# Role of inter- and intraband current in laser interaction with
bi-chromatic quasi-periodic crystals

Amol R. Holkundkar ID<EMAIL_ADDRESS>Nivash R  Jayendra N. Bandyopadhyay
ID Department of Physics, Birla Institute of Technology and Science - Pilani,
Rajasthan, 333031, India

###### Abstract

We study the role of the inter- and intraband current in the laser interaction
with the bi-chromatic quasi-periodic crystals. The interaction dynamics are
simulated by solving the time-dependent Schrödinger equation in the $k$-space,
and time evolution of the inter- and intraband current is obtained in a gauge
invariant form. We observed that for certain bi-chromatic potential ratios,
the energy band structure of the ‘valence band’ and the ‘conduction band’
facilitate the interband transitions only at the center or at the edge of the
Brillouin zone, which leads to a very interesting population transfer
mechanism between the bands. The temporal profile of the inter- and intraband
current gives a detailed account of the interaction. The higher-order harmonic
generation (HHG) is also studied for these bi-chromatic optical lattices, and
the resultant harmonic yield is commented upon.

##  I Introduction

High-harmonic generation (HHG) from solids is gaining consistent traction and
a field of contemporary interest around the globe because of the applications
it promises in strong-field and attosecond physics. Though the HHG by the
atomic gases formed the basis for the attosecond science, the necessity of the
complex setups with vacuum environments and sophisticated optics, along with
the lower conversion efficiency of the HHG, poses significant challenges from
the applied view. Solid-state HHG, from this perspective, has simplified
operational details, with lower laser intensities and strong electron dynamics
within the bands, which is remedied after the advent of the HHG by the Bloch
oscillations in the solids [1, 2, 3, 4, 5, 6]. As a result, solid-state HHG
promises a compact source of the XUV radiations and attosecond spectroscopy
[7, 8, 9, 10, 11]. The pioneering work on the HHG in the bulk ZnO crystal [1]
has opened the avenues in this vast field, and later the HHG is demonstrated
in a wide range of materials such as larg-bandgap dielectrics [7], metasurface
[12], graphene [13], transition metal dichalcogenide [14], topological
insulators [15] and many more. For harmonic cutoff enhancement, yield, and
optimization, numerous studies have reported wherein synthesized laser fields
are used [16, 17, 18].

The solid-state HHG is mainly understood in terms of the delicate interplay
between the two prominent physical mechanisms: interband polarization and the
laser-driven intraband currents [19, 20, 21, 14, 22, 23, 24, 22, 25, 26]. It
has been observed that the harmonics caused by the interband current always
dominate the harmonics generated by the intraband current in the
nonperturbative regime (harmonics above the minimum band gap energy), however
in the perturbative regime, the intraband and interband harmonics are
comparable with intraband harmonics being a slightly stronger [19, 27]. The
gauge-independent (velocity or length) formulation of the inter- and intraband
current for the HHG in the solids is presented in [22, 25]. The inter- and
intraband transitions can also be understood from the motion of the Bloch
electrons moving with the phase and group velocities in the coordinate space
under the influence of the laser fields [26].

The interband transitions typically happen when the electron passes through
the region where the corresponding band gap between valence and conduction (or
any other neighboring bands) is minimal, resulting in rapid changes in the
electron population. In the context of the light-matter interaction, time-
dependent population inversion between two energy levels is referred to as
Rabi oscillations. The time evolution of the conduction band population in the
context of the Rabi flopping/oscillations are previously reported [28, 29].
The carrier-envelope phase (CEP) of the driving laser pulses are also found to
significantly affect the band population [28, 30]. The control over the energy
states of the electron using some external agency is desirable for
understanding the underlying quantum dynamics.

Figure 1: The normalized potential denoted by Eq. (1) for 0:1, 1:2, 1:3 and
5:8 are presented in (a). The corresponding band structure showing the first
four bands is shown for the ratios 1:2 (c), 1:3 (d), and 5:8 (e). The minimum
bandgap between the VB2 and CB1 is 9.4 eV for 1:2 (c), 0.85 eV for 1:3 (d),
and 0.012 eV for 5:8 ratio (e). The ratio 0:1 is pretty standard Mathieu-type
potential [31], and hence the associated band structure is not presented. The
$k$ dependent matrix element of transition from second band (VB2) to the third
band (CB1) is presented in (b), i.e.
$p_{k}^{23}=\braket{\phi_{k}^{2}}{\hat{p}}{\phi_{k}^{3}}$.

For a given periodic crystal, the control over the dynamics or outcome of the
solid-state HHG relies on tweaking the inter- and intraband currents using the
synthesized laser fields. The band structure of the periodic crystal
significantly affects the inter- and intraband current dynamics, and so the
HHG [32]. The intraband current can have the clear signature of the band
structure of the periodic crystal under study [33]. To this end, the optical
lattices provide flexibility in terms of tweaking the lattice spacing and so
the band structure [34, 35].

In this work, we have explored the interaction of the laser pulse with quasi-
periodic crystals and the role the inter- and intraband current plays in
populating higher energy bands and the HHG. We have solved the one-dimensional
(1D) time-dependent Schrödinger equation (TDSE) in the quasi-momemtum space or
$k$-space. The inter- and intraband current are calculated in a gauge
invariant form along with the time-dependent population of the valence and
conduction band. It is observed that the interference of the inter- and
intraband current plays a very crucial role in suppressing high-energy
harmonics. The paper is organized as follows: in Sec. II, we discuss the
theoretical and simulation details, followed by the results in Sec. III and
summary in Sec. IV. Throughout the manuscript, we have used the atomic units
(a.u.) i.e., $|e|=m_{e}=\hbar=1$.

