††thanks: These authors contribute equally to this work.††thanks: These
authors contribute equally to this work.

# Extracting the Quantum Geometric Tensor of an Optical Raman Lattice by Bloch
State Tomography

Chang-Rui Yi  Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  Shanghai 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
Hefei National Laboratory, University of Science and Technology of China,
Hefei 230088, China  Jinlong Yu  Center for Theoretical Physics, Hainan
University, Haikou 570228, China  School of Science, Hainan University, Haikou
570228, China  Huan Yuan  Hefei National Research Center for Physical Sciences
at the Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  Shanghai 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
Hefei National Laboratory, University of Science and Technology of China,
Hefei 230088, China  Rui-Heng Jiao  Hefei National Research Center for
Physical Sciences at the Microscale and School of Physical Sciences,
University of Science and Technology of China, Hefei 230026, China  Shanghai
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  Hefei National Laboratory, University of
Science and Technology of China, Hefei 230088, China  Yu-Meng Yang  Xiao Jiang
Jin-Yi Zhang  Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  Shanghai 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
Hefei National Laboratory, University of Science and Technology of China,
Hefei 230088, China  Shuai Chen  Hefei National Research Center for Physical
Sciences at the Microscale and School of Physical Sciences, University of
Science and Technology of China, Hefei 230026, China  Shanghai 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  Hefei National Laboratory, University of Science and Technology
of China, Hefei 230088, China  Jian-Wei Pan  Hefei National Research Center
for Physical Sciences at the Microscale and School of Physical Sciences,
University of Science and Technology of China, Hefei 230026, China  Shanghai
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  Hefei National Laboratory, University of
Science and Technology of China, Hefei 230088, China

###### Abstract

In Hilbert space, the geometry of the quantum state is identified by the
quantum geometric tensor (QGT), whose imaginary part is the Berry curvature
and real part is the quantum metric tensor. Here, we experimentally realize a
complete Bloch state tomography to directly measure eigenfunction of an
optical Raman lattice for ultracold atoms. Through the measured eigenfunction,
the distribution of the complete QGT in the Brillouin zone is reconstructed,
with which the topological invariants are extracted by the Berry curvature and
the distances of quantum states in momentum space are measured by the quantum
metric tensor. Further, we experimentally test a predicted inequality between
the Berry curvature and quantum metric tensor, which reveals a deep connection
between topology and geometry.

In quantum physics, quantum states are defined in Hilbert spaces, in which
Berry phases profoundly reflect the fundamental geometric structure of Hilbert
spaces Berry (1984). In 1989, Berry further pointed out that the geometry of
Hilbert space is fully determined by the quantum geometry tensor (QGT) Wilczek
and Shapere (1989). The QGT has an antisymmetric imaginary part, i.e. the
well-known Berry curvature, which unveils the geometric phase difference of a
wave fuction between two parameter points in the parameter space. Its
associated topological phenomena have been extensively studied in many
systems, such as the Ahoranov-Bohn effect Aharonov and Bohm (1959), the
quantum Hall effect von Klitzing (1986), topological insulators and
topological superconductors Hasan and Kane (2010); Qi and Zhang (2011).
Moreover, the QGT has a symmetric real part termed as the quantum metric
tensor Provost and Vallee (1980); Anandan and Aharonov (1990), which describes
the distance between two quantum states in the parameter space. The quantum
metric tensor can give the famous Euler characteristic number Ma  _et al._
(2013) and is of great importance on obtaining the quantum Fisher information
Braunstein and Caves (1994); Hauke  _et al._ (2016) as well as depicting
quantum phase transitions Zanardi  _et al._ (2007).

The broadening of topological knowledge from condensed matter physics has
rendered the possibility of quantum control towards measuring the QGT.
Recently, several experiments have been conducted for the measurements of the
QGT of two-level systems in a superconducting qubit Tan  _et al._ (2019), an
NV center qubit Yu  _et al._ (2019, 2022) and exciton-polaritons Gianfrate
_et al._ (2020). Nevertheless, in lattice systems, the complete quantum
geometric tensor of Bloch bands is yet to be measured. Specially, in ultracold
atoms, optical lattices have been developed as a highly controllable and
tunable quantum simulator to investigate topological band physics Cooper  _et
al._ (2019), which offers an ideal platform for the measurement of the QGT
Asteria  _et al._ (2019). In addition, there are several tomography
experiments in cold atom systems, based on the dynamics after a projection
onto flat bands Hauke  _et al._ (2014); Fläschner  _et al._ (2016); Tarnowski
_et al._ (2019), based on projection on different momenta by fast acceleration
Li  _et al._ (2016), based on the interference of paths surrounding Duca  _et
al._ (2015) or crossing band closing points Brown  _et al._ (2022), and based
on off-resonant coupling to higher bands Tarnowski  _et al._ (2017). Whereas
these tomography protocols probe Bloch states in optical lattices in different
aspects, none of them focus on studying the quantum geometric tensor.

Here, we experimentally measure the complete QGT of a topological band using
ultracold atoms in an optical lattice. We implement a complete Bloch state
tomography using a well-controlled momentum-transferred Raman laser pulse in a
two-dimensional (2D) optical Raman lattice Wu  _et al._ (2016); Sun  _et al._
(2018a). Based on the Bloch state tomography, the eigenfunction of two-band
model in the Raman lattices is directly measured through the expectation
values of three Pauli matrices. Thus, the momentum-resolved Berry curvature
and the quantum metric tensor, relying only on the eigenfunction, are
simultaneously reconstructed, which probes the complete geometry of Bloch
states in the Raman lattices. Meanwhile, the Chern number Hasan and Kane
(2010) (quantum volume Ozawa and Mera (2021); Mera  _et al._ (2022)) is
obtained from the integral of the Berry curvature (the quantum metric tensor)
over the full Brillouin zone. An inequality proved in Refs. Ozawa and Mera
(2021); Mera  _et al._ (2022) between the quantum volume and the Chern number
is also tested experimentally.

