capbtabboxtable[][]

# Differentiable Histogram with Hard-Binning

Ibrahim Yusuf
InstaDeep
Lagos, Nigeria
<EMAIL_ADDRESS>
George Igwegbe
InstaDeep
Lagos, Nigeria
<EMAIL_ADDRESS>
Oluwafemi Azeez
InstaDeep
Lagos, Nigeria
<EMAIL_ADDRESS>

###### Abstract

The simplicity and expressiveness of a histogram render it a useful feature in
different contexts including deep learning. Although the process of computing
a histogram is non-differentiable, researchers have proposed differentiable
approximations, which have some limitations. A differentiable histogram that
directly approximates the hard-binning operation in conventional histograms is
proposed. It combines the strength of existing differentiable histograms and
overcomes their individual challenges. In comparison to a histogram computed
using Numpy, the proposed histogram has an absolute approximation error of
0.000158.

##  1 Introduction

A histogram is a statistical feature that captures the distribution of sample
values. It finds useful applications in a variety of domains including deep
learning. Several recent works have incorporated histograms as a learnable
feature in a deep neural network for tasks such as texture analysis,
steganalysis, object detection, semantic segmentation, etc. Because a
conventional histogram involves binning, which can be viewed as an indicator
function whereby a sample returns one (1) if it falls within a bin and zero
(0) otherwise. The works on learnable histograms have tried to approximate
this hard-binning process with a soft-binning version. In the soft-binning
version, a sample distributes its maximum vote of 1 across several bins,
although it is expected to have the maximum vote in the bin it belongs to.
However, the result of soft-binning operation falls short of what is expected
from the computation of a histogram. A particular work that closely
approximates a conventional histogram used Kernel Density Estimation with an
appropriate kernel, but unlike others it cannot be implemented with existing
CNN layers and it has a tuning parameter. In this work, a differentiable
histogram that directly approximates the hard-binning used in conventional
histogram computation is proposed. The proposed histogram has a simple
functional form it can be implemented using CNN layers.

##  2 Related Works

Wang Z. et. al. [4] incorporated histogram features into a deep convolutional
neural network for object detection and semantic segmentation, resulting in
networks namely HistNet-OD and HistNet-SS respectively. The networks add a
learnable histogram layer on top of conventional CNNs and they report superior
performance. The learnable histogram layer was modeled using a linear basis
function. J. Peeples et al. [3] also proposed a learnable histogram layer for
texture analysis. The learnable histogram layer is used to extract spatial
distribution of feature values for characterizing and distinguishing textures.
Avi-Aharon, M. et al. [1] proposed a differentiable histogram for transferring
the color of a source image to a generated image in the context of image-to-
image translation with GANs. The differentiable histogram was modeled using
Kernel Density Estimation (KDE) with the derivative of the logistic regression
function as the kernel. The functional form of the histogram contains a
parameter B of KDE which needs to be properly tuned for a good approximation.
It also cannot be implemented using a single convolutional layer. In this
work, we propose a differentiable histogram that approximates the hard-binning
operation of a conventional histogram. It combines the strengths of existing
differentiable histograms and overcomes the limitations of soft-binning and
parameter tuning.

##  3 Proposed Differentiable Histogram

The proposed histogram layer overcomes the limitations of existing
differentiable histograms by approximating the actual hard-binning operation
found in conventional histograms using equation 1.

$\Phi(1.01^{\omega_{k}-|x_{i}-\mu_{k}|},1,0)\approx\Big{\\{}^{1\ if\ x_{i,j}\
\in\ (\mu_{k}-\omega_{k},\mu_{k}+\omega_{k})}_{0\ otherwise}$ (1)
$\Phi(x,1,0)\approx\Big{\\{}^{x\ if\ x\ >\ 1}_{1\ otherwise}$ (2)

According to equation 1, $\forall x\ \in\
(\mu_{k}-\omega_{k},\mu_{k}+\omega_{k})$ the value of
$\omega_{k}-|x-\mu_{k}|\in(0,\omega_{k})$, otherwise it is $<0$. Consequently,
the value of $1.01^{\omega_{k}-|x-\mu_{k}|}\ \in\ (1,1.01^{\omega_{k}})\
\forall x\ \in\ ((\mu_{k}-\omega_{k},\mu_{k}+\omega_{k})$ and $<1$ otherwise.
At this point, any sample outside the bin interval votes with a value strictly
less than 1. To discard these unwanted votes, we simply apply a “relu at 1”
using the threshold function of equation 2. Fig. 1 shows a comparison of the
histogram computed using differentiable histograms and Numpy histogram
function [2]. Similar to [3] and [4], equation 1 can be easily implemented
using existing CNN layers as illustrated in Fig 2.

Figure 1: Histogram of a standard normal distribution with 20 equally spaced
bins between -1 and 1  Figure 2: Modelling of HistLayer using convolutional
layers

##  4 Experiment

To quantify the approximation error of equation 1, we compute the absolute
error between histogram extracted using equation 1 and that extracted in a
conventional manner, using a function in Numpy [2]. Table 4 shows the error
for different differentiable histograms in Fig. 1.

Figure 3: Histogram of DC-GAN generator output before and after learning. Differentiable Histogram | Absolute Error
---|---
LBF | 36.55150
RBF | 0.407948
KDE | 0.001676
HistLayer (Ours) | 0.000158
Figure 4: A table of Absolute Errors of various differentiable histogram
algorithms

##  5 Conclusion

We propose a differentiable histogram layer that directly approximates the
hard-binning operation in conventional histogram and show that it can be used
to learn a generator that maps an input noise vector to any target
distribution expressible by a histogram.