##  II Theoretical methods

###  II.1 Bi-chromatic potential

We study the interaction of the linearly polarized laser with the one-
dimensional bi-chromatic periodic crystal. The laser polarization direction is
considered to be along the optical lattice, and the bi-chromatic potential is
modeled as [36]:

$V(x)=-V_{0}[A+B-A\cos(g\sigma_{1}x)-B\cos(g\sigma_{2}x)]$ (1)

here, $V_{0}$ denotes the depth of the potential, $g\equiv 2\pi/d$,
$\sigma_{1}$ and $\sigma_{2}$ determines the respective frequencies of the bi-
chromatic potential. Moreover, $d=8$ a.u. is the lattice constant, $A$ and $B$
control the depth of the bi-chromatic potential. However, in our work we have
set $A=B=1$. There are two characteristic lengths in a unit cell: inter-atomic
separation and the lattice constant $d$. In Fig. 1(a), we present the
normalized potential for a unit cell with different frequency ratios
$\sigma_{1}:\sigma_{2}$. The energy band structure for the frequency ratios
1:2, 1:3, and 5:8 is presented in Fig. 1(c), (d), and (e), respectively, where
$V_{0}=0.3$ a.u. is used. Throughout the manuscript, we have considered that
the electron is initially in the second band referred to as ‘Valence-Band’
(VB2). However, the higher lying bands will be referred to as ‘Conduction-
Band’ CB1 (third band), CB2 (fourth band) etc. The lower bandgap between VB2
and CB1 near $k=0$ for the ratio 5:8 makes it very interesting from the inter-
and intraband current dynamics perspective, which we will discuss later in the
paper. As can be understood from Fig. 1(a), the ratio 0:1 signifies 1 atom per
cell, 1:2 denotes 2 atoms per cell, 1:3 denotes 3 atoms per cell, and 5:8
denotes 8 atoms per cell. The results are checked for convergence.

###  II.2 TDSE solver

The electron wave function can be expanded in Bloch state basis
$\ket{\phi_{k}^{n}}$ for a particular value of the crystal quasimomentum $k$,
and band index $n$ in order to numerically solve the TDSE in the velocity
gauge [37, 38]. The Bloch states are evaluated by solving the single-electron
stationary Schrödinger equation with field-free Hamiltonian
$\hat{H}_{\text{o}}=\hat{p}^{2}/2+V(x)$:

$\hat{H}_{\text{o}}\ket{\phi_{k}^{n}}=E_{k}^{n}\ket{\phi_{k}^{n}}.$ (2)

In position basis, the Bloch states can be written as:

$\braket{x}{\phi_{k}^{n}}\equiv\phi_{k}^{n}(x)=\sum_{\ell=1}^{N_{\text{max}}}C_{k,\ell}^{n}\
\text{e}^{i(k+2\pi\ell/d)x},$ (3)

where $N_{\text{max}}=19$ are used throughout the work and respective
convergence is checked. The TDSE can be solved for electronic wavefunction
$\ket{\psi_{k}(t)}$ as :

$i\frac{\partial}{\partial
t}\ket{\psi_{k}(t)}=[\hat{H}_{\text{o}}+\hat{H}_{\text{int}}]\ket{\psi_{k}(t)},$
(4)

where, $\hat{H}_{\text{int}}=\mathbf{A}(t)\cdot\hat{p}$ and
$\mathbf{A}(t)\equiv-\int\limits^{t}\mathbf{E}(t^{\prime})dt^{\prime}$ is the
vector potential associated with the laser pulse under dipole approximation
with the electric field $\mathbf{E}(t)$ polarized along the $x$ direction as:
$\mathbf{E}(t)=E_{0}\sin^{4}(\pi t/T)\sin(\omega_{0}t+\theta)\
\mathbf{e}_{x},$ here, $E_{0}\text{[a.u.]}\sim 5.342\times
10^{-9}\sqrt{I_{0}}$ is the field amplitude with $I_{0}$ being the peak
intensity of the pulse in W/cm2, $T$ is the pulse duration, $\omega_{0}$ is
the fundamental frequency of the laser pulse and $\theta$ is the carrier-
envelope phase (CEP) of the pulse (considered to be zero unless stated
otherwise). Furthermore, at any given time instant, the time-evolving state
$\ket{\psi_{k}(t)}$ can be expanded in the Bloch basis

$\ket{\psi_{k}(t)}=\sum_{n=1}^{N_{\text{max}}}\alpha_{k}^{n}(t)\ket{\phi_{k}^{n}},$
(5)

where $\alpha_{k}^{n}(t)$ are the time-dependent expansion coefficients. Using
Eq. (5) in Eq. (4), the following coupled differential equations need to be
solved [37]:

$i\frac{\partial\alpha_{k}^{s}(t)}{\partial
t}=E_{k}^{s}\alpha_{k}^{s}(t)+A(t)\sum_{u=1}^{N_{\text{max}}}p_{k}^{su}\alpha_{k}^{u}(t).$
(6)

Here, $p_{k}^{su}$ is the matrix element of the momentum operator, which can
be calculated as:

$p_{k}^{su}=\braket{\phi_{k}^{s}}{\hat{p}}{\phi_{k}^{u}}=\sum_{\ell=1}^{N_{\text{max}}}(k+2\pi\ell/d)\left(C_{k,\ell}^{s}\right)^{*}C_{k,\ell}^{u}.$
(7)

If we consider the electron initially in the band $q$, then the initial
condition for solving Eq. (6) is $\alpha_{k}^{s}(0)=\delta_{qs}$. Finally, the
single electron current density for a particular channel $k$ can be calculated
as:

$j_{ks}(t)=-\text{Re}[\braket{\psi_{ks}}{\hat{p}+A(t)}{\psi_{ks}}].$ (8)