Figure 1: Bloch state tomography in 2D optical Raman lattices. (a) Sketch map
of Raman lattices. (b) The distribution of Bose-Einstein condensates (BECs) in
momentum space. BEC is prepared in $\left|\uparrow\right\rangle$ and stay at
momentum point $\bm{k}=(0,0)$. Four BECs distributing at momenta $\bm{k}=(\pm
k_{0},\pm k_{0})$ are in $\left|\downarrow\right\rangle$ due to the Raman
potential. Here, $k_{0}$ is the recoil momentum. Other BECs are in
$\left|\uparrow\right\rangle$ and diffracted by lattice potential without
spin-flipping. (c) Tomography principle. The atoms firstly are in a
superposition state $|\Psi\rangle$ of $\left|\bm{q},\uparrow\right\rangle$ and
$\left|\bm{q}+\bm{k}_{0},\downarrow\right\rangle$ (yellow arrows). Applying a
$\pi/2$ Raman pulse, $|\Psi\rangle$ evolves to a new superposition state
$|\tilde{\Psi}\rangle$ (magenta arrows) along the sky blue trajectories.
$\Delta\varphi$ is exactly the angle between the gray axis and $x$-axis. The
gray axis rotates in the equatorial plane by tuning $\Delta\varphi$. After
ToF, $\left\langle\sigma_{x}\right\rangle$
($\left\langle\sigma_{y}\right\rangle$) is obtained when $\Delta\varphi=\pi/2$
($0$). $\left\langle\sigma_{z}\right\rangle$ is measured directly by ToF,
without applying the Raman pulse. (d) The experimental sequence. The atoms are
loaded in Raman lattices in the preparation stage. The Raman pulse is realized
by suddenly turning off reflected beams of $\bm{E}_{x,y}$, setting Raman pulse
strength to $\Omega_{\text{R0}}$, the relative phase $\Delta\varphi$ to a
certain value and $\delta_{\text{R}}=0$. After ToF, the spin polarization is
measured. (e) The spin polarization $P_{\text{m}}$ at the quasi-momentum
$\Gamma$ and M as a function of the relative phase $\Delta\varphi$. The
circles with error bar (blue curves) are from experimental data (numerical
calculations). The experimental data is fitted by sinusoidal function (red
curves). The insets mark $\Gamma$ and M points in the FBZ. Parameters: the
lattice depth $V_{0}=4.0E_{\rm{r}}$, Raman potential strength
$\Omega_{0}=1.0E_{\rm{r}}$, $\delta=-0.2E_{\rm{r}}$, $t_{\text{R}}=10\mu s$
and $\delta_{\text{R}}=0$. The recoil energy $E_{\rm{r}}\approx 2\pi\times
3.7$kHz.

Our Bloch state tomography is realized in a 2D Raman lattice with ultracold
87Rb atoms Sun  _et al._ (2018a); Liu  _et al._ (2014); sup , as shown in
Fig.1(a). The Raman lattices are constructed by two laser beams $\bm{E}_{x,y}$
with wavelength $\lambda=787$nm that propagate in opposite directions as well
as couple two magnetic levels $\left|F=1,m_{\rm
F}=-1\right\rangle\equiv\left|\uparrow\right\rangle$ and $\left|F=1,m_{\rm
F}=0\right\rangle\equiv\left|\downarrow\right\rangle$. The Hamiltonian is
($\hbar=1$)

$H=\frac{\bm{k}^{2}}{2m}+V_{\rm
latt}+\frac{\delta}{2}\sigma_{z}+\Omega_{1}\sigma_{x}+\Omega_{2}\sigma_{y},$
(1)

where $\bm{k}=(k_{x},k_{y}),m,\delta$ and $\sigma_{x,y,z}$ are the momentum,
mass of the atom, two-photon detuning and Pauli matrices, respectively. The
Raman potential $\Omega_{1,2}$ (lattice potential $V_{\rm latt}$) induces
spin-flipped (spin-conserved) hopping with the strength $\Omega_{0}$ ($V_{0}$)
and transfers momentum of $\bm{k}_{0}=(\pm k_{0},\pm k_{0})$ ($2\bm{k}_{0}$),
which is observed from the atomic distribution of ground state in the momentum
space [Fig.1(b)] Wu  _et al._ (2016); Sun  _et al._ (2018a). Then, the
eigenstate $\left|\Psi(\bm{q})\right\rangle$ of Hamiltonian Eq.(1) can be
denoted as the superposition of $\left|\bm{q},\uparrow\right\rangle$ and
$\left|\bm{q}+\bm{k}_{0},\downarrow\right\rangle$, i.e.,
$\left|\Psi(\bm{q})\right\rangle=c_{\uparrow}(\bm{q})\left|\bm{q},\uparrow\right\rangle+c_{\downarrow}(\bm{q})\left|\bm{q}+\bm{k}_{0},\downarrow\right\rangle$
sup . Here, $\bm{q}=(q_{x},q_{y})$ and $c_{\uparrow,\downarrow}$ are the
quasi-momentum and the normalized complex coefficient, respectively. The
eigenstate $\left|\Psi(\bm{q})\right\rangle$ can be represented on the Bloch
sphere, whose poles are $\left|\bm{q},\uparrow\right\rangle$ and
$\left|\bm{q}+\bm{k}_{0},\downarrow\right\rangle$ [Fig.1(c)].

The main idea of the tomography is to measure the expectation values of three
Pauli matrices $\left\langle\sigma_{x,y,z}(\bm{q})\right\rangle$ for Raman
lattices by rotating the measurement basis (See sup  for details). We directly
obtain the expectation value $\left\langle\sigma_{z}(\bm{q})\right\rangle$ by
spin-resolved time-of-flight (ToF) imaging, which has been applied in Wu  _et
al._ (2016); Sun  _et al._ (2018a). After rotating the measurement basis
$|\bm{q},\uparrow\rangle$ and $|\bm{q}+\bm{k}_{0},\downarrow\rangle$ to the
$x$-axis ( $y$-axis) on the Bloch sphere, the expectation value
$\left\langle\sigma_{x}(\bm{q})\right\rangle$
($\left\langle\sigma_{y}(\bm{q})\right\rangle$) is observed by spin-resolved
ToF imaging Alba  _et al._ (2011). Such rotation is achieved by a coherent
Raman pulse. The pulse acting on $\left|\Psi(\bm{q})\right\rangle$ requires
transferring the momentum $\bm{k}_{0}=(-k_{0},-k_{0})$ between
$\mid\bm{q},\uparrow\rangle$ and $\mid\bm{q}+\bm{k_{0}},\downarrow\rangle$ as
well as maintaining a fixed relative phase $\Delta\varphi$ between the Raman
pulse and the Raman lattices, which ensures accurate determination of
$\left\langle\sigma_{x,y}(\bm{q})\right\rangle$ [Fig.1(c)].

Concretely, the Raman pulse is made up of incident laser beams of
$\bm{E}_{x,y}$ with retroreflective beams switched off by acousto-optic
modulators (AOMs) [Fig.1(a)]. Thus, the Hamiltonian of the Raman pulse is sup

$H_{\text{R}}=\begin{pmatrix}\bm{k}^{2}/2m+\delta_{\text{R}}/2&\Omega_{\text{R}}\\\
\Omega_{\text{R}}^{*}&\bm{k}^{2}/2m-\delta_{\text{R}}/2\end{pmatrix}.$ (2)

In the experiment, a sufficient short Raman pulse with two-photon detuning
$\delta_{\text{R}}=0$ is applied such that the kinetic energy is ignored and
Raman coupling $\Omega_{\text{R}}$ dominates. And
$\Omega_{\text{R}}=\Omega_{\text{R0}}e^{-i[k_{0}(x+y)-\Delta\varphi]}$ with
the strength $\Omega_{\text{R0}}$ launches a momentum transfer of
$\bm{k_{0}}=(-k_{0},-k_{0})$ between $\mid\bm{q},\uparrow\rangle$ and
$\mid\bm{q}+\bm{k_{0}}\downarrow\rangle$. The relative phase $\Delta\varphi$
is controlled by the initial phase difference of the input lasers
$\bm{E}_{x,y}$ that generates the Raman pulse and the Raman lattices sup .

