# Tuning anomalous Floquet topological bands with ultracold atoms

Jin-Yi Zhang1,2,3 Chang-Rui Yi1,2,3 Long Zhang8,4,5 Rui-Heng Jiao1,2,3 Kai-Ye
Shi6 Huan Yuan1,2,3 Wei Zhang6 Xiong-Jun Liu4,5,7 Shuai Chen1,2,3 Jian-Wei
Pan1,2,3 1Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China
2Shanghai Research Center for Quantum Science and CAS Center for Excellence in
Quantum Information and Quantum Physics, University of Science and Technology
of China, Shanghai 201315, China
3Hefei National Laboratory, University of Science and Technology of China,
Hefei 230088, China
4International Center for Quantum Materials, School of Physics, Peking
University, Beijing 100871, China
5Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
6Department of Physics, Renmin University of China, Beijing 100872, China
7Shenzhen Institute for Quantum Science and Engineering, Southern University
of Science and Technology, Shenzhen 518055, China
8School of Physics and Institute for Quantum Science and Engineering, Huazhong
University of Science and Technology, Wuhan 430074, China

(August 27, 2024)

###### Abstract

The Floquet engineering opens the way to create new topological states without
counterparts in static systems. Here, we report the experimental realization
and characterization of new anomalous topological states with high-precision
Floquet engineering for ultracold atoms trapped in a shaking optical Raman
lattice. The Floquet band topology is manipulated by tuning the driving-
induced band crossings referred to as band inversion surfaces (BISs), whose
configurations fully characterize the topology of the underlying states. We
uncover various exotic anomalous topological states by measuring the
configurations of BISs which correspond to the bulk Floquet topology. In
particular, we identify an unprecedented anomalous Floquet valley-Hall state
that possesses anomalous helical-like edge modes protected by valleys and a
chiral state with high Chern number.

Over one century ago, Gaston Floquet established a formalism to describe time-
periodic systems Floquet (1883), which, now called the Floquet theory Shirley
(1965), has advanced the very broad studies of non-equilibrium physics in
quantum systems and played a vital role in the recent developments of
simulating exotic quantum phases out of equilibrium Eckardt (2017); Oka and
Kitamura (2019); Rudner and Lindner (2020); Harper  _et al._ (2020).
Periodical driving introduces an extra degree of freedom in time domain, which
extends the characterization of equilibrium quantum phases to dynamically
steady states. Such a generalization leads to new concepts which attract
considerable attention in the recent years, such as time crystals Sacha and
Zakrzewski (2017); Else  _et al._ (2020) and Floquet topological phases
Lindner  _et al._ (2011); Rechtsman  _et al._ (2013); Jotzu  _et al._ (2014);
Fläschner  _et al._ (2016); Rudner  _et al._ (2013); Nathan and Rudner (2015);
Roy and Harper (2017); Liu  _et al._ (2019); Wintersperger  _et al._ (2020).
The latter is a novel extension of the widely explored topological quantum
phases in equilibrium systems Hasan and Kane (2010); Qi and Zhang (2011), of
which the comprehensive classification and characterization have been
established Kitaev (2009); Chiu  _et al._ (2016).

Floquet engineering provides a versatile tool to manipulate topological phases
beyond the realm of conventional solid-state counterparts Rudner  _et al._
(2013); Nathan and Rudner (2015); Roy and Harper (2017); Yao  _et al._ (2017).
With coherent temporal control, the driven system not only can be tailored
properly to realize the Hamiltonian of novel static systems such as the
Haldane model Jotzu  _et al._ (2014), but also may be exploited to create new
states unreachable in static configurations. For instance, an anomalous
Floquet topological state has been realized experimentally in periodically-
modulated hexagonal lattices Wintersperger  _et al._ (2020). In general, a
spatially $d$-dimensional ($d$D) Floquet phase may acquire nontrivial topology
in the higher $(d+1)$-dimensional space-time domain. This renders a novel
paradigm called anomalous Floquet topological phase, which hosts robust
boundary modes that are unlinked to the bulk topology of $d$D Floquet bands,
hence breaking down the conventional bulk-boundary correspondence.
Conventionally such phases can be characterized with topology defined in the
full $(d+1)$D space-time dimension Rudner  _et al._ (2013), with broad classes
of anomalous Floquet phases having been predicted Nathan and Rudner (2015);
Roy and Harper (2017), albeit many necessitates experimental discovery.
Recently, a theory Zhang  _et al._ (2020) shows that the anomalous Floquet
topological phases can also be fully characterized by sub-dimensional topology
defined on a momentum subspace called band inversion surfaces (BISs) of $d$D
Floquet bands Zhang  _et al._ (2018), without involving the complex evolution
in time dimension. This characterization offers a powerful indicator to
calibrate the engineering of novel Floquet topological phases.

Here, we experimentally realize, manipulate, and detect new anomalous Floquet
topological states for ultracold atoms via a systematic BIS engineering. By
periodically driving a quantum anomalous Hall (QAH) model with 2D spin-orbit
coupling, we control and observe the appearance of BISs as they successively
appear in the Brillouin zone, with which various anomalous Floquet topological
states corresponding to distinct BIS configurations are observed.
Particularly, we realize a novel anomalous Floquet valley-Hall state, which
features helical-like edge states at boundaries, and is in stark difference
with the widely investigated valley-Hall effect in static solid materials
Gorbachev  _et al._ (2014); Mak  _et al._ (2014); Schaibley  _et al._ (2016);
Ren  _et al._ (2016), as well as unlike the anomalous Floquet state in
periodically-modulated hexagonal lattices Wintersperger  _et al._ (2020).

Figure 1: Scheme of shaking Raman lattices. (a) Cartoon of the optical Raman
lattices with frequency modulation of a laser beam. (b) Sketch of the Floquet
bands along the diagonal direction of the quasi-momentum $q_{\rm{diag}}$.
$k_{0}$ is the wave number of the lattice beams, and
$(\left|\uparrow\right\rangle,n\omega)$
($(\left|\downarrow\right\rangle,n\omega)$) is the shifted copy of the static
band $(\left|\uparrow\right\rangle,0)$ ($(\left|\downarrow\right\rangle,0)$).
Solid curves are the Floquet bands which presenting 0-gap at the center of FBZ
(black diamonds) and $\pi$-gap at the boundary of FBZ (green circles). (c)
Experimental protocol and evolution of the spin textures
$\left\langle\sigma_{z}(\bm{q},t)\right\rangle_{z}$. The upper (lower) row is
spin textures without (with) periodic driving where the parameters
$(V_{0},\leavevmode\nobreak\ \Omega_{0},\leavevmode\nobreak\
\delta_{0})=(4.0,\leavevmode\nobreak\ 1.0,\leavevmode\nobreak\
0.4)E_{\text{r}}$ ($(V_{0},\leavevmode\nobreak\
\Omega_{0},\leavevmode\nobreak\
\delta_{0},A_{\text{F}})=(4.0,\leavevmode\nobreak\ 1.0,\leavevmode\nobreak\
0.4,\leavevmode\nobreak\ 0.8)E_{\text{r}}$, $T=400\mu\text{s}$). Here,
$E_{\rm{r}}\approx h\times 3.7$kHz is the recoil energy. $\Gamma$ and M is
high symmetric momenta. The red (blue) color in the spin textures denotes spin
up (down).

Our experimental setup is a 2D periodically driven QAH model realized with
ultracold 87Rb atoms trapped in shaking optical Raman lattices Sup  [see Fig.
1(a)]. The Raman lattices are conducted with magnetic sublevels
$\left|F=1,m_{F}=-1\right\rangle$ (spin-up $\mid\uparrow\rangle$) and
$\left|1,0\right\rangle$ (spin-down $\mid\downarrow\rangle$) in the $F=1$
manifold coupled by two Raman coupling processes Wu  _et al._ (2016); Sun  _et
al._ (2018a, b); Yi  _et al._ (2019). Under the tight-binding approximation,
the time-dependent Bloch Hamiltonian reads

$\displaystyle\leavevmode\nobreak\
\begin{split}\mathcal{H}(\bm{q},t)&=2t_{\text{so}}\sin{q_{y}}\sigma_{x}+2t_{\text{so}}\sin{q_{x}}\sigma_{y}\\\
&+[\delta(t)/2-2t_{0}(\cos{q_{x}}+\cos{q_{y}})]\sigma_{z},\end{split}$ (1)

where ${\bm{q}}=(q_{x},q_{y})$ and $\sigma_{i}$ are respectively the Bloch
momentum and Pauli matrices $\bm{\sigma}$ acting on spin space, and the
lattice constant is taken as unit of length. The spin-conserved (spin-flip)
hopping coefficient $t_{0}$ ($t_{\text{so}}$) is induced by the lattice
(Raman) potential and set by the lattice depth $V_{0}$ (Raman strength
$\Omega_{0}$) Sup . The two-photon detuning is tuned in a form of a sinusoidal
function $\delta(t)=\delta_{0}+A_{\text{F}}\sin\omega t$, which is realized by
periodically modulating the laser frequency (Fig. 1(a)). Here, $\delta_{0}$ is
a constant, $A_{\text{F}}$ denotes the modulation amplitude, and the driving
frequency $\omega=2\pi/T$ with $T$ being the period.

Using the standard Floquet theory, this periodically driven QAH system can be
delineated by an effective time-independent Hamiltonian
$\mathcal{H}_{\text{F}}=\bm{h_{\text{F}}}\cdot\bm{\sigma}=h_{\text{F},x}\sigma_{x}+h_{\text{F},y}\sigma_{y}+h_{\text{F},z}\sigma_{z}$
Zhang  _et al._ (2020), whose eigen-energies are referred as quasienergies
$\varepsilon=\pm|\bm{h}_{\text{F}}|$ (More details see Sup ). Owing to the
temporal periodicity, the quasi-energy band structure can be defined and tuned
via the driving period $T=2\pi/\omega$. Physically, the periodic modulation at
frequency $\omega$ can transfer an energy in portions of $n\omega$ ($n$ is an
integer) to the non-driven system (corresponding to $A_{\text{F}}=0$), such
that the engineered Floquet band structure is formed by copying and shifting
the static $(\mid\uparrow\rangle,0)$ and $(\mid\downarrow\rangle,0)$ bands
within the Floquet Brillouin zone (FBZ)
$\varepsilon\in[-\omega/2,\leavevmode\nobreak\ +\omega/2]$ Rudner  _et al._
(2013); Zhang  _et al._ (2020). At the band crossing points, a finite
$A_{\text{F}}$ and $\Omega_{0}$ will lift the degeneracy and open gaps, which
can be further distinguished as $0$-gaps and $\pi$-gaps as they reside
respectively at the center and the boundary of FBZ. An example of Floquet band
structure is shown in Fig. 1(b). Remarkably, two types of BISs can be
identified at the band crossing, on which $h_{\text{F},z}({\bf q})=0$ Zhang
_et al._ (2020); Sup . While the conventional $0$-gap BIS (or $0$-BIS for
short) is located within the 0-gap as in a static QAH model, an extra
$\pi$-BIS is induced by driving within the $\pi$-gap as an exclusive feature
of periodically driven systems.