In Eq. (8), the subscript ‘$s$’ denotes that the electron was in the band $s$
before the interaction. Total current density can be calculated by summing
over all the bands and integrating over the first Brillouin Zone (BZ) as:

$j_{\text{total}}(t)=\sum_{s\in VB}\int j_{ks}(t)dk.$ (9)

###  II.3 Band population calculation

In order to estimate the Instantaneous Band Population (IBP) of the band $m$
at a given instant of time, the projection operator is expressed in terms of
the field free Bloch basis [25],

$\hat{\Pi}_{mk}=\ket{\phi_{k}^{m}}\bra{\phi_{k}^{m}}.$ (10)

In the velocity gauge the projection operator of Eq. (10) would transform as
[39]:

$\hat{\Pi}_{mk}^{\text{vel}}=\text{e}^{-iA(t)\hat{x}}\ \hat{\Pi}_{mk}\
\text{e}^{iA(t)\hat{x}},$ (11)

where $A(t)$ is the vector potential associated with the driver field. The
time-dependent population of the band $m$ is hence obtained by calculating the
expectation value of the operator given in Eq. (11) as [25]:

$\mathpcal{P}_{m}(t)=\int\limits_{BZ}\bra{\psi_{k}(t)}\hat{\Pi}_{mk}^{\text{vel}}\ket{\psi_{k}(t)}dk.$
(12)

###  II.4 Inter- and intraband current and HHG calculation

In Eq. (9), we obtained temporal dependence of the total current. To
understand the inter- and intraband contribution in total current [Eq. (9)],
typically, one needs to rely on the Houston state basis [4] when working in
the velocity gauge; however, it has some numerical limitations to be used with
a large number of bands. To remedy this, we used the Bloch state basis-based
gauge-independent formulation, and accordingly, the inter- and intraband
currents are calculated [25] in terms of the population operator [Eq. (11)]:

$j_{\text{inter}}(t)=-\sum\limits_{m,m^{\prime}\neq m}\iint_{BZ}dkdk^{\prime}\
\hat{\Pi}_{mk}^{\text{vel}}\ (\hat{p}+A(t))\
\hat{\Pi}_{m^{\prime}k^{\prime}}^{\text{vel}},$ (13)
$j_{\text{intra}}(t)=-\sum\limits_{m}\iint_{BZ}dkdk^{\prime}\
\hat{\Pi}_{mk}^{\text{vel}}\ (\hat{p}+A(t))\
\hat{\Pi}_{mk^{\prime}}^{\text{vel}},$ (14)

such that, $j_{\text{total}}(t)=j_{\text{inter}}(t)+j_{\text{intra}}(t)$.

The spectra of the emitted harmonics can be estimated by doing the Fourier
transform of the current density and are given as:

$S_{\text{total}}(\omega)=\Big{|}\mathpcal{F}_{\omega}[j_{\text{total}}]\Big{|}^{2}$
(15)

where, $\mathpcal{F}_{\omega}[g(t)]=\int g(t)\exp[-i\omega t]dt$ is the
Fourier transform of the time dependent function $g(t)$. However, it can also
be interpreted as the summation of the harmonic spectra from the
$j_{\text{inter}}(t)$, $j_{\text{intra}}(t)$ and the interference of both [24]
as follows:

$S_{\text{total}}(\omega)=S_{\text{inter}}(\omega)+S_{\text{intra}}(\omega)+S_{\text{intfer}}(\omega)$
(16)

where,

$S_{\text{inter,intra}}(\omega)=\Big{|}\mathpcal{F}_{\omega}[j_{\text{inter,intra}}]\Big{|}^{2},$
(17)

and,

$S_{\text{intfer}}(\omega)=\mathpcal{F}_{\omega}^{\ast}[j_{\text{inter}}]\mathpcal{F}_{\omega}[j_{\text{intra}}]+\mathpcal{F}_{\omega}^{\ast}[j_{\text{intra}}]\mathpcal{F}_{\omega}[j_{\text{inter}}],$
(18)

denote the harmonic contribution from the interband, intraband current [Eq.
(17)] and the interference of both is represented as Eq. (18). The harmonic
yield $Y$ for the frequency range $\omega_{1}$ to $\omega_{2}$ is calculated
as: $Y=T^{-1}\int_{\omega_{1}}^{\omega_{2}}S_{\text{total}}(\omega)d\omega$.
Furthermore, the phase of the emitted harmonics is obtained as:
$\varphi_{i}(\omega)=\rm{arctan2}[\rm{Im}\\{S_{i}(\omega)\\},\rm{Re}\\{S_{i}(\omega)\\}]$,
with $i$ associated with either ‘inter’, ‘intra’ or ‘total’ spectra.

Figure 2: The HHG spectra for the frequency ratios 1:2 (a), 1:3 (b) and 5:8
(c) are presented for $S_{\text{inter}}$, $S_{\text{intra}}$, and
$S_{\text{total}}$. The red arrow represents the minimum bandgap between the
VB2 and CB1 for respective cases [refer Fig. 1]. The respective inter- and
intraband currents are presented for 1:2 (d), 1:3 (e), and 5:8 (f). As
mentioned, the inter- and intraband current in (d-f) is plotted on the same
axis with different scaling parameters $a$ and $b$.

##  III Results and discussion

Figure 3: Temporal evolution of the interband current along with the VB2 and
CB1 band population is presented for frequency ratio 1:3 (a) and 5:8 (b). The
zoomed version of (b) in the range $2.6\tau\leq t\leq 3.3\tau$ is shown in
(c), with $\tau$ being an optical cycle. In (c), the temporal profile of the
vector potential (dashed line) is also shown along with the intraband current.
In order to represent band population ($\mathpcal{P}$) and the interband
current on the same scale, an appropriate scale factor $\delta$ is mentioned
for interband current, and the quantity $4\mathpcal{P}-2$ is plotted, such
that $\mathpcal{P}=0$ corresponds to $-2$ and $\mathpcal{P}=1$ corresponds to
$+2$ on the $y$-scale.