## References

  * [1] Mor Avi-Aharon, Assaf Arbelle and Tammy Riklin Raviv  “DeepHist: Differentiable Joint and Color Histogram Layers for Image-to-Image Translation”  In _arXiv preprint arXiv:2005.03995_ , 2020
  * [2] Charles R. Harris et al.  “Array programming with NumPy”  In _Nature_ 585.7825 Springer ScienceBusiness Media LLC, 2020, pp. 357–362  DOI: 10.1038/s41586-020-2649-2
  * [3] Joshua Peeples, Weihuang Xu and Alina Zare  “Histogram Layers for Texture Analysis”  In _arXiv preprint arXiv:2001.00215_ , 2020
  * [4] Zhe Wang, Hongsheng Li, Wanli Ouyang and Xiaogang Wang  “Learnable histogram: Statistical context features for deep neural networks”  In _European Conference on Computer Vision_ , 2016, pp. 246–262  Springer


# Multiple plateaus of high-sideband generation from Floquet matters

Yu-Xuan Chen  1 Gan Wang  1 Mingjie Li  3,4,5 Tao-Yuan Du1,2  1School of
Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
2Key Laboratory of Computational Physical Sciences (MOE), Fudan University,
Shanghai 200433, China
3Department of Applied Physics, The Hong Kong Polytechnic University, Hung
Hom, Kowloon, Hong Kong, China 4 Shenzhen Research Institute, The Hong Kong
Polytechnic University, Shenzhen, Guangdong, 518057, China 5 ming-
<EMAIL_ADDRESS>*Corresponding author<EMAIL_ADDRESS>
###### Abstract

We theoretically report that high-order sideband generation (HSG) from Floquet
matters driven by a strong terahertz light while engineered by weak infrared
light can achieve multiple plateau HSG. The Floquet-engineering systems
exhibit distinctive spectroscopic characteristics that go beyond the HSG
processes in field-free band-structure systems. The spatial-temporal dynamics
analyses under Floquet-Bloch and time-reversal-symmetry theories clarify the
spectral and its odd-even characteristics in the HSG spectrum. Our work
demonstrates the HSG of Floquet matters via Floquet engineering and indicates
a promising way to extract Floquet material parameters in future experiments.

††journal: oe

.

##  1 Introduction

The interaction between light field and matter can regulate the dynamic
process of Floquet materials [1, 2, 3], with particular emphasis on modifying
and controlling quantum states through periodic light fields, known as Floquet
engineering [4, 6, 7, 8, 5]. This method shows promise in tailoring the
physical properties of Floquet materials due to the existence of light-dressed
band structures, facilitating exploration into non-trivial phases in
topological insulators. Many recent experiments have also demonstrated the
practical feasibility of such a Floquet engineering [9, 10, 11, 12, 13]. The
promising nature of Floquet engineering inspires us to explore the underlying
superiority of the aspect of the generation of strong-field coherent light
sources.

Under the driving of the periodic light field, the process of high-harmonic
generation (HHG) constitutes a particular Floquet process, which is observable
across both perturbative and non-perturbative generation regimes. HHG could
provide a unique table-top source of coherent emission in extreme UV and X-ray
spectral regions and serves for producing attosecond pulses for ultrafast
spectroscopy [14, 15, 16, 17, 18]. The mechanism of HHG can be specified by
the three-step model [14, 15, 19]. High-harmonic spectroscopy also paves an
avenue to probe the ultrafast quasiparticle dynamics in condensed matters and
to reconstruct Bloch-band structure, Berry curvature, electron correlation and
shift vector of systems [20, 21, 22, 23, 24]. High-order sideband generation
(HSG), which is analog to HHG, has also attracted much attention over the past
decade [25, 26, 27, 28]. For HSG, the near-resonant excitation by a weak field
is utilized to generate electron-hole pairs around the band edge, then the
electron-hole pairs are driven by a strong long-wavelength light field, and
when the driving field direction is reversed the electrons will recombine with
holes to lead the radiations of high-order sideband photons [25, 26].

HSG can provide a way for exploring the topological properties of non-trivial
states in condensed matters [30, 29, 31, 32]. In addition, compared with HHG,
the electronic structure of Floquet matters can be delicately dressed by the
laser parameters of the periodic infrared field. The HSG in experiments
reflected dynamics in an open quantum system, in which quantum decoherence
resulted in the extreme attenuation of its high-sideband signals [26] and
further limited the HSG spectrum to appear as a single plateau. The aspect
that how to extend the HSG spectral plateau is significant for probing the
highly non-trivial quantum state. The electron-hole pairs are elementary
excitations caused by infrared lasers. Previously, the process of electron-
hole pair generation is depicted by the perturbative nonlinear optical
framework [25, 26]. In HSG, the non-perturbative effect in light-dressed
states of matters has not got enough attentions and has even been neglected.
In this work, we revisit the significance of light-dressed states and find
that these light-dressed states give rise to the multiple plateaus of the HSG
spectrum. The plateau extension of HSG spectrum is also useful in many
electro-optical applications such as wide-band optical multiplexers, optical
pulses with ultra-high repetition rate, and optical communication with THz
bandwidth.

This paper is organized as follows. In Sec. 2, we present the Floquet-
engineering systems, Lindblad master equation and Semiconductor bloch
equation. In Sec. 3, we systematically investigate the HSG spectra in Floquet
matters. We summarize our paper in Sec. 4. Atomic units are used throughout
this paper, unless specified otherwise.

##  2 Theoretical method

###  2.1 Floquet-engineering systems

We commence by directing our attention to the Su-Schrieffer-Heeger (SSH)
system, notable for the emergence of topological edge states [34, 33, 35, 36].
Specifically, we examine linear chains comprising 500 lattice sites subject to
open-boundary conditions. The application of such a system in HHG has been
well studied [37, 38]. In Fig. 1(a) we give a schematic of the system. The
system’s Hamiltonian reads

$\hat{H}_{0}=\sum_{j}^{N}\left(h_{1}\hat{c}_{j,A}^{\dagger}\hat{c}_{j,B}+\text{H.c.}\right)+\sum_{j}^{N-1}\left(h_{2}\hat{c}_{j,B}^{\dagger}\hat{c}_{j+1,A}+\text{H.c}.\right),$
(1)

where $\hat{c}_{j,A/B}^{\dagger}(\hat{c}_{j,A/B})$ is the generation
(annihilation) operator in two sublattices A/B after the quadratic
quantization. $h_{1}$ = -0.01827 a.u. and $h_{2}$ = -0.01003 a.u. respectively
represent intra- and inter-cellular hopping energies, and characterize two
staggered hopping terms of the trivial phase. These parameters are applicable
as models for quasi-one-dimensional systems, including conjugated polymers,
organic crystals, carbon nanotubes, ferromagnetic perovskites, carbon chains,
transition metal complexes, and organic charge transfer salts [39, 40]. By
diagonalizing the Hamiltonian in position representation, all eigenstates can
be obtained, in which the in-gap edge state is absent in the trivial phase
involving in our simulations.