Figure 2: Tunable topological states. The emergence of ring patterns from one
to four in the measured (first row) spin textures
$\left\langle\sigma_{z}(\bm{q},t)\right\rangle$ and the Floquet bands along
$q_{\rm{diag}}$ as $T$ varies with parameters $(V_{0},\leavevmode\nobreak\
\Omega_{0},\leavevmode\nobreak\ \delta_{0})=(4.0,\leavevmode\nobreak\
1.0,\leavevmode\nobreak\ 0.1)E_{\text{r}}$. And $(T,A_{\rm{F}},t)=(280\mu
s,0.8E_{\rm{r}},2T)$, $(400\mu s,0.8E_{\rm{r}},3T)$, $(520\mu
s,0.8E_{\rm{r}},2T)$, $(740\mu s,1.2E_{\rm{r}},2T)$ from left to right. The
dashed black (green) curves mark the 0-BISs ($\pi$-BISs).

In this study, we synthesize unconventional Floquet topological bands by
engineering the BIS configurations systematically and identify their topology
by quench dynamics with the following protocol Sun  _et al._ (2018b); Yi  _et
al._ (2019). The 87Rb atoms are initially prepared in the
$\mid\uparrow\rangle$ state at a temperature around 100nK and the initial
detuning of $-200E_{\rm{r}}$. At the beginning of the experimental sequence,
the detuning is quenched to the modulated value $\delta(t)$ within 200ns and
$\delta_{0}\in[-1,1]E_{\rm{r}}$ (Fig. 1(c)). Thereupon, atoms are driven out
of equilibrium and evolve under the post-quench time-dependent Hamiltonian Eq.
(1). After periodically modulating $\delta(t)$ for a certain time $t=nT$, the
number of atoms of $\mid\uparrow\rangle$ ($N_{\uparrow}$) and
$\mid\downarrow\rangle$ ($N_{\downarrow}$) at quasi-momentum $\bm{q}$ are
measured by spin-resolved time-of-flight (TOF) imaging, and the dynamics of
spin polarization
$\left\langle\sigma_{z}(\bm{q},t)\right\rangle_{z}=[N_{\uparrow}(\bm{q},t)-N_{\downarrow}(\bm{q},t)]/[N_{\uparrow}(\bm{q},t)+N_{\downarrow}(\bm{q},t)]$
is extracted. Here the subscript $z$ of $\langle\cdot\rangle_{z}$ denotes
quenching $\delta$ at the beginning, corresponding to a process of quenching
$h_{{\rm F},z}$ Zhang  _et al._ (2018); Yi  _et al._ (2019). The BISs can be
determined as the locations where full spin-flip takes place during such spin
dynamics, and work as the key feature to characterize the topology Sun  _et
al._ (2018b); Zhang  _et al._ (2018). As an example, we show the spin textures
$\left\langle\sigma_{z}(\bm{q},t=nT)\right\rangle_{z}$ in the lower panel of
Fig. 1(c), in comparison with the upper panel where no periodic driving is
applied. In the upper panel, there exists only one ring surrounding $\Gamma$
point of the Brillouin zone, corresponding to the BIS formed by the spin-orbit
coupling of non-driven QAH model Sun  _et al._ (2018b). The periodic driving
induces an additional ring around the M point, indicating the emergence of a
driving-induced BIS in the $\pi$-gap (Fig. 1(b)).

Figure 3: Characterization of topological states. (a) Time-averaged spin
textures $\overline{\left\langle\sigma_{z}(\bm{q},t)\right\rangle}_{i}$
obtained in the experimental measurements (upper) and numerics with parameters
in experiment (lower). The black and green curves represent the location of
BISs. The dashed curves in the upper row are the guides to the eye. The arrows
along the BISs display the dynamical field. The BISs divide the Brillouin Zone
into three regions: two of them with $h_{\text{F},z}<0$ as labeled by “$-$”,
and one of them with $h_{\text{F},z}>0$ as denoted by “$+$”. $\nu_{0}^{j}$
($\nu_{\pi}^{j}$) is the topological number of the $j$-th BIS in 0-gap
($\pi$-gap). (b) The numerical band structures. The orange (green) curves
denote edge states in the left (right) boundary. The parameters are
($V_{0},\Omega_{0},\delta_{0},A_{\text{F}},T$)=($4.0E_{\text{r}},1.0E_{\text{r}},0.4E_{\text{r}},0.8E_{\text{r}},400\mu\text{s}$).

Now we demonstrate the scheme to realize novel Floquet topological states by
engineering the BISs of Floquet bands, and identify the topology by measuring
the BIS configurations. The engineering and characterisation of a targeted
topological state are implemented in two steps. We first identify in which gap
each BIS lives. This is achieved by monitoring the BISs as gradually
increasing the period of the driving term applied to an initial determined
static phase. Then we measure by quench dynamics the topological invariant of
each emerged BIS from the BIS circling configuration.

In Fig. 2, observed spin textures
$\left\langle\sigma_{z}({\bm{q}},t)\right\rangle_{z}$ with different driving
period are depicted in the first row. For small $T$ (280$\mu$s), there exits
only one BIS circling the $\Gamma$ point (with number of rings
$\mathcal{N}_{\rm BIS}=1$), which is adopted from the static band structure
and corresponds to the crossing of $(\mid\uparrow\rangle,0)$ and
$(\mid\downarrow\rangle,0)$ bands in the FBZ at $\varepsilon=0$, corresponding
to static 0-BIS. The Floquet band structure is continuously connected to the
static QAH bands, and the topology is not changed by the applied driving. With
increasing $T$, new BISs emerge in $0$-gap or $\pi$-gap due to the new band
crossings between the shifted copies of the static bands
$(\mid\uparrow\rangle,n\omega)$ and $(\mid\downarrow\rangle,n\omega)$, and
surround either the $\Gamma$ or M point. There exists a BIS that maintains as
$T$ increases, corresponding to the static 0-BIS. Topological transitions thus
occur with the open and close of gaps at the band crossing ring. From the
theoretical analysis, one concludes a property (P1): the two types of BISs
always appear alternatively along the diagonal direction of quasi-momentum
$q_{\rm diag}$ for the present system (More details see Sup ). With property
(P1) and the static 0-BIS, we precisely identify whether the BIS is a 0-BIS or
a $\pi$-BIS. In experiment, we tune $T$ up to $740\mu\text{s}$, and encounter
four different regions with $\mathcal{N}_{\rm{BIS}}$ up to 4.

To fully characterize the underlying topology by the configurations of BISs,
$\mathcal{N}_{\rm{BIS}}$ alone is not enough, and an extra set of topological
invariants $\nu_{0,\pi}^{j}$ associated with the $j$-th BIS have to be
employed to determine the winding number of the corresponding gap as
$\mathcal{W}_{0,\pi}=\sum_{j}\nu_{0,\pi}^{j}$ Zhang  _et al._ (2020). The
value of $\nu_{0,\pi}^{j}$ can be extracted by the dynamical field, defined as
the variation of time-averaged spin textures
$\overline{\left\langle\sigma_{z}(\bm{q})\right\rangle}_{i=x,y,z}$ across the
BIS. Such $\overline{\left\langle\sigma_{z}(\bm{q})\right\rangle}_{i}$ is
obtained by quenching $h_{\text{F},i}$ Zhang  _et al._ (2020); Yi  _et al._
(2019); Sup . Fig. 3(a) shows
$\overline{\left\langle\sigma_{z}(\bm{q})\right\rangle}_{x,y,z}$ and the
corresponding numerical calculation for ${\cal N}_{\rm BIS}=2$. Two BISs are
identified as two rings circling the $\Gamma$ and M points, respectively.
Based on the sequence of their emergence, we determine that the inner BIS
surrounding the $\Gamma$ point corresponds to the static $0$-BIS, while the
outer one is a driving-induced $\pi$-BIS. These two BISs divide the Brillouin
zone into three regions with $h_{\text{F},z}>0$ or $h_{\text{F},z}<0$, which
can be judged by the properties of band inversions Sup . The two-component
dynamical field is obtained by calculating the gradient from the region “$-$”
to “+” (More details see Sup ). From the top right corner of the BIS
surrounding M point in spin texture
$\overline{\left\langle\sigma_{z}\right\rangle}_{x}$, the difference of the
spin polarisation between region A and region B is negative, hence the
direction of the $x$-component dynamical field is along the opposite direction
of $\sigma_{x}$. The $y$-component dynamical field is obtained similarly from
$\overline{\left\langle\sigma_{z}\right\rangle}_{y}$. The combined dynamical
field on BISs is then determined by superimposing the two components and
depicted as arrows in $\overline{\left\langle\sigma_{z}\right\rangle}_{z}$,
which gives a nonzero $\nu_{0,\pi}^{j}$ for each BIS in the $0$ and $\pi$
gaps. In this case, we get $\nu_{0}^{1}=+1$ for the 0-BIS surrounding the
$\Gamma$ point and $\nu_{\pi}^{1}=-1$ for the $\pi$-BIS circling the M point
(More details see Sup ). The winding numbers are then
$\mathcal{W}_{0}=\nu_{0}^{1}=+1$ and $\mathcal{W}_{\pi}=\nu_{\pi}^{1}=-1$,
respectively, showing a high Chern number of the anomalous Floquet band as
$\mathcal{C}=\mathcal{W}_{0}-\mathcal{W}_{\pi}=+2$. As expected, the Chern
number of the present anomalous Floquet topological band does not directly
correspond to the edge states of each gap under open boundary condition (Fig.
3). Instead, the chiral edge states are directly determined by
$\nu_{0,\pi}^{j}$ (see the right panel of Fig. 3(b) and Sup ).

