$ (or $n$) is the band indice,
$\rho_{mn}$ is the density matrix element, with $m=n$ representing interband
polarization and $m\neq n$ population, $E$ is the band dispersion,
$\mathbf{F}(t)$ is the electric field, $\mathbf{d}$ is the transitional dipole
moment (TDM), $T_{2}$ is the empirical dephasing time and
$\mathbf{k}^{\prime}=\mathbf{k}+\mathbf{A}(t)$ is the crystal momentum in the
moving frame. Note that $T_{2}$ is set to infinity unless otherwise stated and
the influence of shorter dephasing time [53] will be illustrated later.

The total current can be split into the interband and intraband contributions:

$\displaystyle\mathbf{J}_{\mathrm{inter}}\left(t\right)=-\int_{BZ}{d}\mathbf{k}\sum_{m\neq
n}{\rho_{mn}^{\mathbf{k}}\mathbf{p}_{nm}^{\mathbf{k}}},$ (3)
$\displaystyle\mathbf{J}_{\mathrm{intra}}\left(t\right)=-\int_{BZ}{d}\mathbf{k}\sum_{m=n}{\rho_{mn}^{\mathbf{k}}\mathbf{p}_{nm}^{\mathbf{k}}},$
(4)

where the momentum matrix element (MTM) $\mathbf{p}_{nm}^{\mathbf{k}}$ is
related to the TDM $\mathbf{d}_{nm}^{\mathbf{k}}$ through canonical
relation[54]

$\mathbf{d}^{\mathbf{k}}_{nm}=\frac{-i\mathbf{p}^{\mathbf{k}}_{nm}}{E^{n}_{\mathbf{k}}-E^{m}_{\mathbf{k}}}\
(\mathrm{for}\ m\neq n).$ (5)

The calculation of one dimensional band dispersion and TDM are described in
other works[50] for model crystal ZnO along $\Gamma$-M.

The harmonic spectrum is calculated by Fourier transformation

$\mathrm{Y}(\omega)=\left|\int_{0}^{T}\left[\frac{\partial\mathbf{J}(t)}{\partial
t}w(t)\right]e^{i\omega t}dt\right|^{2},$ (6)

where $T$ is the calculation time, $w(t)$ is the absorption function given by

$\displaystyle w\left(t\right)=\left\\{\begin{array}[]{l}1\ \ \ \ \ \ \ \ \ \
\ \ \ ,\ t<T_{0}\\\ e^{\frac{-4\ln 2\left(t-T_{0}\right)^{2}}{\sigma^{2}}},\
t\geq T_{0}\\\ \end{array}\right.,$ (9)

with $T-T_{0}\approx 10$fs. Fourier transformation in Eq. (6) is conducted
using Fast Fourier Transformation (FFT) algorithm.

Time-frequency analysis of the total current is calculated by continuous
wavelet transformation

$\mathrm{S}(t,\omega)=\left|\frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}\frac{\partial}{\partial
t}\mathbf{J}\left(t^{\prime}\right)\alpha\left(\frac{t^{\prime}-t}{a}\right)dt^{\prime}\right|^{2},$
(10)

where $a$ is the scale factor, $\alpha(x)$ is the mother wavelet, i.e. Morlet
wavelet in our case with the time-bandwidth set to 60 to achieve better
resolution in time and space.

##  3 Results and Discussion

Figure 1:  Sketch of the extended three-step model and the general recollision
model. Left panel: acceleration ($0^{\prime}$), injection ($1^{\prime}$),
acceleration ($2^{\prime}$), tunneling ($3^{\prime}$), acceleration
($4^{\prime}$) and recombination ($5^{\prime}$); Right panel: tunneling ($1$),
acceleration ($2$) and recombination ($3$). In the extended three-step model,
excited electrons on band C2 originate far from the BZ center (magenta solid
circle) on band V1, followed by injection to band C1 through properly-delayed
UV pulse.