Since the SSH system has spatial translation periodicity, the Fourier
transform of its coordinate-space Hamilton can be taken, i.e,
$\hat{c}_{j,A}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{k_{0}}\hat{c}^{\dagger}_{k_{0},A}e^{iak_{0}j}$
and
$\hat{c}_{j,B}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{k_{0}}\hat{c}^{\dagger}_{k_{0},B}e^{iak_{0}j}$.
$a$ is the primitive cell constant [33, 37, 34]. Thus we obtain the momentum-
space Hamiltonian
$\hat{H}(k_{0})=[h_{1}+h_{2}\cos(ak_{0})]\sigma_{x}+[h_{2}\sin(ak_{0})]\sigma_{y}$,
in which $\sigma_{x,y}$ is the Pauli matrix. The sum of all $k_{0}$ will give
the bulk Hamiltonian of the system. Given that the SSH system of bulk states
is a two-band system, we straightforwardly designate the upper conduction band
as the "c-band" and the lower valence band as the "v-band". Notably, the
minimal bandgap for the bulk states is 0.448 eV, and the system is adopted to
better construct the light-dressed Floquet-Bloch (FB) state [41].

The time-dependent expression of $\hat{H}(k)$ can be derived by substituting
$k$ with $k(t)=k_{0}+A_{\text{ex}}(t)$. Note that the discussions of Floquet
theory only involve the excitation field $A_{\text{ex}}(t)$. In this context,
$k_{0}$ denotes the initial crystal momentum. Under the driving terahertz
(THz) field, the time-dependent evolution of systems can be numerically
achieved by solving the quantum master equation, which will be specified in
the following subsection. It’s worth noting that the external-field vector
potential $A(t)$ here include both of the excitation $A_{\text{ex}}(t)$ and
THz driving $A_{\text{dr}}(t)$ fields. For the real-space Hamiltonian, its
time-dependent form is satisfied as ${\hat{H}(t)}$ =
${\text{e}^{-iA(t)\hat{x}}}{\hat{H}_{0}}{\text{e}^{iA(t)\hat{x}}}$ under the
velocity gauge, in which $\hat{x}$ is the coordinate operator.

To reveal the role of non-perturbative effect in this light-dressed system, we
turn to its Floquet Hamiltonian [42]. According to Floquet’s theory, a time-
periodic system satisfies $\psi(t)=\varphi(t)e^{-i{\varepsilon}t}$, where
$\varphi(t)$ represents the periodic component. Simultaneously, the time-
periodic Hamiltonian ${\hat{H}(t)}$ can be expanded as the Fourier series of a
fundamental frequency $\Omega$, where $\Omega$ is the light frequency of the
coupling light field, thus constructing a Hilbert space and determining a
series of FB states in a certain frequency range [43, 44, 45] (see Appendix A
for more details). The following Floquet Hamiltonian can be constructed as

$\hat{\mathcal{H}}_{\mathcal{F}}=\left(\begin{array}[]{cccc}\ddots&\hat{H}_{-1}&\hat{H}_{-2}&\\\
\hat{H}_{1}&\hat{H}_{0}-m\hbar\Omega&\hat{H}_{(-1)}&\hat{H}_{-2}\\\
\hat{H}_{2}&\hat{H}_{1}&\hat{H}_{0}-(m+1)\hbar\Omega&\hat{H}_{-1}\\\
&\hat{H}_{2}&\hat{H}_{1}&\ddots\end{array}\right),$ (2)

where $\hat{\mathcal{H}}_{\mathcal{F}}$ satisfies
$\hat{\mathcal{H}}_{\mathcal{F}}\varphi_{n}$ = $\varepsilon_{n}\varphi_{n}$
and $\varphi_{n}$ =
$\begin{pmatrix}...&\phi^{n}_{m-1}&\phi^{n}_{m}&\phi^{n}_{m+1}&...\end{pmatrix}^{T}$.
$\hat{H}_{l}$ is the Fourier expansion coefficient of $\hat{H}(t)$ =
$\sum_{l}e^{il{\Omega}t}\hat{H}_{l}$, and $\phi_{m}$ is the $m$-th Fourier
coefficient of $\varphi(t)$ [46, 47, 48, 49, 50, 51, 52].
$\hat{\mathcal{H}}_{\mathcal{F}}$ represents an extended Hamiltonian, we
truncated it and traversed the whole momentum space to determine the FB states
we need. As presented in Fig. 1(b), we define the spectral function of the FB
band structure as $S(k,E)$ = $\sum_{n}\delta(E-\varepsilon_{n})\left|\langle
k|\varphi_{n}\rangle\right|^{2}$ [24], in which $\left|\langle
k|\varphi_{n}\rangle\right|^{2}$ =
$\sum_{j}\langle\varphi_{j,n}|e^{-ikja_{0}}\sum_{j^{{}^{\prime}}}e^{ikj^{\prime}a_{0}}|\varphi_{j^{\prime},n}\rangle$.
The system’s eigenfunction $\left|\varphi_{n}\right\rangle$ with eigenvalue
$\varepsilon_{n}$ is obtained from the diagonalization of Eq. (2), and
$\left|\varphi_{j,n}\right\rangle$ is the value of eigenfunction
$\left|\varphi_{n}\right\rangle$ at the lattice site $j$. $k$ is the
quasimomentum. The Dirac function is broadened by using a Lorentzian shape
$\frac{1}{\pi}\frac{\eta}{(E-\varepsilon_{n})^{2}+\eta^{2}}$ with a proper
broadening factor $\eta$ = 0.001$|h_{1}|$.

Figure 1: (a) Schematic diagram of SSH chain under external field. (b)
Floquet-Bloch bands (or spectral function) of the trivial bulk phase of SSH
systems. $(\\{\text{c},\text{v}\\},m)$ represent the field-dressed bands under
number $m$ photons. The vertically double-headed arrows in (b) distinguish the
electron-hole recombination between various Floquet-Bloch bands. A small panel
in (b) highlights the characteristic gap opening. (c) The extent of the
characteristic gap opening varies with the strength of the excitation field,
with the vertical dashed line indicating the gap opening achieved under the
adopted strength.