Figure 4:  Topological phase diagram. (a) Topological phase diagram with
$(V_{0},\Omega_{0})=(4.0,1.0)E_{\text{r}}$ in the $T$-$\delta_{0}$ plane. The
blue circles with error bars are experimental measurements. The solid and
dashed curves are calculated phase boundaries induced by 0-gap and $\pi$-gap,
respectively. Roman numerals i to vi label 6 topological phases within the
range of experimental parameters, whose topological invariants are listed in
the table. (b) Spin textures
$\left\langle\sigma_{z}(\bm{q},t)\right\rangle_{z}$ marked by ✩ with typical
phases in (a). To clearly observe vi, the raw data are shown in Fig. S8 of Sup
. The BIS configurations labeled by dashed black and green curves fully
distinguish the topology of all phases. The following gray (orange) frame
contains the bulk band structures, the aerial view of the lowest band and the
band structures with open boundaries corresponding to the spin texture iv
(vi). The orange and green curves denote “helical” edge modes.

The above observation shows the three basic features which enable us to
realize, manipulate and detect the anomalous Floquet topological states
through a systematic engineering of the BIS configurations Sup ; Zhang and Liu
(2021). First, the precise location in the $0$-gap or $\pi$-gap of the
alternatively emerging BISs is determined by gradually increasing the driving
period $T$ and the property (P1). Second, $\nu_{0,\pi}^{j}$ is uniquely
determined by the BIS circling configuration. Namely, $\nu_{0,\pi}^{j}$ is
$+1$ ($-1$) when the BIS surrounds the $\Gamma$ (M) point (P2). Third, each
BIS with $\nu_{0,\pi}^{j}\neq 0$ corresponds to a chiral edge modes if the
open boundary is considered (P3). With these features we map out the rich
phase diagram of the present system in Fig. 4(a) with $\delta_{0}<0$ and the
complete phase diagram is shown in the supplementary materials Sup . The phase
boundaries are determined by fitting the size of the rings in the spin
textures as a function of $T$ or $\delta_{0}$ Sup .

The most striking topological state uncovered in the current experiment is the
one labeled by “vi”, in which four BISs are observed in the spin textures
depicted in Fig. 4(b). The inner two BISs surround the $\Gamma$ point and the
outer two wind around the M point. Through the regulation of driving period
(Fig. 2), we determine that the outer black ring is the static $0$-BIS, while
the other three BISs alternatively lives in the 0-gap and $\pi$-gap according
to the property (P1). The circling configuration further determines
$\nu_{0}^{1}=+1$, $\nu_{\pi}^{1}=+1$, $\nu_{0}^{2}=-1$ and $\nu_{\pi}^{2}=-1$
from center to lattice Brillouin zone edge, leading to the vanishing winding
numbers in each quasienergy gap $\mathcal{W}_{0}=\nu_{0}^{1}+\nu_{0}^{2}=0$
and $\mathcal{W}_{\pi}=\nu_{\pi}^{1}+\nu_{\pi}^{2}=0$, and Chern number
$\mathcal{C}=0$. However, from the BIS-boundary correspondence (P3) the four
BISs correspond to two pairs of edge modes with opposite chirality residing in
the $0$ and $\pi$ gaps, respectively, despite the trivial global topological
invariants, consistent with the numerical results under the same parameter
conditions in experiment (see the lowest panel of Fig. 4(b)).

The counter-propagating edge modes within each gap are protected by local
topological structures (BISs) in two different Brillouin zone regions, i.e.
the valleys, and are thus robust against long-range disorders which do not
induce inter-valley scatterings Ren  _et al._ (2016); Sup , rendering an
exotic anomalous Floquet valley-Hall state with multiple “helical” edge modes.
Such modes, similar to “spin-helical” edge modes in the quantum spin Hall
effect, have a valley degree of freedom now coupled to the direction of motion
(see Fig. 4(b)). This topological state is beyond the classification and
characterization of anomalous topological phases using global topological
invariants of Floquet bands and winding number of each gap Lababidi  _et al._
(2014); Umer  _et al._ (2020), but is fully captured by the BIS–boundary
correspondence. A similar analysis on the phase labeled by “iv” with ${\cal
N}_{\rm BIS}=3$ gives $\mathcal{W}_{0}=-1$, $\mathcal{W}_{\pi}=0$ and
$\mathcal{C}=-1$, which protects counter-propagating edge modes inside the
$\pi$-gap (the middle panel of Fig. 4(b)).

In conclusion, we experimentally implemented a systematic and powerful
technique of spatiotemporal control to realize, engineer and detect novel
anomalous Floquet topological states. From quench measurement, we observe that
the topology of an anomalous Floquet state in the space-time $(d+1)$-dimension
uniquely corresponds to the nontrivial configuration of $(d-1)$-dimensional
BISs. This enables a highly feasible experimental approach to realize and
engineer new Floquet topological states, including a novel anomalous valley-
Hall state, which is a new paradigm achieved here and beyond the existing
anomalous Floquet phases Rudner  _et al._ (2013); Kitagawa  _et al._ (2012);
Hu  _et al._ (2015); Maczewsky  _et al._ (2017); Mukherjee  _et al._ (2017);
Wintersperger  _et al._ (2020). The technique can be directly applied to
fermionic systems and extended to high dimensions Zhang and Liu (2021), and
shall open up a broad avenue of space-time manipulation towards exploring new
fundamental topological states of matter.

###### Acknowledgements.

We acknowledge insightful discussions with Zong-Yao Wang, Xiang-Can Cheng, Wei
Zheng and Hui Zhai. This work was supported by the Innovation Program for
Quantum Science and Technology (Grant No. 2021ZD0302000), the National Key
Research and Development Program of China (Grant No. 2021YFA1400900 and
2018YFA0306501), the National Natural Science Foundation of China (Grant No.
12025406, 11825401, 12074428 and 12104445), Anhui Initiative in Quantum
Information Technologies (Grant No. AHY120000), Shanghai Municipal Science and
Technology Major Project (Grant No. 2019SHZDZX01), and the Strategic Priority
Research Program of Chinese Academy of Science (Grant No. XDB28000000).
J.-Y.Z. acknowledges support from the startup grant of University of Science
and Technology of China (Grant No. KY2340000152) and the Sponsored by Shanghai
Pujiang Program (Grant No. 21PJ1413600). L. Z. acknowledges support from the
startup grant of Huazhong University of Science and Technology (Grant No.
3004012191). W.Z. acknowledges support from the Beijing Natural Science
Foundation (Grant No. Z180013). X.-J.L. acknowledges support from the Open
Project of Shenzhen Institute of Quantum Science and Engineering (Grant No.
SIQSE202003).

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Supplemental Material for:
Tuning anomalous Floquet topological bands with ultracold atoms

##  .1  2D optical Raman lattices for ultracold Bose gases

The experimental setup is demonstrated in Fig. S1. Ultracold atomic cloud of
87Rb is prepared in a dipole optical trap. Laser beam $\bm{E}_{x}$
($\bm{E}_{y}$) with wavelength $\lambda=787$nm along $\hat{x}$ ($\hat{y}$)
direction is incident on the clouds, then it is reflected back to atomic
clouds by the mirror $\text{M}_{1}$ ($\text{M}_{2}$), passing through a
$\lambda/2$ wave plate and electro-optic modulator (EOM). The $\lambda/2$ wave
plate splits the beam $\bm{E}_{x}$ ($\bm{E}_{y}$) into two orthogonally
polarised components $\bm{E}_{xy}$ $(\bm{E}_{yx})$ and $\bm{E}_{xz}$
$(\bm{E}_{yz})$, and the $\text{EOM}_{1}$ ($\text{EOM}_{2}$) induces the phase
shift $\varphi_{1}$ $(\varphi_{2})$ between the two components $\bm{E}_{xy}$
and $\bm{E}_{xz}$ $(\bm{E}_{yx}$ and $\bm{E}_{yz})$. Moreover, two magnetic
sublevels $\mid\uparrow\rangle\equiv\left|F=1,m_{F}=-1\right\rangle$ and
$\mid\downarrow\rangle\equiv\left|1,0\right\rangle$ are subjected to a bias
magnetic field B along $\hat{z}$ direction, generating a Zeeman splitting of
$10.2\text{MHz}$. Thus, the squared optical lattice potentials
$V_{\text{latt}}(x,y)$ and the Raman lattices $\Omega_{x,y}(x,y)$ can be tuned
by $\varphi_{1,2}$ Yi  _et al._ (2019), i.e.,

$\displaystyle V_{\text{latt}}(x,y)$
$\displaystyle=V_{0}^{\prime}[|\bm{E}_{xz}|^{2}\cos^{2}(k_{0}x-\varphi_{1})+|\bm{E}_{xy}|^{2}\cos^{2}(k_{0}x)+|\bm{E}_{yz}|^{2}\cos^{2}(k_{0}y-\varphi_{2})+|\bm{E}_{yx}|^{2}\cos^{2}(k_{0}y)],$
(S1) $\displaystyle\Omega_{x}(x,y)$
$\displaystyle=\Omega_{0x}\cos(k_{0}x-\varphi_{1})\cos(k_{0}y)\exp(-i\varphi_{1}),\Omega_{y}(x,y)=\Omega_{0y}\cos(k_{0}x)\cos(k_{0}y-\varphi_{2})\exp(i\varphi_{2}).$

Here, $k_{0}=2\pi/\lambda$, $V_{0}^{\prime}$ is determined by the level
structure of 87Rb atoms and the Raman coupling strength
$\Omega_{0x,0y}\propto|\bm{E}_{yx,xy}||\bm{E}_{xz,yz}|$. In the experiment, we
set $|\bm{E}_{xy}|=|\bm{E}_{yx}|$ and $|\bm{E}_{xz}|=|\bm{E}_{yz}|$, then
$\Omega_{0x}=\Omega_{0y}=\Omega_{0}$. And the total Hamiltonian reads

$H=\begin{pmatrix}\frac{\bm{p}^{2}}{2m}+V_{\text{latt}}(x,y)+\frac{\delta}{2}&\Omega_{x}-i\Omega_{y}\\\
\Omega_{x}+i\Omega_{y}&\frac{\bm{p}^{2}}{2m}+V_{\text{latt}}(x,y)-\frac{\delta}{2}\end{pmatrix},$
(S2)

where $\delta$ is the two-photon detuning. With the tight-binding
approximation, the above Hamiltonian generally can be written as