When the Raman pulse is applied to $\left|\Psi(\bm{q})\right\rangle$ with the
duration time $t_{\text{R}}$, the relative phase $\Delta\varphi$ is imprinted
onto the time-dependent state $|\widetilde{\Psi}(\bm{q},t_{\text{R}})\rangle$.
The state $|\widetilde{\Psi}\rangle$ precesses around an axis on the Bloch
sphere with frequency $\Omega_{\text{R0}}$. For instance,
$|\widetilde{\Psi}\rangle$ precesses around $y$ ($x$)-axis in the equatorial
plane when $\Delta\varphi=\pi/2\;(0)$ [Fig.1(c)]. After ToF imaging, the
expectation values of Pauli matrices $\left\langle\sigma_{x,y,z}\right\rangle$
are obtained by the spin polarization sup

$\displaystyle\langle\widetilde{\Psi}|\sigma_{z}|\widetilde{\Psi}\rangle$
$\displaystyle=\left\langle\sigma_{z}\right\rangle\cos(2\Omega_{\text{R0}}t_{\text{R}})$
(3)
$\displaystyle+(\left\langle\sigma_{y}\right\rangle\cos\Delta\varphi+\left\langle\sigma_{x}\right\rangle\sin\Delta\varphi)\sin(2\Omega_{\text{R0}}t_{\text{R}}).$

When $t_{\text{R}}=0$, $\left\langle\sigma_{z}\right\rangle$ is obtained; when
$t_{\text{R}}=\pi/(4\Omega_{R0})$, $\left\langle\sigma_{x}\right\rangle$
($\left\langle\sigma_{y}\right\rangle$) is extracted by a $\pi/2$ Raman pulse
with $\Delta\varphi=\pi/2$ ($0$) [Fig.1(c)].

Figure 2: The expectation value of three Pauli matrices
$\langle\sigma_{i=x,y,z}\rangle$ in the FBZ. (a) Normalized
$\langle\sigma_{x,y,z}(\mathbf{q})\rangle$ with $\delta=-0.2E_{\rm{r}}$ for
the lowest band. Results from experimental data (upper row) are compared with
numerical calculations (lower row). (b) The distribution of
$\langle\bm{\sigma}(\mathbf{q})\rangle$ in the FBZ takes a skyrmion
configuration. Parameters:
$(V_{0},\Omega_{0},\delta)=(4.0,1.0,-0.2)E_{\rm{r}}$.

Now, we clarify this technique in the experiment sup . The experimental
protocol is depicted in Fig.1(d). First, BEC of 87Rb atoms is adiabatically
loaded into the Raman lattices. The BECs condense at $\Gamma$ or M in the
first Brillouin zone (FBZ), which depends on BEC prepared in
$\left|\uparrow\right\rangle$ or $\left|\downarrow\right\rangle$. Meanwhile,
atoms stay at the lowest band of Raman lattices. Second, the $\pi$/2 Raman
pulse is switched on within 200ns to couple the atoms (See sup  for the
sensitivities of the pulse), which is accomplished by simultaneously executing
the following manipulations on the beams $\bm{E}_{x,y}$: (i) Turning off the
retroreflective beams of $\bm{E}_{x,y}$; (ii) Turning on the strength of Raman
pulse to $\Omega_{\text{R0}}\approx 3.4E_{\rm{r}}$ by tuning the intensity of
$\bm{E}_{x,y}$; (iii) Setting the relative phase $\Delta\varphi$ to a certain
value by changing the initial phase of $\bm{E}_{x,y}$; (iv) Setting
$\delta_{\text{R}}=0$ by adjusting the frequency of $\bm{E}_{x,y}$. Finally,
after holding the Raman pulse for a certain duration $t_{\text{R}}=10\mu s$,
we measure the atomic numbers $n_{\uparrow,\downarrow}$ using spin-resolved
ToF imaging to obtain spin polarization, i.e.,
$P_{\rm{m}}=(n_{\uparrow}-n_{\downarrow})/(n_{\uparrow}+n_{\downarrow})$.

The measured spin polarizations $P_{\rm{m}}$ versus the relative phase
$\Delta\varphi$ are shown in Fig.1(e). The spin polarizations at $\Gamma$ and
M are fitted by sinusoidal functions, which demonstrates the superposition
between $\left\langle\sigma_{x}\right\rangle$ and
$\left\langle\sigma_{y}\right\rangle$ [cf., Eq. (3)]. Thus,
$\left\langle\sigma_{x}\right\rangle$ ($\left\langle\sigma_{y}\right\rangle$)
are extracted from the fittings when $\Delta\varphi=\pi/2$ ($0$) at $\Gamma$
and M, marked by diamonds. For comparison, the numerical simulations of the
spin polarizations with the same parameters are also plotted in Fig.1(e), in
consistent with the experimental data. All these indicate that Bloch state
tomography in the Raman lattices is achieved.

Next, we explore geometry and topology with this novel Bloch state tomography
to directly map out the quantum geometric tensor from the eigenfunction based
on measuring $\left\langle\sigma_{x,y,z}(\bm{q})\right\rangle$ of Bose gases
in the FBZ. In that case, almost identical procedure used in the
aforementioned BEC measurement is employed, except that the atomic cloud is
cooled to a temperature around $100$nK sup . Then, the entire lowest band is
occupied by a sufficient number of atoms; simultaneously, around one third of
the atoms populate to the higher bands, which reduces the contrast of the spin
polarizations being the same as Refs. Wu  _et al._ (2016); Sun  _et al._
(2018a); Wang  _et al._ (2021). Whereupon, we subtract the high-band
contribution based on Bose distribution determined by the numerical
calculations to obtain $\left\langle\sigma_{x,y,z}(\bm{q})\right\rangle$ for
the lowest band (See sup  for details).

Figure 3: Experimental reconstruction of the eigenfunction in the FBZ: The
amplitude of the eigenfunction $u_{\uparrow}$ (first column) and
$u_{\downarrow}$ (second column), together with the relative phase between
$u_{\uparrow}$ and $u_{\downarrow}$ (third column). Red squares mark the
positions of the phase vortices. Results from experimental data (upper row)
are compared with numerical calculations (lower row). Parameters:
$(V_{0},~{}\Omega_{0},~{}\delta)=(4.0,1.0,-0.2)E_{\rm{r}}$.

The typical normalized $\left\langle\sigma_{x,y,z}(\bm{q})\right\rangle$ are
drawn in Fig.2(a), which can be divided into two regions with
$\langle\sigma_{x,y,z}\rangle<0$ and $\langle\sigma_{x,y,z}\rangle>0$. For
$\langle\sigma_{x}(\bm{q})\rangle$ ($\langle\sigma_{y}(\bm{q})\rangle$), the
two regions are distributed in the upper and lower (left and right) of the
FBZ, respectively. For $\langle\sigma_{z}(\bm{q})\rangle$, one of the regions
is centered on $\Gamma$ and the other region is centered on M for
$\delta=-0.2E_{\rm{r}}$. And the two regions are demarcated by a ring
structure with $\langle\sigma_{z}\rangle=0$ around M, which is a feature of
band topology in our system Wu  _et al._ (2016); Sun  _et al._ (2018a). The
numerical calculations coincide with the experimental measurements. In
addition, the distribution of the vectors
$\langle\bm{\sigma}(\mathbf{q})\rangle=(\left\langle\sigma_{x}(\bm{q})\right\rangle,\left\langle\sigma_{y}(\bm{q})\right\rangle,\left\langle\sigma_{z}(\bm{q})\right\rangle)$
shapes a skyrmion Skyrme (1962) structure in the momentum space, and the
twisted skyrmion texture is demonstrated in Fig.2(b). Such skyrmion
configuration offers the direct evidence for complete tomography.