The Floquet engineering system undergoes precise manipulation through the
utilization of a near-resonant excitation field with frequency $\omega_{ex}$
($\lambda\simeq$ 2700 nm), where the excitation field is considered to be a
continuous wave. The strength of the excitation light pulses is fixed at 200
kV/cm, however its small vector potential makes it challenging to induce
motion of electrons and holes. In Fig. 1(b), the FB bands of the SSH systems
could be obtained from the diagonalization of the Hamiltonian’s matrix of Eq.
(2) under the frequency space.

The index $(\\{\text{c},\text{v}\\},m)$ is marked as the FB state with index
$m$. Under such excitation field strength, it is sufficient to produce a non-
perturbative modulation of the bulk states, and a characteristic gap opening
emerges in their respective modes of FB bands as shown in Fig. 1(b). Moreover,
in experiments this infrared laser field with such light strength was also
sufficient to produce the light-dressed FB states [9, 10, 11]. In Fig. 1(c),
the variation of this light-dressed energy gap between bulk state (v, 1) and
(c, 0) with the field strength of the excitation field is given. As the
excitation field strength increases, the degree of energy gap opening
gradually increases, convincingly exhibiting the characteristics of Floquet
materials. The excitation field not only creates a Floquet system but also
serves as a nearly resonant excitation field in the HSG process.
Simultaneously, the driving THz field with the frequency $\omega$ (= 1 THz)
shines the Floquet engineering systems. However, the magnitude of this gap in
Fig. 1(b) is insufficient to significantly influence the overall resonant
excitation process or the structure of the HSG plateaus.

Figure 2: (a) Red dashed and blue solid curves respectively show the HSG
spectra without and with decoherence, in which $T_{2}$ = 250 fs is considered.
(b) Enlargement of the magenta spectral region II in (a). Inset panels in (a)
and (b) give the odd-even order details of high-order sidebands. The zeros in
(a) and (b) represent the reference point or baseline for the plotted HSG
spectrum, in which the energies of the coupled photon number have been
subtracted separately. The sidebands in the spectral I and II regions
respectively present the even- and odd-order features. The symbols of ① and ②
respectively denote two HSG plateaus in each spectral region.

###  2.2 Lindblad master equation

In our simulations, the directions of the linearly polarized THz and infrared
lights are mutually parallel. In the time-dependent evolution process, we
consider $A(t)$ = $A_{\text{ex}}(t)$ \+ $A_{\text{dr}}(t)$. $A(t)$ denotes the
total vector potential including the excitation field $A_{\text{ex}}$ and
driving field $A_{\text{dr}}$, where the THz driving field irradiates the
Floquet systems tailored by infrared excitation light. The density matrix
evolution of the system can be conducted either in position or momentum space.
Initially, in position space, we have conducted the evolution under the
framework of the Lindblad master equation. The time-dependent evolution of the
density matrix satisfies $\frac{d\hat{\rho}(t)}{dt}$ =
-$i[\hat{H}(t),\hat{\rho}]$ \+ $D(\hat{\rho})$. All states in valence band
were adopted as the initial states in the trivial phase of SSH system. For the
various interband relaxation mechanisms, we incorporate the pure dephasing
term into the Von Neumann equation and indicate it as $\mathit{D}(\hat{\rho})$
= $-\frac{{(1-{\delta_{mn}})}}{{{T_{2}}}}{\hat{\rho}_{mn}}$. $T_{2}$ is set as
250 fs. The current operator is denoted as

$\hat{j}=-i\sum_{m}\left(r_{m}-r_{m+1}\right)\hat{H}_{m,m+1}\hat{c}_{m}^{\dagger}\hat{c}_{m+1}-\text{H.c.},$
(3)

where $r_{m}$ denotes the coordinate of the sublattice site. The light-induced
current is written as $\mathbf{J}(t)$ =
$\operatorname{Tr}\left(\hat{\rho}(t)\hat{j}\right)$. HSG spectrum can be
obtained from the Fourier transform of the current. The emitted spectrum is
characterized as the high-order sideband $2N\omega$ around the excitation
light frequency $\omega_{\text{ex}}$.

###  2.3 Semiconductor Bloch equation

For the initial states, their evolution can also follow the formulation of
semiconductor Bloch equations (SBEs) in momentum space [53, 55, 54]. Time-
dependent Hamiltonian in momentum space is written as

$\hat{H}[k(t)]=\begin{pmatrix}0&h_{1}+h_{2}e^{ia[k(t)]}\\\
h_{1}+h_{2}e^{-ia[k(t)]}&0\end{pmatrix}.$ (4)

After diagonalization of $\hat{H}[k(t)]$, we have the instantaneous
eigenstates and eigenvalues of the system. The time-dependent evolution of the
density matrix $\rho_{mn}^{{k}}$ can be obtained by the semiconductor Bloch
equation under the Huston basis [56, 57, 55].

$\begin{split}\partial_{t}\rho_{mn}^{{k}}=&-i\epsilon_{mn}^{{k_{0}}+{A}(t)}\rho_{mn}^{{k}}-\left(1-\delta_{mn}\right)\frac{\rho_{mn}^{{k}}}{T_{2}}\\\
&+i{F}(t)\cdot\left[\sum_{l}\left({d}_{ml}^{{k_{0}}+{A}(t)}\rho_{ln}^{{k}}-{d}_{ln}^{{k_{0}}+{A}(t)}\rho_{ml}^{{k}}\right)\right].\end{split}$
(5)

Here $\epsilon_{mn}$ is the energy gap between band indexes $m$ and $n$.
${d}_{mn}$ is the dipole matrix elements calculated as ${d}_{mn}=i\left\langle
u_{m}|\bigtriangledown_{k}|u_{n}\right\rangle$, where $u_{m}$ and $u_{n}$ are
the eigenstates after diagonalization of the Hamiltonian of Eq. (4). $F(t)$ =
$-\frac{dA(t)}{dt}$ is the electric field of laser. In addition, to improve
the efficiency and convergence of the calculation, Eq. (5) is rewritten as the
following form