The general recollision model in two-band approximation [31, 30, 55] is shown
on the right panel of Fig. 1. Near the extreme of electric field, electrons
locating initially around high symmetry points are partially excited to the
lowest conduction band and form electron hole pairs. Then, electrons (holes)
are accelerated by the laser field, with harmonics emitted from intraband
current. Finally, when the electric field reverses its direction, excited
electrons are driven back towards the holes and complete recombination with
emission of harmonics, commonly referred to as the interband mechanism. For
direct band-gap materials like ZnO[29] along specific crystal direction,
excited electrons locate generally near the center of Brillouin zone (BZ),
i.e. $\mathbf{k}=0$, which impedes the following propagation to the band edge
to be further pumped to higher-lying band C2.

In order to extend the cutoff, an additional ultraviolet (UV) pulse can be
introduced and the dynamics of electrons are shown on the left panel of Fig.
1. Electrons locating initially far from the BZ center are driven by the IR
field (step $0^{\prime}$) to the center, followed by excitation to the lowest
conduction band C1 through one-photon absorption at the instant of UV
injection (step $1^{\prime}$). Then these electrons can be accelerated closer
to the BZ boundary (step $2^{\prime}$) and tunnel to band C2 (step
$3^{\prime}$) for further acceleration (step $4^{\prime}$). Finally, the
electrons on C2 recombine with the hole pairs and emit high energy photons
(step $5^{\prime}$). Note that though the injection scheme has been employed
by Sun et al[48], the motion of electrons in their work is limited to only two
bands, which differs from the extended three-step model.

Figure 2:  (a) Harmonic spectrum under various injection intensities with IR
fixed at $2\times 10^{11}\mathrm{W/cm}^{2}$. The dashed lines represent energy
gaps between the highest valence band and the lowest ($11\omega_{L}$ to
$36\omega_{L}$) and second lowest ($38\omega_{L}$ to $68\omega_{L}$)
conduction band, respectively; (b) The dependence of cutoff intensity on the
injection intensity (dark pentagrams). Inset shows the separately-rescaled
pulse shape of IR (red) and UV (magenta) fields. Absorption of one UV photon
is indicated by the linear fit (magenta dashed line).

The pulse shape of both IR and UV is given by

$\displaystyle
F(t)=F_{0}\cos^{2}\left[\pi\left(\frac{t-t_{\mathrm{delay}}}{T}-\frac{1}{2}\right)\right]\sin\left[\omega\left(\frac{t-t_{\mathrm{delay}}}{T}-\frac{1}{2}\right)\right],$
(11)

with $F_{0}$ the electric amplitude, $T$ the total duration, $\omega$ the
angular frequency and $t_{\mathrm{delay}}$ the delay time. The IR pulse is
centered at 3200 nm ($\omega_{L}=0.39$ eV), with a total duration of 8 cycles
(8$T_{L}$) and an intensity of $2\ \times 10^{11}\mathrm{W/cm}^{2}$. The
injection UV pulse is centered at 291 nm ($11\omega_{L}$), with a duration of
3 cycles at $-0.25T_{L}$ delay, shown in the inset of Fig. 2(b). Fig. 2(a)
illustrates the harmonic spectra at various injection intensities. It can be
seen that the cutoff in the IR scheme is
$\omega_{\mathrm{cutoff}}^{\mathrm{IR}}\approx 25\omega_{L}$, while that in
the IR+UV scheme is extended to
$\omega_{\mathrm{cutoff}}^{\mathrm{IR+UV}}\approx 60\omega_{L}$ in the
secondary plateau.

Figure 3:  (a) Dependence of harmonic spectrum on two-color delays, with the
second plateau presented in logarithmic scale; Population difference of band
C2 $\Delta N_{\mathrm{C2}}(t,k)$ between the two schemes at the delay of (b)
$-0.25T_{\mathrm{L}}$, (c) $0T_{\mathrm{L}}$ and (d) $0.25T_{\mathrm{L}}$,
with $\Delta
N_{\mathrm{C2}}(t,k)=N_{\mathrm{C2}}^{\mathrm{IR+UV}}(t,k)-N_{\mathrm{C2}}^{\mathrm{IR}}(t,k)$.
The generation of radiation in the extended plateau demands properly-delayed
UV near the zero-crossing of electric field at $-0.25T_{L}$ or $0.25T_{L}$,
and the highly-excited electrons on C2 locates far from the center at
$k_{0}\approx-0.6\pi/a_{0}$ (white dashed line) and $0.63\pi/a_{0}$ (white
dashed line), respectively, as predicted by the extended three-step model. The
emergence of electrons on C2 (red dashed lines) is delayed to the injection
instant (magenta dashed line) by $0.35T_{L}$ and proves the failure of channel
3 (see main text).