Figure 4: The quantum geometric tensor in the FBZ. (a) The Berry curvature
$\mathcal{F}_{xy}(\bm{q})$, together with the three components of the quantum
metric tensor $g_{xx}(\bm{q})$, $g_{yy}(\bm{q})$, $g_{xy}(\bm{q})$. Results
from experimental data (upper row) are compared with numerical calculations
(lower row). Parameters:
$(V_{0},~{}\Omega_{0},~{}\delta)=(4.0,1.0,-0.2)E_{\rm{r}}$. (b) The Chern
number $\mathcal{C}$ and the quantum volume $\text{vol}_{g}$ versus the
detuning $\delta$. The circles and squares with error bars are from
experimental results. The dash pink and green curves are from numerical
calculations. Parameters: $(V_{0},~{}\Omega_{0})=(4.0,1.0)E_{\rm{r}}$.

Moreover, the eigenfunction can also be extracted from the normalized
$\left\langle\sigma_{x,y,z}\right\rangle$. To this end, we consider the Bloch
Hamiltonian under the two-band tight-binding approximation of the Eq.(1) Wu
_et al._ (2016); Sun  _et al._ (2018b). After diagonalizing the Bloch
Hamiltonian, the eigenfunction in the lowest band can be written as
$\left|u(\bm{q})\right\rangle=\left(\sin(\theta_{\bm{q}}/2)e^{-i\phi_{\bm{q}}},-\cos(\theta_{\bm{q}}/2)\right)^{\rm{T}}$,
where the parameters
$\theta_{\bm{q}}=\arccos(-\left\langle\sigma_{z}\right\rangle)\in[0,\pi]$ and
$\phi_{\bm{q}}=\arg(\left\langle\sigma_{x}\right\rangle+i\left\langle\sigma_{y}\right\rangle)\in[0,2\pi)$.
The amplitudes of the eigenfunction $u_{\uparrow,\downarrow}$ and their
relative phase for $\delta=-0.2E_{\rm{r}}$ in the FBZ are exhibited in Fig.3.
The amplitude of the eigenfunction $u_{\uparrow}$ has a value of $\sim$1 at
the $\Gamma$ point and is close to zero at the M point, whereas the amplitude
of $u_{\downarrow}$ is opposite. The relative phase is significantly different
from the amplitude and can be roughly divided into four regions: the blue-gray
region ($0\leq\varphi<0.5\pi$), the gray region ($0.5\pi\leq\varphi<\pi$) on
the left of the phase map, and the light-green region
($\pi\leq\varphi<1.5\pi$), the green region ($1.5\pi\leq\varphi<2\pi$) on the
right of the phase map. The junctions of these four regions emerge phase
vortices, which implies that the vectors $\langle\bm{\sigma}\rangle$ point to
north or south poles of the Bloch sphere Yu (2017). Our experimental
measurements basically agree with numerical calculations (second row of
Fig.3). Note that the phase vortices deviate from the high-symmetry points of
the FBZ {$\Gamma$,M,$X_{1,2}$}, which mainly stems from the coherence between
the bands sup .

Thanks to the complete observation of the eigenfunction $|u(\bm{q})\rangle$,
we can identify the geometric structures of the quantum states by the gauge-
invariant quantum geometric tensor $\chi_{\alpha\beta}$ Kolodrubetz  _et al._
(2017)

$\chi_{\alpha\beta}=\langle\partial_{q_{\alpha}}u|(1-|u\rangle\langle
u|)|\partial_{q_{\beta}}u\rangle=g_{\alpha\beta}-i\mathcal{F}_{\alpha\beta}/2.$
(4)

Here, the quantum metric tensor $g_{\alpha\beta}=\rm{Re}(\chi_{\alpha\beta})$,
as the real part of QGT, measures a distance between quantum states in quasi-
momentum space Provost and Vallee (1980). The Berry curvature
$\mathcal{F}_{\alpha\beta}=-2\rm{Im}(\chi_{\alpha\beta})$, being the imaginary
part of QGT, acts as an effective ‘electromagnetic’ tensor in quasi-momentum
space Hasan and Kane (2010). Hence, the momentum-resolved quantum metric
tensor $g_{\alpha\beta}$ and Berry curvature $\mathcal{F}_{xy}$ are extracted
from the measurement of eigenfunction in Fig.3 using Eq.(4), displayed in
Fig.4(a). The Berry curvature $\mathcal{F}_{xy}$ is mainly localized near the
ring structure, the sign and direction of which is negative and perpendicular
to the plane $q_{x}$-$q_{y}$, respectively. Thereby, Chern number
$\mathcal{C}=\int_{\text{BZ}}\mathcal{F}_{xy}d\bm{q}/2\pi=-1.00\pm 0.02$ at
$\delta=-0.2E_{\text{r}}$, signifying the band is topologically non-trivial.
Tuning the detuning from $\delta=-1.0E_{\text{r}}$ to
$\delta=1.0E_{\text{r}}$, in Fig.4(b), the topological phase diagram is
obtained by the Chern number, in consistent with our previous measurements via
quench dynamics Sun  _et al._ (2018b); Yi  _et al._ (2019a); Zhang  _et al._
(2018).

In addition, the quantum metric tensor with non-vanishing value in Fig.4(a)
also appears on the ring structure around M, indicating the distance between
quantum states on the ring is larger than other areas. Moreover, the integral
of the quantum metric tensor over the full Brillouin zone gives a quantum
volume not
$\text{vol}_{g}=\int_{\text{BZ}}\sqrt{g_{xx}g_{yy}-g_{xy}^{2}}d\bm{q}/\pi$,
which yields an inequality $\text{vol}_{g}\geq|\mathcal{C}|$ for Chern
insulators Ozawa and Mera (2021); Mera  _et al._ (2022). Figure 4(b) shows the
quantum volume and the Chern number as functions of the detuning, validating
such an inequality experimentally. The inequality is related to superfluidity
or superconductivity Peotta and Törmä (2015). Thereafter, we may use the
inequality to roughly evaluate the topological properties in 2D Raman
lattices: If $\text{vol}_{g}<1$, $\mathcal{C}=0$; If $\text{vol}_{g}\geq 1$
and $\delta\neq 0$, $|\mathcal{C}|=1$. Note that the quantum volume takes non-
integer value that can be straightforwardly interpreted from the positive
semi-definite of the the quantum metric tensor as elaborated in Ma (2020).