$\frac{\partial}{\partial t}\begin{pmatrix}n_{\text{cv}}^{{k}}\\\
\rho_{\text{cv}}^{{k}}\\\
\rho_{\text{vc}}^{{k}}\end{pmatrix}=\begin{pmatrix}0&-2iF(t){d}_{\text{cv}}({k})&2iF(t){d}_{\text{cv}}({k})\\\
-iF(t){d}_{\text{cv}}({k})&i\varepsilon_{\text{cv}}+iF(t)[{d}_{\text{cc}}({k})-{d}_{\text{vv}}({k})]-\frac{1}{T_{2}}&0\\\
iF(t){d}_{\text{vc}}({k})&0&i\varepsilon_{\text{vc}}+iF(t)[{d}_{\text{vv}}({k})-{d}_{\text{cc}}({k})]-\frac{1}{T_{2}}\end{pmatrix}\begin{pmatrix}n_{\text{cv}}^{{k}}\\\
\rho_{\text{cv}}^{{k}}\\\ \rho_{\text{vc}}^{{k}}\end{pmatrix}.$ (6)

The evolution of Eq. (6) can be computed using the Crank-Nicolson method [58,
59, 60]. $n_{\text{cv}}$ is the population difference between two bands and we
set the $n_{\text{cv}}$ value at the initial moment to be $-1$. The resulting
current can be expressed by the following term

$\displaystyle\mathbf{J}(t)$
$\displaystyle=\sum_{n}\int_{BZ}\rho_{nn}^{{k}}\nabla_{{k}}\varepsilon_{n}({k})d^{3}k$
(7) $\displaystyle+\sum_{n\neq
m}\int_{BZ}i\varepsilon_{nm}({k})\rho_{mn}^{{k}}{d}_{nm}({k})d^{3}k.$

In the right of Eq. (7), the first term is the intraband current, and the
second term is the interband current. It is worth noting that SBEs requiring
the periodic boundary condition only works in the momentum space and can not
address the open-boundary systems with in-gap edge state.

##  3 Discussion

In Fig. 2(a), we present the HSG spectra in the trivial phase of the SSH
system, where the role of dephasing characterizes the multiple HSG plateaus
clearer. The THz driving field strength is 38.5 kV/cm with a duration of
sixteen optical cycles. For analytical convenience, we categorize the spectra
into three distinct regions: the region below the gap, region I, and region
II. While the observation of the two-plateau structure in spectral region I
aligns with previous finding [27], the emergence of spectral region II
represents a novel discovery. In this spectral region II, the underlying
mechanism pave an avenue to regulate HSG.

Considering the fact that the input optical field $\omega_{\text{ex}}$ creates
the electron-hole pair and then the THz field drives the electron to recombine
with the hole leading to the emissions of high-order sidebands, we shift the
sideband orders by $\omega_{\text{ex}}$ in Fig. 2(a). Due to the system’s
inversion symmetry the sideband orders of spectral region I satisfy the even
orders $2N\omega$, which is exhibited by the inset panel in Fig. 2(a). To
unravel the spectral characteristics in region II, we delicately shift their
sidebands by 2$\omega_{\text{ex}}$ and further display this spectral region in
Figs. 2(b). Contrary to the even-order sidebands in the spectral region I, the
sidebands of region II are odd orders in Fig. 2(b), in which this odd-order
feature is robust under various detuning of excitation light. Its mechanism
will be discussed subsequently. Then we turn to illustrate the cutoff
frequencies in two spectral regions. One sees conventional HSG spectra with a
two-plateau structure in both region I and region II, and further confirms
their two cutoff frequencies $\Omega_{1}$ = 3.17$U_{p}$ and $\Omega_{2}$ =
8$U_{p}$, where $U_{p}$ = ${A_{0}}^{2}/4m_{R}$ is the ponderomotive energy.
$m_{R}$ (= 0.35 a.u.) is the electron-hole reduced mass of the bulk bands, and
$A_{0}$ is the peak of the vector potential of the THz field. The formation of
a two-plateau structure in spectral regions I and II is attributed to the
quantum coherence among the recombination wave packets of electron-hole pair
[61, 27, 62]. Intriguingly, in Fig. 2(a) region I extending delicately to
region II holds a uniform spectral structure among them, which implies that
this light-dressed system becomes the Floquet matter.

Figure 3: (a) HSG spectrum in Floquet matters. Under the case with a shorter
dephasing time ($T_{2}$ = 150 fs), only the single plateau survives in each
spectral region. (b) Time-frequency analysis of the HSG spectra in (a), gray
curves represent the predicted emission timings from the semi-classical model.

Then, we will delve into the HSG of Floquet matters within the momentum space.
To clarify the fact that the two-plateau structure in Fig. 2 attributes to the
enough dephasing time, a shorter dephasing time ($T_{2}$ = 150 fs) has adopted
in Fig. 3. Here a stronger THz field with a strength of 51.5 kV/cm is involved
in Fig. 3(a). One could find that the two-plateau spectral structure in Fig.
2(a) becomes the single spectral plateau in Fig. 3(a). And in Fig. 3(a) the
odd-even characteristics of sidebands are robust. For the electron or hole in
the FB band, its group velocity $v^{g}_{n}(n=e,h)$ can be written as
$\upsilon_{n}^{g}(k)$ = $\frac{\partial\varepsilon_{n}(k)}{\partial k}|_{k}$.
The relative displacement between electron and hole is denoted as $\Delta x$ =
$\int_{t^{{}^{\prime}}}^{t}\\{v_{e}^{g}[k(\tau)]-v_{h}^{g}[k(\tau)]\\}d\tau$.
Zero displacement signifies the electron-hole recombination process occurring
between the ionization timing $t^{\prime}$ and the recollision timing $t$,
resulting in the emission of high-order sidebands with bandgap energy. This
allows one to derive the trajectories of electron-hole recombination in a
semi-classical prediction [63]. In Fig. 3(b) we also perform the time-
frequency analysis on the HSG spectrum of Fig. 3(a). The spectral regions I
and II characterize the same temporal trajectories of the electron-hole
recombination, which are also verified by the semi-classical prediction marked
by gray curves. Furthermore, in Fig. 1(b) we could distinguish the
contributions between FB bands. One could confirm that the spectral regions I
and II are respectively dominated by $\\{(\text{c},0)$ \- $(\text{v},0)\\}$
and $\\{(\text{c},1)$ \- $(\text{v},0)$ or $(\text{c},0)$ \-
$(\text{v},-1)\\}$ interband recombination of electron-hole pair. We can
generalize the above rule to $\\{(\text{c},m)$ \- $(\text{v},n)\\}$, when the
index ($m-n$) = 1, it equivalently contributes to the spectral region II.
Higher sideband plateaus are determined by electron-hole recollisions between
FB bands with larger index of ($m-n$).