Though it is evident that emissions in the secondary region origin from
significant population on band C2, there are actually three possible channels
in the IR+UV setup:

  1. 1.

V1 to C1 and C1 to C2, both involve the absorption of one UV photon,
hereinafter referred to as channel 1;

  2. 2.

V1 to C1 through absorption of one UV photon and C1 to C2 through tunnelling,
hereinafter referred to as channel 2;

  3. 3.

V1 to C1 through tunnelling and C1 to C2 through absorption of one UV photon,
hereinafter referred to as channel 3.

Through detailed analysis of the intensity dependence and population
variation, channel 2 is validated to contribute significantly to the
population on C2. Firstly, as shown in Fig. 2(b), the linear dependence of the
cutoff intensity on the injection intensity $I_{11\omega_{L}}$ implies that
only one UV photon participates in the transition. In result, contribution
from channel 1 can be easily omitted for the expected quadratic dependence on
UV intensity. In channel 2, i.e. the extended three-step model, it can be
predicted that injection at the extreme of electric field fails to populate
electrons up to band C2, because additionally excited electrons still locate
near the BZ center. Fig. 3(a) illustrates the dependence of secondary plateau
on the injection delays. As can be seen, only injections near the delay of
$-0.25T_{L}$ and $0.25T_{L}$ produce evident radiation. Moreover, Fig.
3(b)-(d) illustrate the evolution of population variation on band C2 at three
typical delays, which show clearly that the excited electrons for the two
delays locate initially at $k_{0}=-0.6\pi/a_{0}$ and $0.63\pi/a_{0}$,
respectively, while the injection at zero delay fails to pump electrons to
band C2. Finally, because the absorption of one UV photon is needed for
transition from C1 to C2 in channel 3, obvious increasement of electrons on C2
should be seen at the instant of injection. However, as shown in Fig.
3(b)-(d), growth of population on C2 is delayed to the instant of UV injection
by $0.35T_{L}$. As a result, the extended three-step model is proved to
contribute mainly to the generation of harmonics in the secondary plateau.

Figure 4:  (a) Time-frequency analysis of the two-color delay of $-0.25T_{L}$,
with the result of shorter dephasing time $T_{2}=T_{L}$ presented in the
inset. Strong dephasing leads to obvious suppress of of intrinsic emissions at
both the minimal gap and $50\omega_{L}$ (gap between V1 and C2 at the crystal
momentum of highly-excited electrons); Interband dynamics in the (b) first and
(c) second plateaus employing acceleration theorem, with $t_{i}$ the
ionization instant, $t_{r}$ the recombination time and $t_{t}$ the tunneling
instant. Positive chirp in the primary plateau is linked to electrons that
ionize later but recombine earlier, i.e. the so-called short trajectory, while
that in the secondary plateau is related to electrons that tunnel and
recombine earlier, i.e. the ’general’ trajectory.

Though the transition channel has been validated, the multiband dynamics
remain folded so far. Fig. 4(a) shows the time-frequency analysis, where
harmonic emission in the primary plateau exhibits clearly a period of half
cycle of the fundamental field. The lack of clearness in the spectrum is
mainly caused by long dephasing time, which also results in the quantum beat
signal at the minimal gap after the end of laser field due to strong coherence
of Bloch states near the gap[56]. With the decrease of dephasing time down to
$T_{L}$, the beat signal gradually disappears as shown in the inset.
Meanwhile, significant emission at $t=4.75T_{L}$ can also be seen clearly in
the second plateau and is delayed by approximately one optical cycle to the
injection instant at $3.75T_{L}$. Surprisingly, slightly weakened harmonics
are also generated at $t=6.75T_{L}$, which is emitted by the coherent Bloch
states at the crystal momentum ($-0.6\pi/a_{0}$) of excited electrons, see
Fig. 3(b). The intrinsic emission can also be suppressed by shorter dephasing
time as shown in the inset.