The interaction of the 10 cycles, 3.2 $\mu$m laser with a peak intensity of
$6\times 10^{11}$ W/cm2 is considered with the quasi-periodic crystals with
frequency ratios 1:2, 1:3, and 5:8, having $V_{0}=0.3$ a.u, and $10\%$
contribution around $k=0$ in the BZ is considered i.e. $-0.1<kd/\pi<0.1$
(unless otherwise stated) in order to understand the underlying interaction
dynamics.

###  III.1 HHG for different ratio $\sigma_{1}:\sigma_{2}$

The decomposition of the HHG spectra with inter- and intraband contributions
is presented in Fig. 2(a-c). In the perturbative regime of the solid HHG,
inter- and intraband HHG are dominant with intraband current, and so the
harmonics are stronger. However, in the nonperturbative, regime mainly
interband current contributes, though both the inter- and intraband currents
show the plateau structures [19, 24, 26]. Here, we can corroborate these
findings in Fig. 2(a) for frequency ratio 1:2, wherein the below bandgap
harmonics ($\lesssim 9.4$ eV) the intraband contribution ($S_{\text{intra}}$)
is comparable to the interband ($S_{\text{inter}}$) one. However, in the range
$\gtrsim 9.4$ eV, $S_{\text{inter}}$ overlap $S_{\text{total}}$ (total
contribution), implying the interference ($S_{\text{intfer}}$) and the
$S_{\text{intra}}$ terms are less significant. Furthermore, the bichromatic
frequency ratio is 1:3 [Fig. 2(b)] show a very crucial role played by the
$S_{\text{intfer}}$ terms in the total spectra. It can be seen from Fig. 2(b)
that in the perturbative regime ($\lesssim 0.85$ eV), the $S_{\text{intra}}$
is slightly dominant and in the nonperturbative regime up to harmonic cutoff
($\gtrsim 0.85$ eV and $\lesssim 10$ eV) the $S_{\text{inter}}$ is slightly
stronger than the $S_{\text{intra}}$. However, in this case,
$S_{\text{intfer}}$ is very strong and significantly affects the HHG process,
as can be seen from Fig. 2(b) after the cutoff, $S_{\text{intfer}}$ negates
the contribution from $S_{\text{inter}}$ and $S_{\text{intra}}$, giving a
clear harmonic cutoff around $\sim 10$ eV. Moreover, the dominance of the
interference term $S_{\text{intfer}}$ is found to be very stark for the case
when the frequency ratio is changed to 5:8, i.e., Fig. 2(c). In this case,
mostly all the HHG spectra is in the nonperturbative regime as the minimum
bandgap of VB2 and CB1 is $\sim 0.01$ eV [refer Fig. 1]. It can be seen from
Fig. 2(c) that the contribution of $S_{\text{inter}}$ and $S_{\text{intra}}$
are exactly identical and the contribution from $S_{\text{intfer}}$ term
reduces the combined contribution of $S_{\text{inter}}+S_{\text{intra}}$
significantly and give rise to the clean harmonic cutoff at $\sim 12$ eV.
Later, we will discuss how the minute phase difference between
$S_{\text{inter}}$ and $S_{\text{intra}}$ exactly mimics the total HHG
obtained and reinforces the importance of the interference of the inter- and
intraband currents.

The inter- and intraband current for all these three cases is also illustrated
in Fig. 2(d-f) and for visual appeal $j_{\text{inter}}$ and $j_{\text{intra}}$
are plotted on the same scale with appropriate scaling factors. It should be
worth noting that as the minimum bandgap reduces for cases from Fig. 2(a) -
(c), the interband currents gain very fast Rabi-like oscillations, which
effectively cause the rapid oscillations of the population between VB2 and
CB1. As a result, the distinction between $S_{\text{inter}}$ and
$S_{\text{intra}}$ (which are Fourier transforms of the $j_{\text{inter}}(t)$
and $j_{\text{intra}}(t)$) is completely lost in Fig. 2(c), because say the
fast depletion in one band is fast accumulation in another band in time
domain. Hence, in the spectral domain, these two events are the same. These
fast oscillations in the interband current [Fig. 2(f)] occurs near the point
where the vector potential $A(t)\sim 0$, which happens near $k\sim 0$. In
order to further elucidate this fast oscillating nature of the inter- and
intraband current near zero crossing of the vector potential, we have
presented the zoomed version of Fig. 2(f) along with the respective band
population in Fig. 3.

The detailed inter- and intraband current features for the frequency ratio 1:3
and 5:8 are presented in Fig. 3. The band population is calculated using Eq.
(12), and it can be seen from the temporal profile of the band population
that, the dynamics are effectively reduced to two band problem for the given
laser and potential parameters. If we compare the interband current of 1:3 and
5:8 frequency ratio in Fig. 3(a) and (b), then it is observed that the band
population perfectly oscillates between VB2 and CB1 for 5:8 frequency ratio,
the feature which is missing for 1:3 ratio (later we will discuss this
aspect). The zoomed version of Fig. 3(b) is illustrated as Fig. 3(c) and
following observations can be made from the same:

  1. 1.

During the complete cycle of the pulse, these rapid oscillations in the
interband current happen near the point where $A(t)=0$, which corresponds to
the electric field maxima for that particular cycle in the pulse.

  2. 2.

There are two parts in each cycle; in the first half, the population is
transferred from CB1 to VB2, and in the second half, the reverse happens, and
population transfers from VB2 to CB1. In between successive cycle, the VB2 and
CB1 are equally populated with $\mathpcal{P}=0.5$.