To elucidate the odd-even sideband properties in the spectroscopic regions I
and II, we will discuss the dynamic symmetry between the FB states. One can
also refer to the dynamical symmetry theory for further insights [50]. For the
SSH systems in momentum space, its Hamiltonian satisfies the inversion
symmetry, i.e., $\hat{H}_{0}(-k)=\hat{H}_{0}^{\ast}(k)$. Keeping the external
field with temporal symmetry $A(t+T/2)$ = $-A(t)$ in mind, the time-dependent
Hamiltonian obeys $\hat{H}(-k,t+T/2)$ = $\hat{H}^{\ast}(k,t)$, where
$T=2\pi/\Omega$ is the optical period of laser fields. Therefore, for the
time-dependent Hamiltonian $\hat{H}(\pm k,t)$, its Fourier series in temporal
expansion are denoted as $\hat{H}(t)$ = $\sum_{l}e^{il{\Omega}t}\hat{H}_{l}$,
in which the coefficient satisfies $\hat{H}_{l}(-k)$ =
$(-1)^{l}\hat{H}_{l}^{\ast}(k)$ with integer $l$. Based on the mentioned
symmetry differences between $\hat{H}_{l}({\pm}k)$, one can derive a rule as
$\hat{H}_{mn}^{\mathrm{F}}(-k)=(-1)^{m-n}\hat{H}_{mn}^{\mathrm{F}\
{\ast}}(k)$, in which $\hat{H}_{mn}^{F}$ is the matrix element $\hat{H}_{m-n}$
in Eq. (2) and satisfies
$\hat{H}_{mn}^{F}(k)\phi^{\alpha}_{n}(k)=\varepsilon^{\alpha}(k)\phi^{\alpha}_{m}(k)$.
$m$ or $n$ represents the index of FB state. Under the appropriate selected
overall phase, the wave function will obey $\phi_{n}(-k)$ =
$(-1)^{n}\phi_{n}^{\ast}(k)$, where $n$ is the index of FB state [51]. Thus,
the symmetry characteristics encoded in the coupling FB wave functions can
determine the order feature of the sideband.

Figure 4: (a) HSG spectra varies with the driving-field strength. Cutoff
energy of three regions marked by the gray dashed lines is predicted by the
semi-classical model. (b) HSG spectra as a function of the excitation-field
frequency ($\Delta$ = $\omega_{\text{ex}}$ \- $\epsilon_{g}$). Three gray
dashed lines in (b) denote resonance peaks after coupled photons. Dephasing
time is the same as that in Fig. 2.

For the high-order sideband spectrum contributed by electron-hole
recombination between band indexes $\\{(\text{c},m)$ $-$ $(\text{v},n)\\}$,
its odd-even properties of sidebands are determined by the odd or even
characteristic of the difference of the two FB index ($m-n$). When the index
($m-n$) is odd, the symmetries involved in two coupling FB states are
opposite, leading to the final current satisfies:
$J(k,N)=(-1)^{(N+2)}J(-k,N)$, where N represents the frequency component of
current. Obviously, under the odd number $N$, the $\pm{k}$ channels cancel
each other out so that only the even-order frequency component survives
leading to odd-order sidebands [49, 50, 51, 52, 38]. Conversely, if the FB
band index ($m-n$) is even, the final current meets
$J(k,N)=(-1)^{(N+1)}J(-k,N)$, we can deduce the even-order sidebands (See
details in Appendix B).

Further, we investigate HSG in Floquet materials by varying the laser-field
parameters. In Figs. 4(a) and 4(b), we respectively unravel the roles of the
driving-light field strength and the excitation-light detuning ($\Delta$ =
$\omega_{\text{ex}}$ \- $\epsilon_{g}$) on HSG spectra. Here, $\epsilon_{g}$
is the minimum band gap between bulk bands. As depicted in Fig. 4(a), the
cutoff frequencies in spectral regions I, II, and III adhere to the scaling
rule of 3.17$U_{p}$ predicted by the semi-classical model. It is evident from
Fig. 4(a) that the FB modes dominate their respective HSG spectral regions. In
conventional HHG, the broadening of the frequency comb is typically limited by
the maximal gap of the energy band structure [64]. However, such limitations
can be surpassed in Floquet-engineering systems, enabling the attainment of
broader high-order sideband plateaus or continuous spectra. Additionally, in
Fig. 4(b), we observe the FB modes depicted as their respective slope lines,
suggesting that Floquet materials can be finely tailored by the excitation
light field, thereby providing a means to control HSG through Floquet
engineering.

Moreover, in Fig. 4(b) we observe that the high-order sideband plateau
strongly depends on the positive laser detuning $\Delta$. When we tune the
excitation light frequency to deviate from the situation of zero detuning, in
spectral regions I, the spectral plateau width and the cutoff energies are
shrunk from 3.17$U_{p}$ under a positive laser detuning. In spectral region
II, it is noteworthy that the energy gap between the ($\text{c},m$) and
($\text{v},m-1$) FB bands increases with the rising excitation-field
frequency, leading to an extension of the cutoff energy in region II to a
certain extent. Moreover, it is observed that the spectral width of region II
decreases with increasing detuning. The narrowing of the spectral plateaus can
be attributed to the fact that the electron created under the overexcitation
of light possesses an initial velocity, which subsequently hinders its
recombination with the hole. [28]. Obviously, in the cases with negative
detuning, HSG signal can be emitted when the excitation-field photon energy is
near half of the minimal band gap (corresponding to $\Delta\simeq-210$ meV),
which leads to a two-photon excitation process. The FB band provides a step
for the generation of electron-hole pairs between the original valence and
conduction bands of the field-free system. This two-photon excitation results
in its high-sideband emissions around the 2$\omega_{\text{ex}}$ energy zone,
as shown in the left side of Fig. 4(b).