${\cal H}_{0}(\bm{q})=\bm{h}(\bm{q})\cdot\bm{\sigma}=(m_{x}+2t^{x}_{\rm
so}\sin q_{y})\sigma_{x}+(m_{y}+2t^{y}_{\rm so}\sin
q_{x})\sigma_{y}+[\delta/2-2(t^{x}_{0}\cos q_{x}+t_{0}^{y}\cos
q_{y})]\sigma_{z}.$ (S3)

Here, the Zeeman constants $m_{x,y}$ and the spin-conserved (flipped) hopping
coefficients $t_{0}^{x,y}$ ($t_{\rm so}^{x,y}$) are depends on
$\varphi_{1,2}$. The Pauli matrices $\sigma_{x,y,z}$ are defined as

$\sigma_{x}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\sigma_{y}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\sigma_{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

When $\varphi_{1}=\varphi_{2}=\pi/2$, the squared optical lattice potentials
$V_{\text{latt}}(x,y)$ and the Raman lattices $\Omega_{x,y}(x,y)$ can be
written as

$\displaystyle V_{\text{latt}}(x,y)$
$\displaystyle=V_{0x}\cos^{2}k_{0}x+V_{0y}\cos^{2}k_{0}y,$ (S4)
$\displaystyle\Omega_{x}(x,y)$ $\displaystyle=\Omega_{0x}\sin k_{0}x\cos
k_{0}y,$ $\displaystyle\Omega_{y}(x,y)$ $\displaystyle=\Omega_{0y}\cos
k_{0}x\sin k_{0}y.$

where the lattice depth
$V_{0x,0y}\propto\lvert\bm{E}_{xy,yx}\rvert^{2}-\lvert\bm{E}_{xz,yz}\rvert^{2}$.
Then, $V_{0x}=V_{0y}=V_{0}$ due to $|\bm{E}_{xy}|=|\bm{E}_{yx}|$ and
$|\bm{E}_{xz}|=|\bm{E}_{yz}|$, and the Hamiltonian of the Raman lattices reads

$\displaystyle
H=\frac{\bm{p}^{2}}{2m}+V_{\text{latt}}(x,y)+\Omega_{x}\sigma_{x}+\Omega_{y}\sigma_{y}+\frac{\delta}{2}\sigma_{z}.$
(S5)

With the tight-binding approximation, the Hamiltonian is given in the Eq.(1)
of main text, i.e.,

$\mathcal{H}_{0}(\bm{q})=2t_{\text{so}}\sin{q_{y}}\sigma_{x}+2t_{\text{so}}\sin{q_{x}}\sigma_{y}+[\delta/2-2t_{0}(\cos{q_{x}}+\cos{q_{y}})]\sigma_{z}.$
(S6)

Here, $m_{x}=m_{y}=0$, the spin-conserved hopping coefficients
$t_{0}^{x}=t_{0}^{y}=t_{0}$ and spin-flipped hopping coefficients
$t_{\rm{so}}^{x}=t_{\rm{so}}^{y}=t_{\rm{so}}$ are, respectively,

$\displaystyle t_{0}$ $\displaystyle=-\int
d{\bm{r}}\phi_{s}(x,y)\left[\frac{{\bm{k}}^{2}}{2m}+V_{\rm
latt}({\bm{r}})\right]\phi_{s}(x-1,y),$ $\displaystyle t_{\rm so}$
$\displaystyle=\Omega_{0}\int
d{\bm{r}}\phi_{s}(x,y)\cos(k_{0}y)\sin(k_{0}x)\phi_{s}(x-1,y),$ (S7)

where $\phi_{s}(x,y)$ denotes the Wannier function of the lowest band. For the
typical parameters $V_{0}=4E_{\rm{r}}$ and $\Omega_{0}=1E_{\rm{r}}$ in the
Raman lattices, $t_{0}\approx 0.09$ and $t_{\rm{so}}\approx 0.05$.

When ($\varphi_{1},\varphi_{2}$)=($0,\pi/2$) or ($0,\pi/2$), the Bloch
Hamiltonian in the tight-binding limit reads Yi  _et al._ (2019)

${\cal H}_{0}=[\delta/2-2(t^{x}_{0}\cos q_{x}+t_{0}^{y}\cos
q_{y})]\sigma_{z}+2t^{x}_{\rm so}\sin q_{y}\sigma_{x}+m_{y}\sigma_{y},$ (S8)

or

${\cal H}_{0}=[\delta/2-2(t^{x}_{0}\cos q_{x}+t_{0}^{y}\cos
q_{y})]\sigma_{z}+2t^{y}_{\rm so}\sin q_{x}\sigma_{y}+m_{x}\sigma_{x},$ (S9)

where the Zeeman constant

$m_{y}=\Omega_{0x}\int
d{\bm{r}}\phi_{s}({\bm{r}})\cos(k_{0}x)\cos(k_{0}y)\phi_{s}({\bm{r}})\gg
t_{\rm so}^{x},t_{0}^{x,y}$ (S10)

or

$m_{x}=\Omega_{0y}\int
d{\bm{r}}\phi_{s}({\bm{r}})\cos(k_{0}x)\cos(k_{0}y)\phi_{s}({\bm{r}})\gg
t_{\rm so}^{y},t_{0}^{x,y}.$ (S11)

The shaking Raman lattices are realized by modulating the two-photon detuning
$\delta$. To this end, the frequency of the laser beam $\bm{E}_{x}$ is
modulated by a ratio-frequency (RF) signal which drives an acousto-optic
modulators (AOM) (Fig. S1). Meanwhile, the frequency of the laser beam
$\bm{E}_{y}$ is fixed. Thus, the two-photon detuning
$\delta=\delta_{0}+A_{\text{F}}\sin{\omega t}$.

Figure S1: Experimental setup. $\bm{E}_{x}$ ($\bm{E}_{y}$) represents the
laser beam, possessing two orthogonally polarised components $\bm{E}_{xy}$
$(\bm{E}_{yx})$ and $\bm{E}_{xz}$ $(\bm{E}_{yz})$. $\lambda/2$ represents the
half-wave plate; $\text{M}_{1}$ ($\text{M}_{2}$) denotes the mirror;
$\text{EOM}_{1,2}$ are the electro-optic modulators; AOM is the acousto-optic
modulator; $\text{RF}_{1,2}$ are the ratio-frequency signals;
$\text{Switch}_{1,2}$ are RF switches. Bias magnetic field B is along $z$
direction. Inset: level structures and Raman couplings.

##  .2 Quench protocol for three quantized axes

To better understand quench protocol, we shall first present the basic idea of
quench dynamics that was described in the Ref. Sun  _et al._ (2018b); Yi  _et
al._ (2019). Our system is prepared initially in a topological trivial state
via controlling the detuning or the phase of the beams. Thereupon, the system
is quenched to the topological regime and the initial state evolves under the
post-quench Hamiltonian. Effectively, the quench projects the initial state to
the eigenstates of the post-quench Hamiltonian, which are the superposition
states of $\mid\uparrow\rangle$ and $\mid\downarrow\rangle$. Therefore, under
the post-quench Hamiltonian the state at each ${\bf q}$ oscillates between
$\mid\uparrow\rangle$ and $\mid\downarrow\rangle$. The oscillation frequency
is governed by the energy difference of eigen-bands. The quench dynamics is
similar to the Rabi oscillation of a two-level system in quantum optics.

In experiments, we quench three quantized axes of the effective Hamiltonian
$h_{{\rm F},i}$ ($i=x,y,z$) to explore and characterise the novel topological
phases Zhang  _et al._ (2018, 2020); Sun  _et al._ (2018b); Yi  _et al._
(2019). Quenching $h_{\text{F},z}$ is realised by fast switching the two-
photon detuning $\delta$, while quenching $h_{{\rm F},x}(h_{{\rm F},y})$ is
performed by suddenly tuning the phase of Raman couplings
$\varphi_{2}(\varphi_{1})$. For all the quenches, we divide them into three
steps: the preparation of the initial states, quenching to the objective
Hamiltonian and the detection of the states. Here, the objective Hamiltonian
Eq. S6 possesses the parameters
$V_{0}=4.0E_{\rm{r}},\Omega_{0}=1.0E_{\rm{r}}$. Quench protocol is performed
as following:

(1) _The preparation of the initial states._ The 87Rb atoms are prepared
slightly above the critical temperature of Bose-Einstein condensation.
Subsequently, they are adiabatically loaded into the Raman lattices in 100ms
with temperature of 100nK. Regarding quenching $h_{{\rm F},z}$, we set the
initial two-photon detuning $\delta=-200E_{\rm{r}}$ by tuning the frequency of
$\text{RF}_{1}$ and the phase $(\varphi_{1},\varphi_{2})=(\pi/2,\pi/2)$. Then,
$h_{x,y}$ are suppressed due to $|\delta|\gg t_{0,\rm{so}}$, such that the
initial state is polarized to $\mid\uparrow\rangle$ and points to the North
pole on the Bloch sphere Sun  _et al._ (2018b). Hence, the value of the spin
polarization is close to 1 at $t=0$ (See Fig. 1 and Fig S14). Regarding
quenching $h_{{\rm F},x}$ or $h_{{\rm F},y}$, we set the phase
$(\varphi_{1},\varphi_{2})=(0,\pi/2)$ or ($\pi/2,0$), meanwhile the detuning
$\delta$ is a constant by fixing the frequency of $\text{RF}_{1}$, i.e.,
$\delta=\delta_{0}$. Such settings make the Zeeman term constant $m_{x,y}\gg
t_{0,\rm{so}}^{x,y}$, which induces $h_{x}$ or $h_{y}$ dominates as well as
the initial state is prepared to be almost polarized in the $\sigma_{x}$
($\sigma_{y}$) direction Yi  _et al._ (2019). In other words, the initial
state points near the equator on the Bloch sphere. Thus, the value of the spin
polarization is close to 0 (see Fig. S14 and Fig. S6). Before all the
quenchings, the modulated RF is turned off. For all the quenchings, the
frequency of $\text{RF}_{2}$ is a constant and the phase $\varphi_{1,2}$ are
controlled by the $\text{EOM}_{1,2}$. Thus, the atoms populate mainly in the
lowest trivial band of the initial Hamiltonian.