  3. 3.

The rapid time variation of the intraband current is out of phase with the
interband current.

  4. 4.

These time variations of the interband currents only last till the population
transfers complete from VB2 to CB1 and vice-versa.

  5. 5.

The time interval for a complete cycle of populating and then depopulating the
CB2 reduces in the window $3.15\tau<t<3.3\tau$ (with $\tau$ being one optical
cycle) as compared to the time window $2.6\tau<t<2.8\tau$.

  6. 6.

All the above points can easily be understood in terms of the Rabi
oscillations. The Rabi frequency is given by:

$\Omega_{R}(k,t)\propto|\mathbf{E}(t)|\frac{\braket{\phi_{k}^{\text{CB{\hbox{_1}}}}}{\hat{p}}{\phi_{k}^{\text{VB{\hbox{_2}}}}}}{E_{k}^{\text{CB{\hbox{_1}}}}-E_{k}^{\text{VB{\hbox{_2}}}}}$
(19)

The strong field amplitude in the window $3.15\tau<t<3.3\tau$ manifests in
higher Rabi frequency, and hence the population transfer from the VB2 to CB1
and vice-versa takes less time. These features are consistent if we see the
time dependence over a complete laser pulse duration [Fig. 3(b)].

  7. 7.

The transition matrix element between VB2 and CB1, along with the bandgap
between the two for particular $k$ values, also plays a very crucial role in
determining the Rabi frequency. In this particular case of 5:8 ratio, we see
that the transition matrix element in the momentum space
$\braket{\phi_{k}^{\text{CB${}_{1}$}}}{\hat{p}}{\phi_{k}^{\text{VB${}_{2}$}}}$
only peaks near minimum bandgap only, i.e. at $k=0$ [refer Fig. 1(b)].
However, for 1:3 ratio the contribution from the neighboring Bloch states also
plays a role and so the interband current and the band population is not
similar to as it is with 5:8 ratio.

Figure 4: Harmonic spectra associated with the interband, intraband, and
interference terms [refer Eq. (17) and (18)] are presented for the bichromatic
ratio 1:3 ($V_{0}=0.3$ a.u.) (a), 5:8 with $V_{0}=0.3$ a.u. (b) and
$V_{0}=0.6$ a.u. (c). On the right panel, the respective total HHG spectra are
plotted in (d)-(f). The phase difference $\Delta\varphi$ [refer Eq. (20)] is
also shown in the figure.

###  III.2 Interference of inter- and intraband currents

We learned from the Figs. 2 and 3, how the electric field amplitude and the
matrix element of the dipole operator affect the Rabi oscillations between the
VB2 and CB1, which eventually affect the respective band-population. In order
to further understand the nature of the HHG spectra and the role of
interference of inter- and intraband contribution, the HHG spectra for 1:3 and
5:8 cases are shown in Fig. 4. It can be understood from this figure that for
the case of 5:8 ratio, the inter- and intraband harmonics ‘almost’ overlap
with each other, and the interference between the two causes the emergence of
the total HHG spectra with multiple plateaus. As we saw previously in Fig. 3,
the population transfer in case of 5:8 occurs only near the peak of the
electric field (minimum of vector potential), and as a result, the process
repeats in each half cycle, and hence only the odd order harmonics are
prominently visible for inter- and intraband HHG in Fig. 4(b) and (c) for 5:8
case. However, for the 1:3 case, no such observation is made [refer Fig.
3(a)]. The amplitude of the interference term is of the order of addition of
inter- and intraband harmonics, hinting towards strong destructive
interference, and as a result, the clear odd order harmonics are not visible
in total HHG spectra in Fig. 4(d)-(f). Furthermore, the phase difference
between the inter- and intraband harmonics is defined as:

$\Delta\varphi=1-\frac{1}{\pi}\Big{|}\varphi_{\text{inter}}-\varphi_{\text{intra}}\Big{|}.$
(20)

The offset from perfect destructive interference ($\Delta\varphi=0$) between
the inter- and intraband harmonics is observed to accurately manifest in the
observed cutoff energies of full HHG spectra.

Figure 5: Harmonic spectra for 1:3 and 5:8 cases are illustrated in (a) and
(b) respectively for different values of the $V_{0}$ [refer Eq. (1)]. The
energy bands VB2, CB1 and CB2 for $V_{0}=0.3$ and 0.6 a.u. cases are
illustrated for 1:3 (c) and 5:8 ratios (d). The temporal evolution of the
population of CB2 i.e. $\mathpcal{P}_{CB_{2}}$, are compared for $V_{0}=0.3$
and 0.6 a.u. cases for 1:3 (e) and 5:8 (f) ratios.