Our measurements of the quantum geometric tensor via Bloch state tomography in
2D optical Raman lattices exhibit a paradigm to investigate geometry and
topology for quantum states. Since such tomography technique only requires
accurate control over the momentum transfer and the phase of the Raman pulse,
it can be readily generalized to interacting systems Zheng  _et al._ (2020),
multi-band systems Ünal  _et al._ (2020), and 3D topological systems Wang  _et
al._ (2021). It also opens the possibility of spatiotemporal coherent
manipulation towards detecting new dynamical topological states of matter. For
instance, people could apply the tomography to quench dynamics with the goal
of probing the complete structure of Hopf fibration Wang  _et al._ (2017); Yu
(2019); Yi  _et al._ (2019b); Tarnowski  _et al._ (2019). The quantum metric
tensor also provides important information that the Berry curvature incapable
to tell when the Berry curvature vanishes or diverges, including geometric or
topological properties Hwang  _et al._ (2021), non-Hermatian systems
Solnyshkov  _et al._ (2021); Chen  _et al._ (2017), geometric orbital
susceptibility Piéchon  _et al._ (2016) as well as Bose and Fermi superfluid
Peotta and Törmä (2015); Iskin (2018). Such situation can be reached by
adjusting the parameters of the Raman lattices so that there exist gapless
bands and one can detect certain phenomena in topological semi-metals governed
by the quantum metric tensor, such as the dynamics of the wave packet
Gianfrate  _et al._ (2020); Solnyshkov  _et al._ (2021).

###### Acknowledgements.

We acknowledge insightful discussions with Ji-Zhou Wu. This work was supported
by the Innovation Program for Quantum Science and Technology (Grant No.
2021ZD0302001 and 2021ZD0302100), the National Natural Science Foundation of
China (Grant No. 12025406 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). C.-R.Y. acknowledges support from
China Postdoctoral Science Foundation (Grant No. 2021M703112).

## References

  * Berry (1984) M. V. Berry, Proc. R. Soc. A 392, 45 (1984).
  * Wilczek and Shapere (1989) F. Wilczek and A. Shapere,  _Geometric Phases in Physics_ (World Scientific, 1989) pp. 7–28.
  * Aharonov and Bohm (1959) Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
  * von Klitzing (1986) K. von Klitzing, Rev. Mod. Phys. 58, 519 (1986).
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Provost and Vallee (1980) J. P. Provost and G. Vallee, Commun. Math. Phys. 76, 289 (1980).
  * Anandan and Aharonov (1990) J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 (1990).
  * Ma  _et al._ (2013) Y.-Q. Ma, S.-J. Gu, S. Chen, H. Fan, and W.-M. Liu, Europhys. Lett. 103, 10008 (2013).
  * Braunstein and Caves (1994) S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
  * Hauke  _et al._ (2016) P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Nat. Phys. 12, 778 (2016).
  * Zanardi  _et al._ (2007) P. Zanardi, P. Giorda, and M. Cozzini, Phys. Rev. Lett. 99, 100603 (2007).
  * Tan  _et al._ (2019) X. Tan, D.-W. Zhang, Z. Yang, J. Chu, Y.-Q. Zhu, D. Li, X. Yang, S. Song, Z. Han, Z. Li, Y. Dong, H.-F. Yu, H. Yan, S.-L. Zhu, and Y. Yu, Phys. Rev. Lett. 122, 210401 (2019).
  * Yu  _et al._ (2019) M. Yu, P. Yang, M. Gong, Q. Cao, Q. Lu, H. Liu, S. Zhang, M. B. Plenio, F. Jelezko, T. Ozawa, N. Goldman, and J. Cai, Natl. Sci. Rev. 7, 254 (2019).
  * Yu  _et al._ (2022) M. Yu, X. Li, Y. Chu, B. Mera, F. N. Ünal, P. Yang, Y. Liu, N. Goldman, and J. Cai, arXiv:2206.00546 (2022).
  * Gianfrate  _et al._ (2020) A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. W. West, L. N. Pfeiffer, D. D. Solnyshkov, D. Sanvitto, and G. Malpuech, Nature 578, 381 (2020).
  * Cooper  _et al._ (2019) N. R. Cooper, J. Dalibard, and I. B. Spielman, Rev. Mod. Phys. 91, 015005 (2019).
  * Asteria  _et al._ (2019) L. Asteria, D. T. Tran, T. Ozawa, M. Tarnowski, B. S. Rem, N. Fläschner, K. Sengstock, N. Goldman, and C. Weitenberg, Nat. Phys. 15, 449 (2019).
  * Hauke  _et al._ (2014) P. Hauke, M. Lewenstein, and A. Eckardt, Phys. Rev. Lett. 113, 045303 (2014).
  * Fläschner  _et al._ (2016) N. Fläschner, B. S. Rem, M. Tarnowski, D. Vogel, D.-S. Lühmann, K. Sengstock, and C. Weitenberg, Science 352, 1091 (2016).
  * Tarnowski  _et al._ (2019) M. Tarnowski, F. N. Ünal, N. Fläschner, B. S. Rem, A. Eckardt, K. Sengstock, and C. Weitenberg, Nat. Commun. 10, 1728 (2019).
  * Li  _et al._ (2016) T. Li, L. Duca, M. Reitter, F. Grusdt, E. Demler, M. Endres, M. Schleier-Smith, I. Bloch, and U. Schneider, Science 352, 1094 (2016).
  * Duca  _et al._ (2015) L. Duca, T. Li, M. Reitter, I. Bloch, M. Schleier-Smith, and U. Schneider, Science 347, 288 (2015).
  * Brown  _et al._ (2022) C. D. Brown, S.-W. Chang, M. N. Schwarz, T.-H. Leung, V. Kozii, A. Avdoshkin, J. E. Moore, and D. Stamper-Kurn, Science 377, 1319 (2022).
  * Tarnowski  _et al._ (2017) M. Tarnowski, M. Nuske, N. Fläschner, B. Rem, D. Vogel, L. Freystatzky, K. Sengstock, L. Mathey, and C. Weitenberg, Phys. Rev. Lett. 118, 240403 (2017).
  * Wu  _et al._ (2016) Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Science 354, 83 (2016).
  * Sun  _et al._ (2018a) W. Sun, B.-Z. Wang, X.-T. Xu, C.-R. Yi, L. Zhang, Z. Wu, Y. Deng, X.-J. Liu, S. Chen, and J.-W. Pan, Phys. Rev. Lett. 121, 150401 (2018a).
  * Ozawa and Mera (2021) T. Ozawa and B. Mera, Phys. Rev. B 104, 045103 (2021).
  * Mera  _et al._ (2022) B. Mera, A. Zhang, and N. Goldman, SciPost Phys. 12, 18 (2022).
  * Liu  _et al._ (2014) X.-J. Liu, K. T. Law, and T. K. Ng, Phys. Rev. Lett. 112, 086401 (2014).
  * (31) See Supplemental Material for more details including the experimental scheme, the theoretical calculations of Bloch state tomography, the experimental protocol and the data analyzing.
  * Alba  _et al._ (2011) E. Alba, X. Fernandez-Gonzalvo, J. Mur-Petit, J. K. Pachos, and J. J. Garcia-Ripoll, Phys. Rev. Lett. 107, 235301 (2011).
  * Wang  _et al._ (2021) Z.-Y. Wang, X.-C. Cheng, B.-Z. Wang, J.-Y. Zhang, Y.-H. Lu, C.-R. Yi, S. Niu, Y. Deng, X.-J. Liu, S. Chen, and J.-W. Pan, Science 372, 271 (2021).
  * Skyrme (1962) T. Skyrme, Nucl. Phys. 31, 556 (1962).
  * Sun  _et al._ (2018b) W. Sun, C.-R. Yi, B.-Z. Wang, W.-W. Zhang, B. C. Sanders, X.-T. Xu, Z.-Y. Wang, J. Schmiedmayer, Y. Deng, X.-J. Liu, S. Chen, and J.-W. Pan, Phys. Rev. Lett. 121, 250403 (2018b).
  * Yu (2017) J. Yu, Phys. Rev. A 96, 023601 (2017).
  * Kolodrubetz  _et al._ (2017) M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Phys. Rep. 697, 1 (2017).
  * Yi  _et al._ (2019a) C.-R. Yi, L. Zhang, L. Zhang, R.-H. Jiao, X.-C. Cheng, Z.-Y. Wang, X.-T. Xu, W. Sun, X.-J. Liu, S. Chen, and J.-W. Pan, Phys. Rev. Lett. 123, 190603 (2019a).
  * Zhang  _et al._ (2018) L. Zhang, L. Zhang, S. Niu, and X.-J. Liu, Science Bulletin 63, 1385 (2018).
  * (40) Compared with the definition of the quantum volume in Ref. Ozawa and Mera (2021); Mera  _et al._ (2022), we have rescaled its coefficient (by a factor of $\pi$) here to facilitate a direct comparison of this quantity and the Chern number: this convention help to make the inequality look slightly more concise.
  * Peotta and Törmä (2015) S. Peotta and P. Törmä, Nat. Commun. 6, 8944 (2015).
  * Ma (2020) Y.-Q. Ma, arXiv:2001.05946 (2020).
  * Zheng  _et al._ (2020) J.-H. Zheng, B. Irsigler, L. Jiang, C. Weitenberg, and W. Hofstetter, Phys. Rev. A 101, 013631 (2020).
  * Ünal  _et al._ (2020) F. N. Ünal, A. Bouhon, and R.-J. Slager, Phys. Rev. Lett. 125, 053601 (2020).
  * Wang  _et al._ (2017) C. Wang, P. Zhang, X. Chen, J. Yu, and H. Zhai, Phys. Rev. Lett. 118, 185701 (2017).
  * Yu (2019) J. Yu, Phys. Rev. A 99, 043619 (2019).
  * Yi  _et al._ (2019b) C.-R. Yi, J.-L. Yu, W. Sun, X.-T. Xu, S. Chen, and J.-W. Pan, arXiv:1904.11656 (2019b).
  * Hwang  _et al._ (2021) Y. Hwang, J. Jung, J.-W. Rhim, and B.-J. Yang, Phys. Rev. B 103, L241102 (2021).
  * Solnyshkov  _et al._ (2021) D. D. Solnyshkov, C. Leblanc, L. Bessonart, A. Nalitov, J. Ren, Q. Liao, F. Li, and G. Malpuech, Phys. Rev. B 103, 125302 (2021).
  * Chen  _et al._ (2017) W. Chen, Ş. Kaya Özdemir, G. Zhao, J. Wiersig, and L. Yang, Nature 548, 192 (2017).
  * Piéchon  _et al._ (2016) F. Piéchon, A. Raoux, J.-N. Fuchs, and G. Montambaux, Phys. Rev. B 94, 134423 (2016).
  * Iskin (2018) M. Iskin, Phys. Rev. A 97, 063625 (2018).