(2) _Quench to the objective Hamiltonian._ For quenching $h_{{\rm F},z}$, the
initial detuning is switched to the final detuning
$\delta_{0}\in[-1,1]E_{\rm{r}}$ within 200ns by $\text{Switch}_{1}$. Such
final detuning is the same order of magnitude as $t_{0,\rm{so}}$, inducing
$h_{x}$ and $h_{y}$ take effect. For quenching $h_{{\rm F},x}$ or $h_{{\rm
F},y}$, the phase $(\varphi_{1},\varphi_{2})=(\pi/2,0)$ or $(0,\pi/2)$ is
switched to $(\varphi_{1},\varphi_{2})=(\pi/2,\pi/2)$ by tuning the
$\rm{EOM}_{1}$ or $\rm{EOM}_{2}$ within 2$\mu\text{s}$. And the Zeeman
constant is switched suddenly from a large $m_{x}$ or $m_{y}$ to $m_{x}=0$ or
$m_{y}=0$, inducing $h_{x,y,z}$ work. Subsequently, the detuning $\delta$ is
modulated for a certain time $t$, i.e.,
$\delta=\delta_{0}+A_{\rm{F}}\sin(\omega t)$. Hence, the atoms evolve in the
objective Hamiltonian.

(3) _The detection of the states._ We take off all the lasers in less than
1$\mu$s. Then the Stern-Gerlach magnetic field is turned on so that the atoms
in $\mid\uparrow\rangle$ and $\mid\downarrow\rangle$ are separated. After the
atoms expand freely for 25ms, we take photos of atoms to obtain the
distribution in $\mid\uparrow\rangle$ and $\mid\downarrow\rangle$ in momentum
space, $N_{\uparrow}(\bm{k},t)$ and $N_{\downarrow}(\bm{k},t)$. Finally, after
mapping $N_{\uparrow}(\bm{k},t)$ and $N_{\downarrow}(\bm{k},t)$ to the quasi-
momentum space Sun  _et al._ (2018b), atomic number $N_{\uparrow}(\bm{q},t)$
and $N_{\downarrow}(\bm{q},t)$ are obtained.

##  .3 Band inversion surfaces.

Band inversion surfaces (BISs) refer to the quasimomenta where the two spin
bands are inverted, and can be dynamically identified by resonant spin-
flipping oscillations Zhang  _et al._ (2018). For our periodically driven QAH
model, the BISs are all 1D rings that surround the $\Gamma$ or $\rm{M}$ point.
We have the following two properties about BISs to distinguish topological
phases of the present system: (P1) The two types of BISs ($0$\- or $\pi$-BIS)
always appear alternatively along the diagonal direction from $\Gamma$ to
$\rm{M}$. For example, in the case of $\mathcal{N}_{\rm{BIS}}=4$ in Fig.4 of
main text, the rings from the innermost to outermost are $0$-BIS, $\pi$-BIS,
$0$-BIS and $\pi$-BIS, respectively. (P2) Topological invariant
$\nu_{0,\pi}^{j}$ equals $+1$ (or $-1$) for a BIS surrounding the $\Gamma$ (or
$\rm{M}$) point.

Figure S2: The formation of Floquet bands. The dotted curves are the static
$|\\!\uparrow,0\rangle$ and $|\\!\downarrow,0\rangle$ bands, which are copied
and shifted by the periodic driving (dashes curves). The gap-opening at all
band crossings renders the Floquet bands (solid curves).

To demonstrate the two conclusions, we write Eq. (1) as
$\mathcal{H}(t)=\mathcal{H}_{0}+A_{\rm{F}}\sin(\omega t)\sigma_{z}/2$, and
consider the non-driven Hamiltonian
$\mathcal{H}_{0}=\sum_{j=x,y,z}h_{j}({\bm{q}})\sigma_{j}$. Before
demonstrating the two conclusions, let us explain the formation of the Floquet
bands. The spin up and spin down bands are respectively defined as
$(\mid\uparrow\rangle,0)$ and $(\mid\downarrow\rangle,0)$, given by
$h_{z}(\bm{q})$ and $-h_{z}(\bm{q})$, as shown in the first column of Fig. S2.
In the presence of periodic driving, the periodic driving transfers energies
$n\omega$ ($\omega$ is the driving frequency, $n=0,\pm 1,\dots$) to the non-
driven system, such that the static spin bands ($(\mid\uparrow\rangle,0)$ and
$(\mid\downarrow\rangle,0)$ bands) are copied and shifted in steps of
$\omega$. The copying and shifting lead to new band crossings, corresponding
to driving-induced BISs (see the second column of Fig. S2). A finite
modulation amplitude $A_{\rm{F}}$ and Raman coupling strength $\Omega_{0}$
then lift the degeneracy and open band gaps, rendering the Floquet bands shown
in the third column of Fig. S2. Since the quasi-energy is periodic with the
frequency $\omega$, the quasi-energy bands outside the FBZ are redundant
$2\pi$-copies. Similar to the Bloch bands in a lattice system, we only
consider the Floquet bands within the FBZ
$\varepsilon\in[-\omega/2,+\omega/2]$.

We now demonstrate the two conclusions. For the first conclusion, we note that
the emergence of all BISs comes from the crossings of the spin bands in the
Floquet Brillouin zone, determined by $h_{z}({\bm{q}})=n\omega/2$ with $n$
being a nonzero integer. $n$ being even (odd) corresponds to 0-BISs
($\pi$-BISs). In the absence of periodic driving, the BIS exists at the
crossing of $(\mid\uparrow\rangle,0)$ and $(\mid\downarrow\rangle,0)$ bands,
where $h_{z}(\bm{q})=0$. Between two adjacent $\pi$-BISs where
$h_{z}({\bm{q}})=(2m-1)\omega/2$ and $(2m+1)\omega/2$, there must exist one
$0$-BIS that corresponds to $h_{z}({\bm{q}})=m\omega$ ($m$ is a integer.),
which is ensured by the monotonicity of the spin band $h_{z}(|{\bm{q}}|)$.
(See Fig. 1(b) and Fig. 2 in the main text.)

The second conclusion can be obtained by the aid of topological charges. For
the static Hamiltonian $\mathcal{H}_{0}$, the topological charges are located
at ${\bm{q}}_{\rm c}$ with $h_{x}({\bm{q}}_{\rm c})=h_{y}({\bm{q}}_{\rm
c})=0$, and the topological invariant defined on a BIS reflects the total
charges enclosed Zhang  _et al._ (2018); Yi  _et al._ (2019). For the non-
driven QAH model, The topological charges are located at four highly symmetric
momentum points: $\Gamma$ (charge value ${\cal C}_{\Gamma}=+1$), $\rm{M}$
(${\cal C}_{\rm M}=+1$) and ${\rm X}_{1,2}$ (${\cal C}_{{\rm X}_{1,2}}=-1$)
Zhang  _et al._ (2018); Yi  _et al._ (2019), as shown in Fig. S3a. When the
periodic driving is added to the $h_{z}$ term of $\mathcal{H}_{0}$, it does
not change the locations and values of the four topological charges (Fig.
S3b). Hence, as in the non-driven case, the BIS surrounding the $\Gamma$ point
has the topological invariant $\nu_{0,\pi}^{j}={\cal C}_{\Gamma}=+1$, while
the BIS circling the $\rm{M}$ point has $\nu_{0,\pi}^{j}={\cal
C}_{\Gamma}+{\cal C}_{{\rm X}_{1}}+{\cal C}_{{\rm X}_{2}}=-1$.

Figure S3: Correspondence between BISs and edge states. BISs (upper) and
energy spectrum under open boundary conditions (lower) for $\textbf{a}.$
$\delta_{0}=0.4E_{\text{r}}$ (left) and $\delta_{0}=-0.4E_{\text{r}}$ (right)
without driving; and $\textbf{b}.$
$\delta_{0}=0.4E_{\text{r}},T=400\mu\text{s}$.

##  .4 BIS-boundary correspondence.

Here we demonstrate the property (P3): one-to-one correspondence between BISs
and edge states for the 2D periodically driven QAH model under open boundary
conditions (OBCs). The proof consists of two steps: (i) This one-to-one
correspondence holds for the non-driven QAH system (Fig. S3a). (ii) For the
driven model, the Floquet band structure can be regarded as a combination of
several static bands with different parameters (Fig. S3b).

The demonstration of Step (i) is straightforward. According to Refs. Zhang
_et al._ (2018), the so-called bulk-surface duality reveals a correspondence
between the bulk topology and the winding defined on BISs. For the 2D QAH
model, the Chern number ${\cal C}=\pm 1$ is characterised by the topological
invariant $\nu=\pm 1$ defined on the BIS surrounding the $\Gamma$ (for ${\cal
C}=+1$) or $\rm{M}$ (${\cal C}=-1$) point. Besides, the bulk-edge
correspondence obviously holds. We then obtain a one-to-one correspondence
between the topological invariant on BIS and the chirality of chiral edge
states.

For Step (ii), we examine an approximate expression for the effective
Hamiltonian by applying the Floquet-Magnus expansion. Consider a BIS formed by
two shifted bands $(\left|\downarrow\right\rangle,m_{1}\omega)$ and
$(\left|\uparrow\right\rangle,m_{2}\omega)$ with $\zeta\equiv m_{1}-m_{2}$. We
apply a rotation ${\cal O}(t)=\exp(i\zeta\omega t\sigma_{z}/2)$ to the time-
dependent Hamiltonian ${\cal H}(t)$, and write ${\cal H}_{\rm rot}(t)={\cal
O}(t){\cal H}(t){\cal O}^{\dagger}(t)=\sum_{-\infty}^{\infty}H_{n}e^{in\omega
t}$. According to the Floquet-Magnus expansion Eckardt (2017), the effective
Hamiltonian near this BIS can be obtained by ${\cal H}_{\rm F}=\sum_{n\neq
0}(H_{n}H_{-n}+[H_{0},H_{n}])/n\omega$, which yields $h_{{\rm
F},x/y}\approx(-1)^{m}J_{m}\left(\frac{4A_{\rm{F}}}{\omega}\right)h_{x/y}\propto
h_{x/y}$ with the Bessel function $J_{m}(z)$ and $h_{{\rm F},z}\approx
h_{z}-\zeta\omega/2$ for the driven QAH model. Hence, the local band structure
near a driving-induced BIS can be approximated as the static one with a
shifted Zeeman constant $m_{\rm eff}=m_{z}-\zeta\omega/2$.

With the two results (i) and (ii), we finally come to the conclusion that for
our periodically driven QAH model with OBCs, a 0-BIS (or $\pi$-BIS) indicates
a chiral edge state within the 0-gap (or $\pi$-gap). For example, for the
high-Chern-number phase in Fig. 3, the 0-BIS around the $\Gamma$ point
corresponds to a chiral edge state in the 0-gap near $q_{y}=0$, while the
$\pi$-BIS around the $\rm M$ point corresponds to a chiral edge state in the
$\pi$-gap near $\left|q_{y}\right|=k_{0}$. The fact that the BIS surrounds the
$\Gamma$ or $\rm M$ point determines the chirality of the corresponding edge
state.