We then examine the subcycle electron dynamics in both the first and second
plateau. Within the acceleration theorem, the spacial displacement of electron
hole pairs can be determined by

$R_{e,h}(t)=\int^{t_{r}}_{t_{i}}\nabla_{\mathbf{k}}E(\mathbf{k}_{0}+A(\tau)-A(t_{i}))d\tau,$
(12)

where $\mathbf{k}_{0}$ is the initial crystal momentum of excited electrons,
$t_{i}$ and $t_{r}$ is the ionization and recombination instant, respectively.
$t_{r}$ and energy of harmonic $\omega$ is given by

$\displaystyle R_{e}-R_{h}=$ $\displaystyle 0,$ (13) $\displaystyle
E_{e}\left[\mathbf{k}_{0}+A(t_{r})-A(t_{i})\right]-E_{h}\left[\mathbf{k}_{0}+A(t_{r})-A(t_{i})\right]=$
$\displaystyle\hbar\omega.$ (14)

Taking ionization near $4T_{L}$ as an example, Fig. 4(b) shows the dependence
of harmonic energy and recombination time on ionization time by solving Eq.
(12)-(14) with $\mathbf{k}_{0}=0$. It can be seen that electrons ionized right
after the field extreme ($4T_{L}\leq t\leq 4.06T_{L}$) recombine at delays
larger than $\sim 0.7T_{L}$, while those ionized later ($4.06T_{L}<t\leq
4.25T_{L}$) recombine earlier, i.e. the so-called long and short trajectory
respectively. Meanwhile, emissions from long trajectory exhibit negative chirp
while those from short trajectory generate positive chirp. Similarly, the
displacement of electrons in the extended three-step model is given by a
convenient variation of Eq. (12)

$\displaystyle R_{e}(t)=$
$\displaystyle\int^{t_{t}}_{t_{i}}\nabla_{\mathbf{k}}E_{\mathrm{C1}}(\mathbf{k}_{0}+A(\tau)-A(t_{i}))d\tau+$
(15)
$\displaystyle\int^{t_{r}}_{t_{t}}\nabla_{\mathbf{k}}E_{\mathrm{C2}}(\mathbf{k}_{0}+A(\tau)-A(t_{i}))d\tau,$

with $t_{t}$ the tunneling instant from C1 to C2. Setting the initial crystal
momentum to be $-0.6\pi/a_{0}$ as shown in Fig. 3(b) and the ionization time
$t_{i}$ at the injection instant $3.75T_{L}$, Fig. 4(c) shows the dependence
of harmonic energy and recombination time on the tunneling instant. It can be
seen that the recombination time and energy of harmonic emission shows nearly
perfect match with the time-frequency analysis. Surprisingly, though emissions
in the secondary plateau exhibit both positive and negative chirps, the chirps
are not related to the so-called long or short trajectories in the interband
dynamics of the first plateau. Actually, the positively and negatively chirped
emission in the extended region is radiated by electrons from the ‘general’
trajectory, where the electrons that ionize earlier recombine earlier with a
delay of approximately $0.37\sim 0.6T_{L}$. Varying the band dispersion gives
a delay between $0.1T_{L}$ and $0.7T_{L}$, approaching that of the short
trajectory. As a matter of fact, time interval between tunneling and
recombination depends on the actual band dispersion of band C2, because it
determines the group velocity of recombining electrons, indicated by Eq. (15).
The sensitivity to band dispersion, on the other hand, can be possibly applied
to reconstruct the band structure of higher-lying band.

##  4 Conclusions

In summary, we investigate the multiband dynamics of cut-off extension in the
IR+UV scheme employing 1D SBE. Through linear scaling of cutoff intensity on
injection intensity, delay sensitivity and time-dependent population
difference, we solidify the foundation of extended three-step model in
describing the origin of population increasement on high-lying conduction
band. Surprisingly, deeper exploration into the time-frequency characteristics
shows that though both positive and negative chirps coexists during the
subcyle dynamics in the first and second plateau, the positive chirp in the
first plateau is radiated by electrons that ionize later but recombine earlier
while that in the second plateau is linked to electrons that tunnel earlier
and recombine earlier. Our research may have implications for the generation
of table-top attosecond sources and hold the promise of all-optical
reconstruction of band structure beyond the lowest band edge.

This study is supported by China National Key R&D Program (Grant No.
2019YFA0307703), the NSF of China (Grant No. 12234020, 12274461 and 11904400),
and the Major Research plan of NSF of China (Grant No. 91850201).

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