##  4 Conclusion

In summary, periodic excitation light generates Floquet matters under
relatively weak intensity, and a unique HSG in Floquet matters has been
revealed. The multiple-plateau HSG exhibits unprecedented features beyond the
single-plateau HSG spectra. Furthermore, the electronic structure of Floquet
matters can be regulated by the periodic excitation light, offering a potent
method for controlling the HSG process, although it should be noted that this
Floquet system has a relatively low damage threshold. The Floquet-Bloch and
dynamical-symmetry theories can clarify this multiple-plateau HSG spectrum and
its odd-even characteristics. This work demonstrates how Floquet engineering
offers a new avenue for controlling HSG, providing a promising approach to
explore non-equilibrium quantum states within Floquet matter. Beyond HHG and
HSG studies, light-dressed Floquet matter is an issue of interest and an
extremely active and exciting field. Our work also shows a direct time-domain
insight into Floquet physics and explores the fundamental frontiers of
ultrafast band-structure engineering.

## APPENDIX A: Floquet theory

Consider that most spectroscopic methods satisfy a fundamental property, i.e.,
under the action of periodic external fields. For a periodically driven field
with frequency $\Omega={2\pi/T}$, where $T$ is an optical period. The wave
function satisfies periodicity, and the solution can be written in the
following form [43, 44, 45]

$\psi(t)=\varphi(t)e^{-i{\varepsilon}t},$ (A1)

where $\varphi(t)=\varphi(t+T)$. This is the expression of Floquet theory
which is similar to Bloch theorem. For a solution that satisfies the time
periodicity, we use the discrete Fourier transformation to determine the
frequency distribution, that is $\varphi(t)=\sum_{n}e^{in{\Omega}t}\phi_{n}$.
Then one substitutes Eq. (4) to the Schrödinger equation and to get the
following solution

$\left[\hat{H}(t)-i\hbar\frac{\partial}{\partial
t}\right]\varphi(t)=\varepsilon\varphi(t).$ (A2)

$\hat{H}(t)$ is the time-dependent Hamiltonian and can be expanded as the
discrete Fourier series, i.e.,

$\hat{H}(t)=\sum_{l}e^{il{\Omega}t}\hat{H}_{l}.$ (A3)

We substitutes $\varphi(t)=\sum_{n}e^{in{\Omega}t}\phi_{n}$ to Eq. (A2), and
set $\hat{\mathcal{H}}_{\mathcal{F}}=\hat{H}(t)-n\hbar\Omega$, which satisfies
$\hat{\mathcal{H}}_{\mathcal{F}}\phi_{n}=\varepsilon\phi_{n}$. Thus construct
an extended Hilbert space
$\left|\alpha,n\right\rangle$=$\left|\alpha\right\rangle\otimes\left|n\right\rangle$=$\left|\alpha\right\rangle{e^{in{\Omega}t}}$.
$\left|{\alpha}\right\rangle$ is the internal degree of freedom. In the
momentum space of the two-band model, this degree of freedom is two.
$\left|{n}\right\rangle$ is the degree of freedom that couples with the
external-field frequency. The matrix element of Floquet Hamiltonian
$\hat{\mathcal{H}}_{\mathcal{F}}$ is denoted as

$\displaystyle\hat{H}_{mn}^{F}$
$\displaystyle=\left\langle\alpha^{\prime},m\left|\hat{\mathcal{H}}_{\mathcal{F}}\right|\alpha,n\right\rangle$
(A4)
$\displaystyle=\left\langle\alpha^{\prime}\left|\sum_{l}\hat{H}_{l}e^{il{\Omega}t}e^{-im\Omega
t}e^{in\Omega
t}\right|\alpha\right\rangle-n\hbar\Omega\left\langle\alpha^{\prime},m\mid\alpha,n\right\rangle$
$\displaystyle=\left\langle\alpha^{\prime}\left|\hat{H}_{m-n}\right|\alpha\right\rangle-n\hbar\Omega\delta_{\alpha\alpha^{\prime}}\delta_{nm}.$

Then we can construct a Hamiltonian in Floquet space. After traversing the
whole momentum space, the desired Floquet-Bloch state can be obtained.

## APPENDIX B: symmetry and order characteristic analysis

To begin, we notice the following inversion symmetry of the SSH system [34]:

$\hat{H}_{0}(k)=\hat{H}_{0}^{\ast}(-k)$ (B1)

which implies the time-reversal symmetry of the SSH system. Since the external
field follows $A(t+T_{ex}/2)$ = $-A(t)$ in the time domain,
$T_{ex}=2{\pi}/{\Omega}$ is the excitation field optical period, the time-
dependent Hamiltonian obeys $\hat{H}(k,t)$ = $\hat{H}^{\ast}(-k,t+T_{ex}/2)$.
We give the following proof:

$\hat{H}(k,t)=\sum_{n}e^{in{\Omega}t}\hat{H}_{n}(k),\\\ $ (B2a)
$\hat{H}^{\ast}(-k,t+T_{ex}/2)=\sum_{n}e^{in{\Omega}(t+T_{ex}/2)}\hat{H}^{\ast}_{n}(-k),\\\
$ (B2b) $\hat{H}_{n}(k)=e^{in{\pi}}\hat{H}^{\ast}_{n}(-k).$ (B2c)

The left of the equal sign of (B2a) and (B2b) are equal, thus (B2c) is easy to
get. For the SSH system, it satisfies $\hat{H}_{n}(k)$ =
$(-1)^{n}\hat{H}_{n}^{\ast}(-k)$, and thus the corresponding wave function
also formulates $\phi_{n}(-k)$ = $(-1)^{n}\phi_{n}^{\ast}(k)$. Here n denotes
the index of Floquet-Bloch state. This reveals the temporal inversion symmetry
of the Floquet-Bloch states.

Figure 5: (a) The degree of Floquet-Bloch band opening varies with the
excitation-field strength. $k$ ($\simeq$ -0.235 a.u.) is set at band-crossing
point between Floquet band $(\text{c},-1)$ and $(\text{v},1)$.
$(\\{\text{c},\text{v}\\},m)$ represent valence and conduction bands coupling
with number $m$ photon. (b) The Floquet-Bloch bands under the action of the
THz field with laser strength 50 kV/cm. The field-free bands are denoted by
blue and red curves in (b).