##  .5 Robustness of counterpropagating edge states.

We shall first consider the ideal tight-binding QAH model and numerically
verify that counterpropagating edge states are robust against smooth
impurities. We then argue that the conclusion also holds for our cold atom
realisation beyond the tight-binding limit.

For the periodically driven QAH model, the predicted counterpropagating edge
states are protected by both the particle-hole symmetry (PHS) and large-
momentum separation. The PHS ${\cal P}={\cal K}P$, where ${\cal K}$ is the
complex conjugation operator and $P$ is unitary, acting on the Hamiltonian
gives $P{\cal H}({\bm{q}},t)P^{-1}=-{\cal H}^{*}(-{\bm{q}},t)$ Roy and Harper
(2017); Yao  _et al._ (2017), which leads to
$\varepsilon({\bm{q}})=-\varepsilon(-{\bm{q}})$. Thus, the zero-energy edge
states can only appear at $q_{y}=0$ or $\pi$ under an OBC in
$\hat{x}$-direction. The driven QAH model has a PHS $P=\sigma_{x}$, and the
edge states with opposite chirality are located around $q_{y}=0$ and $\pi$,
respectively. Similar to the quantum valley-Hall effect (see, e.g., Ref. Ren
_et al._ (2016) and references therein), such a large momentum separation
protects the counterpropagating edge states from long-range disorders.

Figure S4: Quasienergy-dependent conductance $G_{0,\pi}$ in the presence of
two kinds of disorders. Here we take the Hamiltonian as the QAH tight-binding
model Eq. (1) with $\delta_{0}=6t_{0}$, $t_{\rm so}=t_{0}$, and the driving
frequency $\omega=4t_{0}$. The lattice size is set as $40\times 40$. Each
point is averaged over 50 disorder configurations.

To examine the disorder effects, we employ scattering matrix invariants to
characterise the topological phase Fulga and Maksymenko (2016); Umer  _et al._
(2020). The quasienergy-dependent Floquet scattering matrix can be expressed
as

$\displaystyle S(\varepsilon)={\sf P}\left[e^{-i\varepsilon}{\bm{1}}-U(1-{\sf
P}^{\rm T}{\sf P})\right]^{-1}U{\sf P}^{\rm T},$

where ${\sf P}$ is the projector matrix (see Ref. Fulga and Maksymenko (2016)
for the definition), the superscript ${\rm T}$ denotes the matrix transpose,
and $U$ is the time-evolution operator under open boundary conditions. The
scattering matrix takes the form

$\displaystyle S(\varepsilon)=\left(\begin{array}[]{cc}r&t\\\
t^{\prime}&r^{\prime}\\\ \end{array}\right).$

From this one can compute the conductance $G_{0,\pi}={\rm Tr}(tt^{\dagger})$,
which characterises the number of edge states within the corresponding
quasienergy gap. We consider two kinds of PHS-preserving disorders applied to
the Zeeman term, i.e., $m_{z}\to m_{z}+V_{\rm disorder}$. The first one is the
random on-site disorder

$\displaystyle V_{\rm
rand}({\bm{r}})=\sum_{j}V_{j}\delta({\bm{r}}-{\bm{r}}_{j}),$

and the second one represents a long-range potential induced by smooth
impurities

$\displaystyle V_{\rm imp}({\bm{r}})=\sum_{l}^{N_{\rm
imp}}\frac{V_{l}}{\sqrt{({\bm{r}}-{\bm{r}}_{l})^{2}+d^{2}}},$

where $N_{\rm imp}$ denotes the number of the randomly distributed impurities
and $d$ controls the potential range. Here we set $d=1$ and take $V_{\rm
rand},V_{\rm imp}\in[-V_{\rm amp},V_{\rm amp}]$, with $V_{\rm amp}$ denoting
the strength of disorders. The numerical results are shown in Fig. S4 for
disorder effects in the phase with ${\cal N}_{\rm BIS}=4$. Without disorders,
two $0$-BIS (or $\pi$-BISs) with opposite topological invariants $\pm 1$
ensure a pair of counterpropagating edge states within the 0-gap (or
$\pi$-gap), due to the correspondence between BISs and edge states. In the
presence of on-site disorders $V_{\rm rand}({\bm{r}})$, our numerical results
show that $G_{0,\pi}$ deviate from their quantized values 2 as the disorder
strength $V_{\rm amp}$ increases, indicating that the edge states with
opposite chirality are hybridized with each other by disorders. In comparison,
the computed values of $G_{0,\pi}$ are almost unaffected by $V_{\rm
imp}({\bm{r}})$, which verifies that the counterpropagating edge states are
immune against long-range disorders.

The above conclusion on the robustness of counterpropagating edge states is
not restricted to the ideal QAH model. For our realistic system beyond the
tight-binding limit, the PHS is broken by the next-nearest-neighbor hopping.
However, the correspondence between BISs and edge states still holds, and each
gapless edge state is protected by a valley-like local topological structure
formed by a BIS Zhang and Liu (2021). Hence, the large momentum separation
between two “valleys” also protects the counterpropagating edge states against
smooth impurities.

Figure S5: The spin textures $\left\langle\sigma_{z}(\bm{q})\right\rangle$ for
different driving period $T$. $\Gamma$ and M are high symmetric momenta. The
black and green dashed curves denote 0-BIS and $\pi$-BIS, respectively. The
red (blue) colour in the spin textures denotes spin up (down). The parameters
$(V_{0},\leavevmode\nobreak\ \Omega_{0},\leavevmode\nobreak\
\delta_{0},A_{\text{F}})=(4.0,\leavevmode\nobreak\ 1.0,\leavevmode\nobreak\
0.1,\leavevmode\nobreak\ 0.8)E_{\text{r}}$.

##  .6  Obtainment of the time-averaged spin textures

To characterise the topology of driven QAH model, the critical step is to
obtain the time-averaged spin textures
$\overline{\left\langle\sigma_{z}(\bm{q},t)\right\rangle}_{x,y,z}$. To this
end, we fit the time-evolved spin textures
$\left\langle\sigma_{z}(\bm{q},t)\right\rangle_{x,y,z}$ at each quasi-momentum
point $\bm{q}$ by the combination function (also see Ref.Yi  _et al._ (2019))

$\displaystyle
P_{z}(\bm{q},t)=\sum_{i=1}^{2}A_{i}(\bm{q})\cos(2\pi\nu_{i}(\bm{q})t+\varphi_{i})e^{-\frac{t}{\tau_{1}}}+B(\bm{q})e^{-\frac{t}{\tau_{2}}}+D(\bm{q}),$
(S12)

where the $A_{i}(\bm{q})$ term represents the damped oscillations with
characteristic frequency $\nu_{i}$ and damping time $\tau_{1}$, the
$B(\bm{q})$ term represents the pure decay with characteristic time
$\tau_{2}$, and the $D(\bm{q})$ term is the offset. The damping and the decay
are induced by the noise of magnetic field and the interaction between the
atoms. Then we remove these relaxation effects and calculate the average of
spin polarization by

Figure S6: a. Time-evolved spin textures for quenching $h_{{\rm F},x}$. b.
Time-evolved spin polarisation for quenching $h_{{\rm F},x}$ (upper) and
$h_{{\rm F},z}$ (lower). The green circles with error bars are experimental
data. The red curves are the fitting curves. The insets are the time-averaged
spin textures $\overline{\left\langle\sigma_{z}(\bm{q},t)\right\rangle}_{x}$
($\overline{\left\langle\sigma_{z}(\bm{q},t)\right\rangle}_{z}$), on which the
quasi-momentum points $\bm{q}$ are marked by green triangles. Parameters are
$(V_{0},\leavevmode\nobreak\ \Omega_{0},\leavevmode\nobreak\
\delta_{0},\leavevmode\nobreak\ A_{\rm F})=(4.0,\leavevmode\nobreak\
1.0,\leavevmode\nobreak\ 0.4,\leavevmode\nobreak\ 0.8)E_{\text{r}}$ and
$T=400\mu\text{s}$.

$\displaystyle\overline{\langle\sigma_{z}(\bm{q})\rangle}_{x,y,z}=\frac{1}{NT}\sum_{t=0}^{NT}\left(\sum_{i=1}^{2}A_{i}(\bm{q})\cos(2\pi\nu_{i}(\bm{q})t+\varphi_{i})\right)+B(\bm{q})+D(\bm{q})$
(S13)

where $T$ and $N$ are the driving period and the integer number, respectively.
For example, for quenching $h_{{\rm F},x}$ (time-evolved spin textures shown
in Fig. S6a), the time-evolved spin polarisation at $\bm{q}=(-0.70,0.51)k_{0}$
is shown in Fig. S6b (upper). According to the fitting parameters and setting
$NT=107\times 280\mu s$, we have
$\overline{\langle\sigma_{z}(\bm{q}=(-0.70,0.51)k_{0})\rangle}_{x}\approx
0.16$. Similarly, for quenching $h_{{\rm F},z}$, we have
$\overline{\langle\sigma_{z}(\bm{q}=(-0.43,-0.19)k_{0})\rangle}_{z}\approx-0.12$
(Fig. S6b (lower)).

##  .7 The sign of $h_{\text{F},z}$

In Fig. 3a of the main text, the two BISs divide the first Brillouin zone into
three regions. The sign of $h_{\text{F},z}$ in each region can be determined
by the formation of the Floquet bands. Consider the static Hamiltonian
$\mathcal{H}_{0}=\sum_{j=x,y,z}h_{j}({\bm{q}})\sigma_{j}$, of which the
$\mid\uparrow\rangle$ and $\mid\downarrow\rangle$ bands cross each other at
the momenta where $h_{z}(\bm{q})=0$. when periodic driving is added to
$\mathcal{H}_{0}$, the spin bands are copied and shifted by $n\omega$ ($n$ is
an integer), forming band crossings where $h_{z}({\bm{q}})=n\omega/2$. The
finite Raman coupling strength $\Omega_{0}$ and driving amplitude $A_{\rm{F}}$
open gaps at these crossings, so that band inversion surfaces (BISs) emerge,
and the spin bands folded in the Floquet Brillouin zone form the Floquet bands
$h_{\text{F},z}$ (Fig. 2). Thus, the sign of $h_{\text{F},z}$ , which is the
same as the sign of spin polarisation $-\left\langle\sigma_{z}\right\rangle$,
must be opposite on both sides of the BIS (Fig. S5). Therefore, in Fig. 3a of
the main text, the sign of $h_{\text{F},z}$ in the three regions from $\Gamma$
to M are ”-”, ”+” and ”-”, respectively.