###  III.3 Effect of potential depth $V_{0}$ on the HHG spectra

So far, we learned that the typical characteristics of the 5:8 ratio make it
interesting from the interaction perspective, as the transition to higher
bands happens either at the center of BZ or at the edge of the same. In order
to further elucidate on this aspect, in Fig. 5, we have presented the HHG
spectra for 1:3 and 5:8 cases for different potential depth $V_{0}$ [refer Eq.
(1)]. It can be seen from Fig. 5(a) and (b) that with the increase in the
potential depth, the efficiency of the secondary plateau $\sim 20-40$ eV
decreases for 1:3 ratio and correspondingly increases for 5:8 ratios. This can
be understood by studying the energy bands for 1:3 and 5:8 ratios for
different $V_{0}$ values. We have presented the VB2, CB1 and CB2 energy bands
for both 1:3 and 5:8 cases in Fig. 5(c) and (d) for $V_{0}=0.3$ and 0.6 a.u.
The photons in the energy range $\sim 20-40$ eV (which comprises the secondary
plateau) are mostly emitted by the transition from the CB2 to the lying band,
and hence the time-dependent band-population of CB2 can be a good measure to
co-relate to the harmonic efficiency of the radiation in the secondary
plateau. It can be seen that for the case of 1:3 ratio, the energy difference
between the CB1 and CB2 increases from $\sim 7.6$ eV to $\sim 12.6$ eV with an
increase in the potential depth from 0.3 a.u. to 0.6 a.u., and so the
instantaneous band population of CB2 reduces implying the less probability to
Zener tunneling to CB2 [refer Fig. 5(e)]. On the contrary, for a 5:8 ratio, as
we increase the $V_{0}$, the bands CB1 and CB2 flatten a bit at the edge of
the BZ. As a result, the Zener tunneling increases, giving enhanced population
for CB2. This enhanced population of CB2 manifests in enhanced efficiency in
HHG yield for the energy range $\sim 20-40$ eV.

Figure 6: The variation of the harmonic yield with the potential depth $V_{0}$
is presented for three different bichromatic ratios in (a). However, the
maximum of the band population $\mathpcal{P}_{CB_{2}}$ dependence on the
$V_{0}$ is shown in (b). In (a), plots above (below), the black dashed line
represents the harmonic yield in the energy range 3 - 10 eV (20 - 30 eV).

In Fig. 6, the variation of the harmonic yield and maximum of the CB2
population with $V_{0}$ is presented for different bi-chromatic ratios. The
harmonic yield in the energy range 20 - 30 eV is not plotted for 1:3, and 3:5
ratio for $V_{0}>0.5$ a.u. [Fig. 6(a)], as the secondary plateau completely
diminishes for $V_{0}>0.5$ a.u. It can be understood from this figure that the
harmonic yield in the energy range belonging to the secondary plateau
increases (decreases) with $V_{0}$ which is reflected by the increase
(decrease) in the population of the band, which contributes to the said energy
range in HHG spectra [Fig. 6(b)]. The harmonics in the energy range 3-10 eV
(primary plateau) arise mainly by the transition from the CB1 to VB2 in all
the cases. For the 5:8 case, as the $V_{0}$ increases, the bandgap between VB2
and CB1 increases, and the band slopes near $k=0$ reduce. This combined effect
causes enhanced Zener tunning to higher bands, enhancing harmonic yield in the
3-10 eV range and the secondary plateau by subsequent promotion to CB2.

Figure 7: Temporal dependence of $j_{\text{inter}}$ is presented three
different values of $V_{0}$ for 5:8 ratio case (a). The zoomed version of (a)
is illustrated in (b), and the variation of a maximum of $j_{\text{inter}}$
with $V_{0}$ is shown in (c). It should be noted that for visual appeal, the
temporal profile in (a) for $V_{0}=0.5$ a.u. and 0.6 a.u. are intentionally
shifted by 10 and 20 cycles, respectively.

We learned that for the case of 5:8 ratio, the Zener tunneling to higher bands
only happens either at the center or at the edge of BZ, which is also quite
visible from the temporal evolution of the interband current
$j_{\text{inter}}(t)$ as discussed in detailed in Fig. 3. This fact brings us
to a question, how does the strength of $j_{\text{inter}}$ affect the HHG
yield? In order to explore this facet, in Fig. 7 we have analyzed
$j_{\text{inter}}$ for different potential depths $V_{0}$. It can be observed
from Fig. 7(a) and (b) that as we increase $V_{0}$, the peak value of the
interband current increases; this is due to the enhanced Zener tunneling among
bands [lower ‘velocity’ $\sim\nabla_{k}\mathpcal{E}(k)$ near the edge of the
BZ] with increase in $V_{0}$ for 5:8 ratio. Furthermore, as $V_{0}$ increases,
the band slope decreases, and so the neighboring $k$ values also contribute to
the transition or in other words, the $p_{k}^{23}(k)$ broadens [refer Fig.
1(b)]. This results in the disappearance of Rabi-type fast oscillations, as we
have seen for $V_{0}=0.3$ a.u. case. Typically, we observe these Rabi type of
oscillations when mostly only two energy levels are involved near the $k=0$.
However, as more and more $k$ values are participating in the transition, this
assembly of ‘two-level’ system at $k=0$ breaks down and rapid oscillations are
replaced by the strong interband current. The strong interband current
eventually populates higher conduction bands, enhancing the harmonic yield in
the secondary plateau region. However, it should be noted that even in this
scenario, the interband current only contributes near the peak of the electric
field or the temporal points where $A(t)\sim 0$. The maximum interband current
variation with the potential depth is also presented in Fig. 7(c); it is
observed that the maximum interband current is positively correlated with the
harmonic yield of the secondary plateau [Fig. 6(a)]. In order to further
understand the HHG spectra and the role of interband current, next we discuss
the time-frequency analysis for two cases of 5:8 with $V_{0}=0.3$ and 0.6 a.u.

Figure 8: The time frequency analysis for 5:8 case with $V_{0}=0.3$ a.u. (a)
and 0.6 a.u. (b) are shown. In both the figures, the horizontal dashed gray
lines show even harmonics. The temporal profile of the $j_{\text{inter}}$ is
also superimposed on these plots.