Supplementary Materials for: Extracting the Quantum Geometric Tensor of an
Optical Raman Lattice by Bloch State Tomography

##  .1 The experimental scheme

The experimental scheme is shown in Fig.S1. An input laser beam with
wavelength $\lambda=787$nm is split into two sub-lasers by the polarized beam
splitter (PBS) and their frequencies are shifted by acousto-optic modulators
($\rm{AOM}_{1,2}$). The two sub-lasers are transferred to the experimental
platform by polarization maintaining fibers, and then pass through a high
extinction ratio PBS separately to generate linear polarized laser beams
$\bm{E}_{x}$ and $\bm{E}_{y}$. $\bm{E}_{x,y}$ irradiate the atomic cloud and
are reflected back by mirrors $M_{x,y}$. Meanwhile, $\bm{E}_{x,y}$ are split
into two orthogonally polarized components
$\bm{E}_{x,y}=\bm{E}_{xy,yx}+\bm{E}_{xz,yz}$ by $\lambda/2$ wave plates and
their phases are delayed by $\lambda/4$ wave plates. The ratio of
$\bm{E}_{x,y}$ reflected back to the atomic cloud, marked as $\gamma$, is
controlled by $\rm{AOM}_{3,4}$. We note that, the polarization of
$\bm{E}_{x,y}$ must be not changed with time after the laser passes through
$\text{AOM}_{3,4}$, so we use $\text{AOM}_{3,4}$ with an acousto-optical
material called “dense flint glass”. Thus, the component of laser beams
$\bm{E}_{xy,xz,yx,yz}$ can be written as

$\displaystyle\begin{split}{\bm{E}_{xy}}&=\left|E_{xy}\right|e^{i(\alpha+\alpha_{1}+\alpha_{L}/2)}\left[e^{i(k_{0}x-\alpha_{L}/2)}+\gamma
e^{-i(k_{0}x-\alpha_{L}/2)}\right],\\\
{\bm{E}_{xz}}&=\left|E_{xz}\right|e^{i(\alpha+\alpha_{1}+\alpha_{L}/2)}\left[e^{i(k_{0}x-\alpha_{L}/2)}-\gamma
e^{-i(k_{0}x-\alpha_{L}/2)}\right],\\\
{\bm{E}_{yx}}&=\left|E_{yx}\right|e^{i(\beta+\beta_{1}+\beta_{L}/2)}\left[e^{-i(k_{0}y+\beta_{L}/2)}+\gamma
e^{i(k_{0}y+\beta_{L}/2)}\right],\\\
{\bm{E}_{xz}}&=\left|E_{yz}\right|e^{i(\beta+\beta_{1}+\beta_{L}/2)}\left[e^{-i(k_{0}y+\beta_{L}/2)}-\gamma
e^{i(k_{0}y+\beta_{L}/2)}\right],\end{split}$ (S1)

where the recoil momentum (wave number) $k_{0}=2\pi/\lambda$,
$\left|E_{xy,xz,yx,yz}\right|$ are amplitudes of $\bm{E}_{xy,xz,yx,yz}$,
$\alpha_{1}$($\beta_{1}$) is the phase controlled by the ratio-frequency (RF)
signal input to $\rm{AOM}_{1}$ ($\rm{AOM}_{2}$), $\alpha$ ($\beta$) are
initial phases of $\bm{E}_{x}$ ($\bm{E}_{y}$), and $\alpha_{L}(\beta_{L})$ are
the phases that $\bm{E}_{x}$ ($\bm{E}_{y}$) passes an additional optical path
from the atom cloud to mirror $M_{x}(M_{y})$, then back to the atom cloud. Two
magnetic sublevels
$\left|F=1,m_{F}=-1\right\rangle\equiv\left|\uparrow\right\rangle$ and
$\left|F=1,m_{F}=0\right\rangle\equiv\left|\downarrow\right\rangle$, split by
a bias magnetic field $\bm{B}=|\bm{B}|\hat{z}$ with strength $|\bm{B}|\approx
23.4$G (corresponding to a Zeeman splitting of 16.5MHz) along $\hat{z}$
direction, are coupled by laser beams $\bm{E}_{x,y}$ Sun  _et al._ (2018a).
Thus, the effective Hamiltonian takes the form of Eq.(1) in the main text
where the square lattice $V_{\rm{latt}}(x,y)$ and Raman potentials
$\Omega_{1,2}(x,y)$ can be written as