##  .8 The dynamical field and the topological invariant $\nu_{0,\pi}^{j}$

Figure S7: The obtainment of $\nu_{0,\pi}^{j}$.

We obtain $\nu^{j}_{0,\pi}$ from the following steps:

(i) Constructing the dynamical field from time-averaged spin textures.
According to Ref. [23], the dynamical field $\bm{g}(\bm{q})=(g_{y},g_{x})$ is
defined with two components $g_{x,y}(\bm{q})\equiv-\frac{1}{{\cal
N}_{\bm{q}}}\partial_{q_{\perp}}\overline{\langle\sigma_{z}(\bm{q})\rangle}_{x,y}$,
where $\mathcal{N}_{\bm{q}}$ is a normalization factor and $q_{\perp}$ denotes
the momentum perpendicular to the BIS and points from “$-$” to “$+$”. Here the
sign “$+$” (“$-$”) denotes the region where $h_{{\rm F},z}>0$ ($h_{{\rm
F},z}<0$), which can be judged by the Floquet band structure and the
properties of band inversion (see the Supplemental Materials for details).
From the definition of $g_{x,y}(\bm{q})$, one can see that the direction of
the dynamical field is determined by the difference of the measured spin
polarization between the two regions divided by the BIS. Due to the minus sign
in the definition, we compute the directional derivative across the BIS by
subtracting the measured spin polarization of the “$+$” region from the
measured spin polarization of the “$-$” region. Take Fig. 3(a) in the main
text as an example. The region B is the “$+$” region and A is the “$-$”
region. The polarization difference across the BIS is thus $P(A)-P(B)<0$,
where $P(A)$ [$P(B)$] denotes the measured spin polarization in region $A$
($B$). Hence, we have $g_{x}(\bm{q})<0$ near the region $A$ and $B$, which
means that the direction of the $x$-component of the dynamical field should be
in the opposite direction of $\sigma_{x}$-axis. Like this, the components of
the dynamical field $g_{x,y}$ are obtained and drawn in the first two columns
of Fig. 3 in the main text. Thus, superimposing the two components $g_{x,y}$,
the combined dynamical field $\bm{g}$ is obtained, i.e.,
$\bm{g}(\bm{q})=g_{x}(\bm{q})\hat{e}_{x}+g_{y}(\bm{q})\hat{e}_{y}$, as shown
in the three column of Fig. 3 in the main text.

(ii) Determining the topological invariant from the dynamical field. According
to Ref. Zhang  _et al._ (2020), the topological invariant associated with the
$j$-th BIS is defined as $\nu^{j}_{0,\pi}=\int_{\rm{BIS}}\bm{g}(\bm{q}){\rm
d}\bm{g}(\bm{q})/2\pi$, where ${\bf g}{\rm d}{\bf g}\equiv g_{y}{\rm
d}g_{x}-g_{x}{\rm d}g_{y}$ and “${\rm d}$” denotes the exterior derivative,
which characterizes the winding number of the dynamical field $\bm{g}(\bm{q})$
on BISs. The expression also tells that the winding orientation of the
dynamical field along the BIS determines whether the associated topological
number is positive or negative. As shown in Fig. S7, we need to not only count
how many times these arrows rotate around along each BIS, but also examine the
direction of their rotation (clockwise or anticlockwise). Note that the
winding orientation should be identified with respect to the same reference
point. Here we take the $\Gamma$ point as the reference point, and first
consider the BIS surrounding $\Gamma$ (black circle). When tracing out the
black circle anticlockwise around $\Gamma$, one can find that the dynamical
field on the BIS winds once also in an anticlockwise direction, which
indicates that the associated topological number should be $\nu^{1}_{0}=+1$.
In contrast, for the BIS surrounding the $M$ point (green circle), the
situation is reversed. When we trace out the green circle anticlockwise with
respect to $\Gamma$, we actually make a circle that travels clockwise around M
(the upper right in Fig. S7). Therefore, the winding orientation of the
dynamical field also becomes clockwise, giving the topological number
$\nu^{1}_{\pi}=-1$.

Based on above descriptions, one sees that the experimental data are used to
determine both the sign of $\nu^{j}_{0,\pi}$ and also its magnitude. Since it
characterizes how many times the dynamical field winds along the BIS (Here,
selecting the same momentum point for the start and end point on the dynamical
field to obtain $\nu^{j}_{0,\pi}$.), the values of $\nu^{j}_{0,\pi}$ must be
quantized without error bars.

##  .9 Four BISs in momentum space

For quenching $h_{\text{F},z}$, the atoms are prepared in the initial
polarized state $\mid\uparrow\rangle$. After quenching $h_{\text{F},z}$, the
spin flips appear and the atoms flip to $\mid\downarrow\rangle$. Those atoms
flipping to $\mid\downarrow\rangle$ form ring structures surrounding $\Gamma$
or M, which is the signature of the BISs. The experimental data with four BISs
in momentum space are shown in Fig. S8. From the momentum distribution of
$\mid\downarrow\rangle$ $N_{\downarrow}(\bm{k},t)$, four rings are clearly
observed.

Figure S8: $N_{\uparrow}(\bm{k},t)$ and $N_{\downarrow}(\bm{k},t)$ in momentum
space. The upper (lower) row corresponds to four BISs data of Fig. 2 (Fig. 4)
before mapping to quasi-momentum space with $t=2T$. The parameters of upper
(lower) row is
$V_{0}=4.0E_{\rm{r}},\Omega_{0}=1.0E_{\rm{r}},\delta_{0}=0.1E_{\rm{r}},T=740\mu
s,A_{\rm{F}}=1.2E_{\rm{r}}$
($V_{0}=4.0E_{\rm{r}},\Omega_{0}=1.0E_{\rm{r}},\delta_{0}=-0.2E_{\rm{r}},T=600\mu
s,A_{\rm{F}}=1.2E_{\rm{r}}$).

##  .10 Determination of the topological phase boundary

Figure S9: The extraction of the size $r$. The spin texture is obtained via
rotating the original spin texture by $45^{\circ}$. The right and top
subfigures are obtained by extracting the spin polarization along the diagonal
and anti-diagonal direction of quasi-momentum $q_{\rm{diag}}$, respectively.
The green dots are experimental data and the solid curves are the fitting by
the numerical smoothing. The sizes $r_{1,2}$ are obtained by measuring the
distance from $\Gamma$ point to the local minimal value of $P_{\rm{m}}$. The
parameters:
$V_{0}=4.0E_{\rm{r}},\Omega_{0}=1.0E_{\rm{r}},\delta_{0}=0.4E_{\rm{r}},T=400\mu
s,A_{\rm{F}}=0.8E_{\rm{r}},t=3T$.

According to the theoretical analysis, BIS is the essence of the topological
phases in the driven QAH model Zhang  _et al._ (2020). The BISs vanish for the
trivial phases, while they exist for the topological phases. Moreover, the
size of BISs change continuously when one of the parameters $V_{0}$,
$\Omega_{0}$, $T$ and $\delta_{0}$ is tuned. Thus, via tuning $T$ and
$\delta_{0}$ as well as fixing $V_{0}$ and $\Omega_{0}$, the topological phase
boundaries in Fig. 4(a) in the main text can be determined where the
corresponding BIS just appears. Phase boundaries with black straight lines in
Fig. 4(a) are identified by measuring the size of BISs as a function of the
detuning $\delta_{0}$, while other phase boundaries are extracted by measuring
the size of BISs as a function of the period $T$. Here, the size of the ring
$r$ is defined as the distance between $\Gamma$ point and the ring.

The size $r$ is extracted by locating the local minimal of the spin
polarization along the diagonal and anti-diagonal direction of the quasi-
momentum $q_{\rm{diag}}$. The procedure of the extraction is the following: I.
Extracting the spin polarization along the diagonal and anti-diagonal
direction of the quasi-momentum $q_{\rm{diag}}$, as shown in Fig. S9. II.
Obtaining four $r_{1}$ and four $r_{2}$ by measuring the distance between the
quasi-momentum point (where the spin polarization $P_{\rm{m}}$ is the local
minimal value) and $\Gamma$ point. III. Repeating the steps I and II to obtain
the expected values and the statistic errors of the sizes for multiple spin
textures with the same parameters. IV. Adjusting the modulation period $T$ or
the detuning $\delta_{0}$, the sizes are extracted, as shown in Fig. S10.

Figure S10: Topological phase boundary. a. Topological phase boundaries for
$(V_{0},\Omega_{0})=(4.0,1.0)E_{\text{r}}$, $\delta_{0}=0.1E_{\text{r}}$
(left) and $\delta_{0}=0.4E_{\text{r}}$ (right). Green circles denote the size
of the ring $r_{1}$ corresponding to the BIS of static QAH bands; purple
triangles and black diamonds represent the sizes of rings $r_{2}$ and $r_{3}$
induced by periodic driving, which are fitted exponentially. The fitting
curves intersect the straight lines $(r=0)$ or $(r=1)$, which gives the
location of phase boundaries (small blocks). Insets are the representative
spin textures with different number of rings. b. Topological phase boundaries
for $(V_{0},\Omega_{0})=(4.0,1.0)E_{\text{r}}$ and $T=260\mu\text{s}$.

As the driving period $T$ increases, there emerge multiple rings in the time-
evolved spin textures (Fig. S10a). The first ring corresponds to BIS formed by
the spin-orbit coupling of static QAH model, of which the size $r_{1}$ is
constant. Other rings correspond to the BISs induced by the driving, of which
the sizes $r_{2,3}$ vary. We fit $r_{2,3}$ with an exponential function

$\displaystyle r=Ae^{-\frac{T}{T_{0}}}+r_{0},$ (S14)

where $r_{0}$ and $T_{0}$ are the fitting parameters. The topological boundary
is the intersection between the fitting curve and the straight line $r=0$ or
$r=1$. $r=0$ ($r=1$) represents the smallest (largest) size of the ring where
the ring just emerges. When the detuning $\delta_{0}=0.1(0.4)E_{\rm{r}}$, The
topological boundaries are $T=(293\pm 4)\mu\text{s}$ ($T=(220\pm
4)\mu\text{s}$) for the number of rings from one to two and $T=(396\pm
8)\mu\text{s}$ ($T=(470\pm 3)\mu\text{s}$) for the number of rings from two to
three.