###  III.4 Time frequency analysis of HHG

As we discussed in Figs. 3 and 4 that the peculiar feature the bi-chromatic
ratio 5:8 is that the transition to higher bands only happens at the center or
at the edge of the BZ, which implies that if you consider the bands VB2 and
CB1 (corresponds to the primary plateau $\lesssim 15$ eV) the interband
transition will only happen near $k\sim 0$ as $p_{k}^{23}$ is maximum only for
$k=0$ [refer Fig. 1(b)]. This condition will be met only when $A(t)\sim 0$ and
the transition is facilitated, giving rise to $j_{\text{inter}}$. This process
will happen at each half cycle, giving rise to the only odd-order harmonics as
seen in Fig. 4(b) and (c), however due to the interference between the inter-
and intraband harmonics, clear odd-order harmonics are not that prominent
[refer Fig. 4(e) and (f)]. In Fig. 8 we have presented the time-frequency
analysis of the HHG spectra for a 5:8 ratio for two different values of
$V_{0}$. It is observed that even in the total HHG spectra, the intensity of
the even harmonics is significantly reduced, as shown by the dashed horizontal
lines in Fig. 8(a) and (b). We have also superimposed the temporal profile of
the interband current and it can be seen that the intensity of the harmonics
between two consecutive peaks of $j_{\text{inter}}$ is smaller than the one
observed at peaks of $j_{\text{inter}}$. This further establishes the fact
that the interband current plays a prominent role in the HHG process from the
(quasi) periodic crystals and reduction of the same is reflected in weaker
harmonic efficiency. These missing even harmonics are not observed for 0:1,
1:3, and 3:5 cases, as the interband current in those scenarios is the
superposition of all different channels corresponding to different $k$ values.

##  IV Summary

In summary, we have studied the role of inter- and intraband current on the
HHG by the laser interaction with the bi-chromatic quasi-periodic crystals
with the frequency ratios $\sigma_{1}:\sigma_{2}$ being 1:3, 3:5, and 5:8 [Eq.
(1)]. It is observed that the typical characteristics of the energy bands
associated with the 5:8 ratio make it possible to have the transition from the
VB2 to CB1 only at the center or at the edge of the BZ, which leads to very
interesting population transfer mechanism between VB2 and CB1. The effect of
the potential depth on the generated harmonics and the interband current is
also studied, and it is observed that the harmonic yield of the secondary and
primary plateau increases with increasing the potential depth for a 5:8 ratio,
which is positively correlated to the strength of the interband current. The
conclusions are vindicated by also studying the time-dependent instantaneous
band populations. The presented analysis is also observed to be true even when
the decoherence effects are included in a phenomenological manner [38, 40].
The laser field parameters and their respective influence on the HHG by these
quasi-periodic crystals are beyond the scope of the current manuscript and are
reserved for future studies. The temporal control of the band-population for
such optical lattices might interest the ‘quantum-computing’ fraternity.

## Acknowledgments

The authors acknowledge the Science and Engineering Research Board (SERB),
India, for funding Project No. CRG/2020/001020.