$\displaystyle\Omega_{1}(x,y)=\frac{\Omega_{0}}{4}e^{-i(\alpha_{1}-\beta_{1})}\left(e^{-ik_{0}x}-\gamma
e^{ik_{0}x}\right)\left(e^{-ik_{0}y}+\gamma e^{ik_{0}y}\right),$
$\displaystyle\Omega_{2}(x,y)=\frac{i\Omega_{0}}{4}e^{-i(\alpha_{1}-\beta_{1})}\left(e^{-ik_{0}x}+\gamma
e^{ik_{0}x}\right)\left(-e^{-ik_{0}y}+\gamma e^{ik_{0}y}\right),$
$\displaystyle V_{\rm{latt}}(x,y)=\gamma
V_{0}\left[\cos^{2}(k_{0}x)+\cos^{2}(k_{0}y)\right].$ (S2)

Here, we have considered a unitary transformation using
$U=\rm{diag}(e^{i\phi},e^{-i\phi})$ with the global phase
$\phi=(\alpha+\alpha_{L}/2-\beta-\beta_{L}/2)/2$ together with a coordinate
transformation: $k_{0}x-\alpha_{L}/2\mapsto k_{0}x$,
$k_{0}y-\beta_{L}/2\mapsto k_{0}y$.

Figure S1: (a) The experimental setup. The laser beams $\bm{E}_{x,y}$ are used
to generate the 2D lattice potential, the Raman potentials and the Raman
pulse. The frequencies, strengths and phases of $\bm{E}_{x,y}$ are controlled
by $\rm{AOM}_{1,2}$, phase-locked radio-frequency signals $\rm{RF}_{1,2,3,4}$
and radio-frequency switches $\rm{Switch}_{1,2}$, respectively.
$\rm{RF}_{1,2,3,4}$ are generated by two dual-channel arbitrary waveform
generators (Keysight 33600A), respectively. $\rm{AOM}_{3,4}$ are used to turn
off the reflected beams of $\bm{E}_{x,y}$. A bias magnetic field
$\bm{B}=|\bm{B}|\hat{z}$ with strength $|\bm{B}|\approx 23.4$G along the
$\hat{z}$ direction, reducing the quadratic Zeeman splitting of $\sim 79$kHz,
guarantees the state $\left|F=1,m_{F}=+1\right\rangle$ is decoupled for strong
Raman strength. (b) Level structure and Raman transitions.

_2D Raman lattices._ –For $\gamma=1$, the optical lattices take the form
$V_{\rm{latt}}(x,y)=V_{0}[\cos^{2}(k_{0}x)+\cos^{2}(k_{0}y)]$ with lattice
depth $V_{0}$, and Raman potentials take the form
$\Omega_{1}(x,y)=\Omega_{0}\cos(k_{0}x)\sin(k_{0}y)$ and
$\Omega_{2}(x,y)=\Omega_{0}\cos(k_{0}y)\sin(k_{0}x)$ with Raman strength
$\Omega_{0}$. Here, without loss of generality, we has set
$\alpha_{1}=\beta_{1}=0$. It is found that the Raman potentials take a
momentum shift of ($\pm k_{0},\pm k_{0}$) between
$\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$. Therefore,
the eigen-state $\left|\Psi(\bm{q})\right\rangle$ can be expanded by a
complete basis of plane waves
$|\bm{q}+n\bm{b}_{1}+l\bm{b}_{2}\rangle=e^{i(q_{x}+2k_{0}n)x}e^{i(q_{y}+2k_{0}l)y}/\sqrt{S}$
and
$|\bm{q}+\bm{k}_{0}+v\bm{b}_{1}+w\bm{b}_{2}\rangle=e^{i(q_{x}+2k_{0}v+k_{0})x}e^{i(q_{y}+2k_{0}w+k_{0})y}/\sqrt{S}$
Wu  _et al._ (2016), that is,

$\displaystyle\begin{split}\left|\Psi(\bm{q})\right\rangle&=\binom{\sum_{n,l}C_{n,l,\uparrow}|\bm{q}+n\bm{b}_{1}+l\bm{b}_{2}\rangle}{\sum_{v,w}C_{v,w,\downarrow}|\bm{q}+\bm{k}_{0}+v\bm{b}_{1}+w\bm{b}_{2}\rangle},\end{split}$
(S3)

where $C_{n,l,\uparrow}$ and $C_{n,l,\downarrow}$ are the complex
coefficients, $\bm{b}_{1,2}$ are the vectors in the reciprocal lattice,
$(n,l,v,w)$ are integers, and $S$ denotes the system area. If only quasi-
momentum in the FBZ is considered, the eigen-state can be simply expressed as
$\left|\Psi(\bm{q})\right\rangle=c_{\uparrow}(\bm{q})\left|\bm{q},\uparrow\right\rangle+c_{\downarrow}(\bm{q})\left|\bm{q}+\bm{k}_{0},\downarrow\right\rangle$.

_Raman pulse_ –For $\gamma=0$, only the incident beams of $\bm{E}_{x,y}$ are
retained, which is used to generate the Raman pulse. The optical lattices
vanish. Raman potentials turns into the Raman coupling
$\Omega_{\text{R}}=\Omega_{\text{R0}}e^{-i[k_{0}(x+y)-\Delta\varphi]}$ with
Raman pulse strength $\Omega_{\text{R0}}=\sqrt{2}\Omega_{0}/4$, which takes a
momentum shift of ($-k_{0},-k_{0}$) between $\left|\uparrow\right\rangle$ and
$\left|\downarrow\right\rangle$. The relative phase between the Raman pulse
and the Raman lattices $\Delta\varphi=-\pi/4-\alpha_{1}+\beta_{1}$ only
depends on the phases $\alpha_{1}$ and $\beta_{1}$. The Hamiltonian of Raman
pulse is then given by

$H_{\rm{R}}=\begin{pmatrix}\bm{k}^{2}/2m+\delta_{\text{R}}/2&\Omega_{\text{R}}\\\
\Omega_{\text{R}}^{*}&\bm{k}^{2}/2m-\delta_{\text{R}}/2\end{pmatrix}.$ (S4)

##  .2 Bloch state tomography in Raman lattices

The purpose of the tomography is to measure the expectation values of three
Pauli matrices $\langle\sigma_{i=x,y,z}\rangle$, which can be written as

$\displaystyle\begin{split}&\left\langle\sigma_{x}\right\rangle=\sum_{n,l}\left(C_{n,l,\uparrow}^{*}C_{n,l,\downarrow}+C_{n,l,\uparrow}C_{n,l,\downarrow}^{*}\right),\\\
&\left\langle\sigma_{y}\right\rangle=\sum_{n,l}\left(-iC_{n,l,\uparrow}^{*}C_{n,l,\downarrow}+iC_{n,l,\uparrow}C_{n,l,\downarrow}^{*}\right),\\\
&\left\langle\sigma_{z}\right\rangle=\sum_{n,l}\left(\left|C_{n,l,\uparrow}\right|^{2}-\left|C_{n,l,\downarrow}\right|^{2}\right),\end{split}$
(S5)

according to Eq.(S3). We apply a momentum-transferred Raman pulse to measure
$\langle\sigma_{x,y,z}\rangle$. When the Raman pulse is applied to the eigen-
state of Raman lattices $\left|\Psi\right\rangle$ with a duration time
$t_{\text{R}}$, the state evolves unitarily to a new state