On the other hand, as the detuning $\delta_{0}$ increases, the size of the
ring $r_{1}$ in the time-evolved spin textures decreases (Fig. S10b). To
obtain the topological boundaries with black straight lines in Fig. 4a, we fit
the size $r_{1}$ by a polynomial function Sun  _et al._ (2018b)

$\displaystyle r=A+B\delta_{0}+C\delta_{0}^{3},$ (S15)

where $A$, $B$ and $C$ are the fitting parameters. The topological boundary is
also the intersection between the fitting curve and the straight line $r=0$ or
$r=1$. Thus, when $T=260\mu\text{s}$, the topological boundaries are
$\delta_{0}=(0.8\pm 0.03)E_{\rm{r}}$ and $\delta_{0}=(-0.79\pm
0.01)E_{\rm{r}}$ (Fig. S10b).

Figure S11: a. Topological phase diagram for
$(V_{0},\Omega_{0})=(4.0,1.0)E_{\text{r}}$. The blue circles with error bars
are experimental measurements. The solid or dashed curves of numerical
calculations denote phase boundaries on which a 0-BIS or a $\pi$-BIS appears,
respectively. The Roman numerals i to xii label twelve topological phases in
the experimental parameters. b. The spin textures
$\left\langle\sigma_{z}\right\rangle_{z}$ for each phase marked by ✩ in a. The
dashed curves with different colours distinguish the twelve phases.
$t=(3,4,3,2,3,2,3,2,3,3,5,3)T$ from the phase i to xii.
$A_{\rm{r}}=0.9E_{\rm{r}}$ for the phase vi and $A_{\rm{r}}=0.8E_{\rm{r}}$ for
other phases. c. Topological invariants of the twelve phases.

##  .11 The topological phase diagram

Based on the method of determined the phase boundaries descried in section
I.C, we summarize the topological phase diagram of driven QAH model in Fig.
S11a, part of which has been shown in Fig.4 in the main text. We can reach
twelve phases by varying driving period $T$ and detuning $\delta_{0}$. The
typical spin textures $\left\langle\sigma_{z}\right\rangle_{z}$ for each phase
are shown in Fig. S11b.

Further, we demonstrate the topological invariants of the twelve phases in
Fig. S11c, including the total number of BISs $\mathcal{N}_{\rm BIS}$, and the
winding number of 0-gap ($\pi$-gap), i.e., $\mathcal{W}_{0}$
$(\mathcal{W}_{\pi})$. And the Chern number is obtained by
$\mathcal{C}=\mathcal{W}_{0}-\mathcal{W}_{\pi}$. The phases (i-v) can be
transformed into ($\text{i}^{\prime}-\text{v}^{\prime}$) by changing the sign
of the winding number $\mathcal{W}_{0}$ or $\mathcal{W}_{\pi}$. And the phases
in $\delta_{0}>0$ plane and $\delta_{0}<0$ plane can be also converted to each
other. For example, the winding number
$(\mathcal{W}_{0},\mathcal{W}_{\pi})=(+1,-1)$ of the phase (iii) with
$\delta_{0}<0$ can be transfers into
$(\mathcal{W}_{0},\mathcal{W}_{\pi})=(-1,+1)$ of the phase
($\text{iii}^{\prime}$) with $\delta_{0}>0$.

Figure S12: (a) Energy gaps to the first excited bands compared to the
modulation frequency. The red and green solid curves are respectively the
upper and lower bounds of the gaps between the first two excited bands
($p_{1,2}$) and the lowest two bands ($s_{1,2}$). The dots represent the
parameter settings in our measurents shown in Fig. 4(a) of the main text.
Inset: The measured spin texture with the temperature of 100nK. Here the
parameters are $\delta_{0}=-0.9E_{\rm{r}}$, the modulation period $T=200\mu
s$, and the amplitude $A_{\text{F}}=0.8E_{\rm{r}}$. The measurement is
performed at time $t=4T$. (b) The calculated band structure of our 2D Raman
lattice with two lowest bands $s_{1,2}$ and two excited bands $p_{1,2}$.

In addition, the energy gaps to the first excited bands are shown in Fig. S12.
The selected experimental parameters are mostly beneath the lower bound (green
curve) of the gaps between the first two excited bands ($p_{1,2}$) and the
lowest two bands ($s_{1,2}$), except one case with
$|\delta_{0}|=0.9E_{\text{r}}$, which locates slightly above the lower bound.
For $\delta_{0}=-0.9E_{\text{r}}$, only the BIS surrounding $\Gamma$ point
appears. The higher band effect only appears near the M point, which, however,
does not affect the extraction of the information of the BIS near the $\Gamma$
point, see the inset in Fig. S12(a). For $\delta_{0}=0.9E_{\text{r}}$, the
case is similar to $\delta_{0}=-0.9E_{\text{r}}$. Thus, the full phase diagram
was mapped.

Figure S13: The Floquet bands with different driving amplitude $A_{\rm{F}}$.
Parameters are $(V_{0},\leavevmode\nobreak\ \Omega_{0},\leavevmode\nobreak\
\delta_{0})=(4.0,\leavevmode\nobreak\ 1.0,\leavevmode\nobreak\
0.4)E_{\text{r}}$ and $T=400\mu\text{s}$. The red (blue) colour in the spin
textures denotes spin up (down).

##  .12  Numerical simulations for quenching three quantized axes

For periodically driven QAH model, the effective Hamiltonian
$\mathcal{H}_{\rm{F}}$ can be written as

$\mathcal{H}_{\rm{F}}=i\log(U(T))/T,$ (S16)

where time-evolution operator $U(t)={\cal
T}\exp\big{[}-i\int_{0}^{t}\mathcal{H}(\bm{q},\tau)d\tau\big{]}$ and ${\cal
T}$ denotes the time ordering. By diagonalizing Hamiltonian Eq. S16 we can
obtain the Floquet bands and the eigenstate $|\Psi_{0}(\bm{q})\rangle$. The
Floquet bands with different driving amplitude $A_{\rm{F}}$ are shown in Fig.
S13, manifesting that $A_{\rm{F}}$ only affects the band gap. According to the
eigenstate, the spin textures $\langle\sigma_{x,y,z}(\bm{q})\rangle$ can be
obtained by
$\langle\sigma_{x,y,z}\rangle=\langle\Psi_{0}(\bm{q})|\sigma_{x,y,z}|\Psi_{0}(\bm{q})\rangle$.
$\langle\sigma_{z}\rangle$ with different modulation periods $T$ are shown in
Fig. S5.

We calculate numerically the time-evolved spin texture
$\langle\sigma_{z}(\bm{q},t)\rangle_{i}$ along $\sigma_{z}$ axis by
$\langle\sigma_{z}(\bm{q},t)\rangle_{i}=\langle\Psi(\bm{q},t)|\sigma_{z}|\Psi(\bm{q},t)\rangle=[N_{\uparrow}(\bm{q},t)-N_{\downarrow}(\bm{q},t)]/[N_{\uparrow}(\bm{q},t)+N_{\downarrow}(\bm{q},t)]$
with
$\left|\Psi(\bm{q},t)\right\rangle=U(T)\left|\Psi_{\text{i}}\right\rangle$.
The initial state $\left|\Psi_{\text{i}}\right\rangle$ is set by the
experimental conditions: For quenching $h_{\text{F},x}$,
$m_{x}\approx-14t_{0}$; For quenching $h_{\text{F},y}$,
$m_{y}\approx-14t_{0}$; For quenching $h_{\text{F},z}$,
$\delta_{0}\approx-50t_{0}$. The time-evolved spin textures from numerical
calculations are shown in Fig. S14. Regarding the quenching $h_{{\rm F},z}$,
the spins are polarized to 1 initially since the initial detuning is far away
from the resonance. As the time $t$ increases, there emerge two rings in the
spin textures due to spin flip from +1 to -1, which is the signature of BISs.
Hence, we can directly observe the rings to identify the BISs. Regarding to
quenching $h_{{\rm F},x}$, the initial spin polarization is not equal to zero
because $m_{x}$ is not infinite relative to $t_{0}$. With the time $t$
increases, spiral patterns form in the spin textures, which is similar to
quenching $h_{\text{F},y}$.

Then, we show numerically the time-averaged spin texture
$\overline{\langle\sigma_{z}(\bm{q})\rangle}_{x,y,z}$ in Fig. 3 (a) of the
main text by calculating the time average of
$\langle\sigma_{z}(\bm{q},t)\rangle_{x,y,z}$. According to Refs. Zhang  _et
al._ (2020); Yi  _et al._ (2019), the time-averaged spin texture
$\overline{\langle\sigma_{z}(\bm{q})\rangle}_{i}$ involves the information of
not only $h_{{\rm F},z}$ but also $h_{{\rm F},x}$ or $h_{{\rm F},y}$. In fact,
analytical calculations prove that for a fully polarized initial state (i.e.,
the spins are initially prepared to be polarized in the $\sigma_{i}$ axis for
quenching $h_{{\rm F},i}$), we have
$\overline{\langle\sigma_{z}(\bm{q})\rangle}_{i}\propto-h_{{\rm
F},z}(\bm{q})h_{{\rm F},i}(\bm{q})$. That is why we use such quench processes
to measure the effective spin-orbit coupling field. For example in left panel
of Fig. 3(a) in the main text, we should have
$\overline{\langle\sigma_{z}(\bm{q})\rangle}_{x}\propto-h_{{\rm F},z}h_{{\rm
F},x}$ under ideal conditions. However, the results numberically finally looks
a bit different because the prepared initial state cannot be fully polarized,
which does not affect obtaining the dynamical field.

Figure S14: Evolution of the spin textures
$\left\langle\sigma_{z}(\bm{q},t)\right\rangle$ for quenching $h_{{\rm F},z}$
(upper) and $h_{{\rm F},x}$ (lower). Parameters are
$(V_{0},\leavevmode\nobreak\ \Omega_{0},\leavevmode\nobreak\
\delta_{0},A_{\text{F}})=(4.0,\leavevmode\nobreak\ 1.0,\leavevmode\nobreak\
0.4,\leavevmode\nobreak\ 0.8)E_{\text{r}},T=400\mu\text{s}$. $\Gamma$ and M
are high symmetric momenta. The red (blue) colour in the spin textures denotes
spin up (down).