## References

  * Ghimire  _et al._ [2011] S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Observation of high-order harmonic generation in a bulk crystal, Nature Physics 7, 138 (2011).
  * Ghimire  _et al._ [2012] S. Ghimire, A. D. DiChiara, E. Sistrunk, G. Ndabashimiye, U. B. Szafruga, A. Mohammad, P. Agostini, L. F. DiMauro, and D. A. Reis, Generation and propagation of high-order harmonics in crystals, Phys. Rev. A 85, 043836 (2012).
  * Ghimire and Reis [2019] S. Ghimire and D. A. Reis, High-harmonic generation from solids, Nat. Phys. 15, 10 (2019).
  * Wu  _et al._ [2015] M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, High-harmonic generation from bloch electrons in solids, Phys. Rev. A 91, 043839 (2015).
  * You  _et al._ [2016] Y. S. You, D. A. Reis, and S. Ghimire, Anisotropic high-harmonic generation in bulk crystals, Nat. Phys. 13, 345 (2016).
  * Ndabashimiye  _et al._ [2016] G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, Solid-state harmonics beyond the atomic limit, Nature 534, 520 (2016).
  * Luu  _et al._ [2015] T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, Extreme ultraviolet high-harmonic spectroscopy of solids, Nature 521, 498 (2015).
  * Kruchinin  _et al._ [2018] S. Y. Kruchinin, F. Krausz, and V. S. Yakovlev, Colloquium: Strong-field phenomena in periodic systems, Rev. Mod. Phys. 90, 021002 (2018).
  * Vampa and Brabec [2017] G. Vampa and T. Brabec, Merge of high harmonic generation from gases and solids and its implications for attosecond science, J. Phys. B: At. Mol. Opt. Phys. 50, 083001 (2017).
  * Floss  _et al._ [2018] I. Floss, C. Lemell, G. Wachter, V. Smejkal, S. A. Sato, X.-M. Tong, K. Yabana, and J. Burgdörfer, Ab initio multiscale simulation of high-order harmonic generation in solids, Phys. Rev. A 97, 011401 (2018).
  * Yue and Gaarde [2022] L. Yue and M. B. Gaarde, Introduction to theory of high-harmonic generation in solids: tutorial, J. Opt. Soc. Am. B 39, 535 (2022).
  * Liu  _et al._ [2018] H. Liu, C. Guo, G. Vampa, J. L. Zhang, T. Sarmiento, M. Xiao, P. H. Bucksbaum, J. Vučković, S. Fan, and D. A. Reis, Enhanced high-harmonic generation from an all-dielectric metasurface, Nature Physics 14, 1006 (2018).
  * Yoshikawa  _et al._ [2017] N. Yoshikawa, T. Tamaya, and K. Tanaka, High-harmonic generation in graphene enhanced by elliptically polarized light excitation, Science 356, 736 (2017).
  * Yoshikawa  _et al._ [2019] N. Yoshikawa, K. Nagai, K. Uchida, Y. Takaguchi, S. Sasaki, Y. Miyata, and K. Tanaka, Interband resonant high-harmonic generation by valley polarized electron–hole pairs, Nature Communications 10, 3709 (2019).
  * Bai  _et al._ [2021] Y. Bai, F. Fei, S. Wang, N. Li, X. Li, F. Song, R. Li, Z. Xu, and P. Liu, High-harmonic generation from topological surface states, Nature Physics 17, 311 (2021).
  * Song  _et al._ [2020] X. Song, S. Yang, R. Zuo, T. Meier, and W. Yang, Enhanced high-order harmonic generation in semiconductors by excitation with multicolor pulses, Phys. Rev. A 101, 033410 (2020).
  * Navarrete and Thumm [2020] F. Navarrete and U. Thumm, Two-color-driven enhanced high-order harmonic generation in solids, Phys. Rev. A 102, 063123 (2020).
  * Huang  _et al._ [2017] T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, High-order-harmonic generation of a doped semiconductor, Phys. Rev. A 96, 043425 (2017).
  * Vampa  _et al._ [2013] G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, Theoretical analysis of high-harmonic generation in solids, Phys. Rev. Lett. 113, 073901 (2013).
  * Vampa  _et al._ [2015a] G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, and P. B. Corkum, Linking high harmonics from gases and solids, Nature 522, 462 (2015a).
  * Bian  _et al._ [2018] X.-B. Bian, T.-Y. Du, G.-R. Jia, and X.-H. Huang, Intra- and interband transitions in high-order harmonic generation from solids, in  _Progress in Ultrafast Intense Laser Science XIV_, edited by K. Yamanouchi, P. Martin, M. Sentis, L. Ruxin, and D. Normand (Springer International Publishing, Cham, 2018) pp. 239–256.
  * Földi [2017] P. Földi, Gauge invariance and interpretation of interband and intraband processes in high-order harmonic generation from bulk solids, Phys. Rev. B 96, 035112 (2017).
  * Tancogne-Dejean  _et al._ [2017a] N. Tancogne-Dejean, O. D. Mücke, F. X. Kärtner, and A. Rubio, Ellipticity dependence of high-harmonic generation in solids originating from coupled intraband and interband dynamics, Nat. Commun. 8, 745 (2017a).
  * Wang  _et al._ [2018] X.-Q. Wang, Y. Xu, X.-H. Huang, and X.-B. Bian, Interference between inter- and intraband currents in high-order harmonic generation in solids, Phys. Rev. A 98, 023427 (2018).
  * Ernotte  _et al._ [2018] G. Ernotte, T. J. Hammond, and M. Taucer, A gauge-invariant formulation of interband and intraband currents in solids, Phys. Rev. B 98, 235202 (2018).
  * Du  _et al._ [2018] T.-Y. Du, X.-H. Huang, and X.-B. Bian, High-order-harmonic generation from solids: The contributions of the bloch wave packets moving at the group and phase velocities, Phys. Rev. A 97, 013403 (2018).
  * Vampa  _et al._ [2015b] G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, Semiclassical analysis of high harmonic generation in bulk crystals, Phys. Rev. B 91, 064302 (2015b).
  * Wismer  _et al._ [2016] M. S. Wismer, S. Y. Kruchinin, M. Ciappina, M. I. Stockman, and V. S. Yakovlev, Strong-field resonant dynamics in semiconductors, Phys. Rev. Lett. 116, 197401 (2016).
  * Kuehn  _et al._ [2010] W. Kuehn, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, and R. Hey, Terahertz-induced interband tunneling of electrons in gaas, Phys. Rev. B 82, 075204 (2010).
  * Mücke  _et al._ [2001] O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, Signatures of carrier-wave rabi flopping in gaas, Phys. Rev. Lett. 87, 057401 (2001).
  * Slater [1952] J. C. Slater, A soluble problem in energy bands, Phys. Rev. 87, 807 (1952).
  * Tancogne-Dejean  _et al._ [2017b] N. Tancogne-Dejean, O. D. Mücke, F. X. Kärtner, and A. Rubio, Impact of the electronic band structure in high-harmonic generation spectra of solids, Phys. Rev. Lett. 118, 087403 (2017b).
  * Lanin  _et al._ [2017] A. A. Lanin, E. A. Stepanov, A. B. Fedotov, and A. M. Zheltikov, Mapping the electron band structure by intraband high-harmonic generation in solids, Optica 4, 516 (2017).
  * Singh  _et al._ [2015] K. Singh, K. Saha, S. A. Parameswaran, and D. M. Weld, Fibonacci optical lattices for tunable quantum quasicrystals, Phys. Rev. A 92, 063426 (2015).
  * Qiu  _et al._ [2020] X. Qiu, J. Zou, X. Qi, and X. Li, Precise programmable quantum simulations with optical lattices, npj Quantum Information 6, 87 (2020).
  * M. S.  _et al._ [2019] M. M. S., A. Pattanayak, M. Ivanov, and G. Dixit, Direct numerical observation of real-space recollision in high-order harmonic generation from solids, Phys. Rev. A 100, 043420 (2019).
  * Korbman  _et al._ [2013] M. Korbman, S. Y. Kruchinin, and V. S. Yakovlev, Quantum beats in the polarization response of a dielectric to intense few-cycle laser pulses, New J. Phys. 15, 013006 (2013), 1210.2238 .
  * R  _et al._ [2023] N. R, A. R. Holkundkar, and J. N. Bandyopadhyay, Temporal control of high-order harmonic cutoffs in periodic crystals, Phys. Rev. A 108, 063503 (2023).
  * Han and Madsen [2010] Y.-C. Han and L. B. Madsen, Comparison between length and velocity gauges in quantum simulations of high-order harmonic generation, Phys. Rev. A 81, 063430 (2010).
  * Wang and Du [2021] G. Wang and T.-Y. Du, Quantum decoherence in high-order harmonic generation from solids, Phys. Rev. A 103, 063109 (2021).