We then perform Floqeut theory analysis on the HSG emission [52, 51]. Taking
the Floquet-Bloch bands, (c, 1) and (v, 0), as representative examples, their
respective wave functions follow the rules ${\phi^{\text{c}}}(k)$ =
$-{\phi^{\text{c}}}^{\ast}(-k)$ and ${\phi^{\text{v}}}(k)$ =
${\phi^{\text{v}}}^{\ast}(-k)$. Under the driving of the THz field with
optical period $T_{\text{THz}}$ = $2{\pi}/{\omega}$, one can also exploit the
Floquet theorem to expand their wave functions by Fourier series and to
analyze the odd-even feature of high-order sidebands, which are denoted as

$\phi_{\text{c}}(k,t)=e^{i\varepsilon_{\text{c}}t}u^{c}(t)=e^{i\varepsilon_{\text{c}}t}\sum_{m}\chi_{m}^{\text{c}}(k)e^{im{\omega}t},$
(B3a)
$\phi_{\text{v}}(k,t)=e^{i\varepsilon_{\text{v}}t}u^{v}(t)=e^{i\varepsilon_{\text{v}}t}\sum_{n}\chi_{n}^{\text{v}}(k)e^{in{\omega}t}.$
(B3b)

The time-dependent wave function satisfies the conditions:
${u^{\text{c}}}(k,t)$ = $-{u^{\text{c}}}^{\ast}(-k,t+T_{\text{THz}}/2)$ and
${u^{\text{v}}}(k,t)$ = ${u^{\text{v}}}^{\ast}(-k,t+T_{\text{THz}}/2))$.
According to the proof of Eq. (B2), the conclusions can also be drawn as

$\chi_{m}^{\text{c}}(k)=(-1)^{m+1}{\chi_{m}^{\text{c}}}^{\ast}(-k),$ (B4a)
$\chi_{n}^{\text{v}}(k)=(-1)^{n}{\chi_{n}^{\text{v}}}^{\ast}(-k).$ (B4b)

To obtain the light-inducing current and extract the frequency components
underlying the extremely nonlinear current, we define the current operator and
perform Fourier series expansion, i.e.,
$\hat{j}(k,t)=\sum_{l}\hat{j}_{l}(k)e^{il{\omega}t}$, here $l$ is an integer.
Since the current operators in ${\pm}k$ are opposite, leading the coefficient
term $\hat{j}_{l}(k)$ possesses the following relationship as

$\hat{j}_{l}(k)=(-1)^{l+1}\hat{j}_{l}(-k).$ (B5)

Its proof process is similar to Eq. (B2). Then, we deduce the nonlinear
current as

$\displaystyle J(t)$
$\displaystyle=\sum_{k}\left\langle\phi(k,t)|\hat{j}(k,t)|\phi(k,t)\right\rangle$
(B6)
$\displaystyle=\sum_{k}\sum_{{\alpha},{\beta}=\\{\text{c},\text{v}\\}}\sum_{m,n,l}\chi_{m}^{\alpha}(k)^{\ast}\hat{j}_{l}(k)\chi_{n}^{\beta}(k)\times
e^{-i[{\varepsilon_{\alpha}(k)-{\varepsilon_{\beta}(k)}}+(m-l-n){\omega}]t}.$

HSG is determined by the electron-hole recollision, we thus concern the
interband recombination between two bands
(${\alpha},{\beta}=\text{c},\text{v}$), in which the energy gap,
$\varepsilon_{\alpha}(k)-{\varepsilon_{\beta}(k)}$, also obeys the time-
reversal symmetry of their energy bands. We perform a substitution $m=n+l+N$
in term $e^{-i(m-l-n){\omega}t}$, which could embody the frequency component
$N$.

Substituting Eqs. (B4) and (B5) into Eq. (B6), one obtains

$J(k,N)={\chi_{n+l+N}^{\text{c}}(k)}^{\ast}\hat{j}_{l}(k){\chi_{n}^{\text{v}}(k)},\\\
$ (B7a)
$J(-k,N)=(-1)^{(n+l+N)+1}(-1)^{(l+1)}(-1)^{n}{\chi_{n+l+N}^{\text{c}}(k)}^{\ast}\hat{j}_{l}(k){\chi_{n}^{\text{v}}(k)},\\\
$ (B7b) $J(k,N)=(-1)^{(N+2)}J(-k,N).$ (B7c)

For odd $N$, the channels of ${\pm}k$ cancel each other out, so that only even
order signals eventually survive. Considering the HSG signal after resonance
excitation, only odd-order sideband signals exist in the corresponding
plateau. Extending the above rule to the electron-hole recombination signal
between $\\{(\text{c},m)$ \- $(\text{v},n)\\}$, it is obvious that when ($m$
\- $n$) is odd, the corresponding HSG plateau signal conforms to the odd-order
characteristics. Similarly, it can be given that for even ($m$ \- $n$), its
current satisfies $J(k,N)=(-1)^{(N+1)}J(-k,N).$ The property of the
corresponding plateau satisfies even order. This indicates that plateau
regions exhibit either odd or even-order characteristics based on the
differences of contributions from ($m$ \- $n$).

## APPENDIX C: Band inversion and Floquet-Bloch state under THz field

Since the different Floquet states of the valence band and conduction band
will form a cross, the study of such a band intersection is also desirable. As
shown in Fig. 5(a), the Floquet band (c, -1) and (v, 1) will cross each other.
However, with the increasing of excitation field strength, the band inversion
is realized to avoid band crossing. This result is consistent with the
experimental results [9, 11]. The signature of band inversion is indicative of
a topological phase transition taking place within a non-equilibrium state.
Future explorations could involve leveraging all-optical detection methods to
delve deeper into these non-equilibrium processes. In addition, we show the
Floquet-Bloch states under the individual action of a THz field with strength
50 kV/cm as presented in Fig. 5(b). The THz field only performs the periodic
energy modulation on the original energy bands, and it is difficult to achieve
the non-trivial band inversion due to its small photon energy. Thus, the THz
fields primarily play the role of driving the motion of electron-hole pairs.

## Funding

T.-Y. Du is supported by the Natural Science Foundation of Hubei Province
China (Grant No. 2023AFB613) and the Fundamental Research Funds for the
Central Universities, China University of Geosciences (Wuhan) with Grant No.
G1323523064; M. Li acknowledges the financial support from the Research Grant
Council of Hong Kong (Project No.PolyU 25301522 and PolyU 15301323), National
Natural Science Foundation of China (22373081).

## Acknowledgement

We thank Prof. Ren-Bao Liu for the fruitful discussions.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

All the data used in this study are available upon reasonable request from the
corresponding author.

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