$|\widetilde{\Psi}(\bm{q},t_{\text{R}})\rangle=\exp(-iH_{\text{R}}t_{\text{R}})\left|\Psi\right\rangle$
(S6)

After spin-resolved ToF image, the spin polarization
$P=\left\langle\widetilde{\Psi}\left|\sigma_{z}\right|\widetilde{\Psi}\right\rangle$
is obtained. In order to get a concise expression for the physical process
above, we neglect the kinetic energy term $\bm{k}^{2}/2m$ (as is justified
below) and set $\delta_{\text{R}}=0$. The Hamiltonian in Eq. (S4) then takes a
simplified off-diagonal form
${H}_{\rm{R}}=\Omega_{\text{R0}}\cos[k_{0}(x+y)-\Delta\varphi]\sigma_{x}+\Omega_{\text{R0}}\sin[k_{0}(x+y)-\Delta\varphi]\sigma_{y}$.
Thus, the spin polarization can be expressed as

$\displaystyle\begin{split}P&=(|c_{\uparrow}|^{2}-|c_{\downarrow}|^{2})\cos(2\Omega_{\text{R0}}t_{\text{R}})\\\
&+\left(-ie^{i\Delta\varphi}c_{\uparrow}c_{\downarrow}^{*}\langle\bm{q}|e^{-ik_{0}(x+y)}|\bm{q}+\bm{k_{0}}\rangle+ie^{-i\Delta\varphi}c_{\uparrow}^{*}c_{\downarrow}\langle\bm{q}+\bm{k}_{0}|e^{ik_{0}(x+y)}|\bm{q}\rangle\right)\sin(2\Omega_{\text{R0}}t_{\text{R}}).\end{split}$
(S7)

Using the relation
$\langle\bm{q}+n\bm{b}_{1}+l\bm{b}_{2}|e^{-ik_{0}(x+y)}|\bm{q}+\bm{k_{0}}+v\bm{b}_{1}+w\bm{b}_{2}\rangle=\delta_{n,v}\delta_{l,w}$
and
$\langle\bm{q}+\bm{k}_{0}+v\bm{b}_{1}+w\bm{b}_{2}|e^{ik_{0}(x+y)}|\bm{q}+n\bm{b}_{1}+l\bm{b}_{2}\rangle=\delta_{n,v}\delta_{l,w}$,
one finds that the momentum transferred by Raman lattices is counteracted by
the Raman pulse. Then, combining Eq.(S5), Eq.(S7) can be written as the
superposition of $\left\langle\sigma_{x,y,z}\right\rangle$, as shown by Eq.
(3) in the main text. Equation. (3) is actually a sinusoidal function that
evolves with $t_{\text{R}}$ and possesses an amplitude of the oscillations
$\text{Amp}=\sqrt{(\langle\sigma_{z}\rangle)^{2}+(\langle\sigma_{y}\rangle\cos\Delta\varphi+\langle\sigma_{x}\rangle\sin\Delta\varphi)^{2}}$.
Therefore, $\left\langle\sigma_{x,y,z}\right\rangle$ are extracted by tuning
$\Delta\varphi$ and $t_{\text{R}}$. When $t_{\text{R}}=0$,
$\left\langle\sigma_{z}\right\rangle$ is obtained. When
$t_{\text{R}}=\pi/(4\Omega_{\text{R0}})$,
$\left\langle\sigma_{x}\right\rangle$ ($\left\langle\sigma_{y}\right\rangle$)
is obtained via a $\pi/2$ Raman pulse with $\Delta\varphi=\pi/2(0)$.

Figure S2: The sensitivity of the Bloch state tomography. Correlation
coefficient $\rho$ between $P(\bm{q},t)$ and
$\left\langle\sigma_{y}(\bm{q},t)\right\rangle$ for the lowest band as a
function of $\Omega_{\rm{R0}}$ (a), $\Delta\varphi$ (b) and $\delta_{\rm{R}}$
(c), respectively. The solid curves represent $\rho$ versus
$\Omega_{\rm{R0}}$, $\Delta\varphi$ and $\delta_{\rm{R}}$. Insets: the typical
$P$ in FBZ is shown and their location is marked by triangles, diamonds and
circles. The parameters of the green triangle, diamond and circle are
$\Omega_{\rm{R0}}\approx 3.4E_{\rm{r}}$, $\delta_{\rm{R}}=0$ and
$\Delta\varphi=\pi/2$, and the corresponding correlation coefficient
$\rho>0.97$. The small shaded gray regions indicate the measurement errors of
$\Omega_{\rm{R0}}$, $\Delta\varphi$ and $\delta_{\rm{R}}$, which are from
multiple experimental observations. Other results in the figure are the
numerical simulations. Parameters:
$(V_{0},\Omega_{0},\delta)=(4,1,0.2)E_{\rm{r}}$; (a) $\Delta\varphi=0$ and
$\delta_{\rm{R}}=0$, (b) $\delta_{\rm{R}}=0$ and $\Omega_{\rm{R0}}\approx
3.4E_{\rm{r}}$, (c) $\Delta\varphi=0$ and $\Omega_{\rm{R0}}\approx
3.4E_{\rm{r}}$.

Next, we estimate the impact of the kinetic energy term in the Raman pulse on
the measurement of $\langle\sigma_{x,y}\rangle$. To this end, we firstly
calculate numerically the spin polarization $P$ for the lowest band. In the
numerical calculation, we set $\gamma=10\%$. Then, the correlation coefficient
$\rho$ between $P$ and $\langle\sigma_{x,y}\rangle$ as a function of
$\Omega_{\text{R0}}$ is calculated to evaluate under how strong Raman pulse
can we ignore the kinetic energy term. The correlation coefficient is written
as
$\rho(P,\langle\sigma_{i}\rangle)=\text{cov}(P,\langle\sigma_{i}\rangle)/{\sigma_{P}\sigma_{\langle\sigma_{i}\rangle}}$,
where $\text{cov}(P,\langle\sigma_{i}\rangle)$ and
$\sigma_{P,\langle\sigma_{i}\rangle}$ are the covariance and the standard
deviation, respectively. For $\rho=0$, $P$ and $\langle\sigma_{i}\rangle$ are
completely uncorrelated; For $\rho=1$, $P$ is exactly same as
$\langle\sigma_{i}\rangle$. As a example, the correlation coefficient $\rho$
versus the strength of Raman pulse $\Omega_{\text{R0}}$ for $\Delta\varphi=0$
is shown in Fig.S2(a). $\rho$ increases from $\sim 0$ to 1 when
$\Omega_{\text{R0}}$ is tuned from $\sim 0$ to $7E_{\rm{r}}$. Meanwhile, the
patterns $P(\bm{q},t_{\text{R}})$ become more and more consistent with
$\left\langle\sigma_{y}\right\rangle$ when increasing $\Omega_{\text{R0}}$. In
experiment, we select $\Omega_{\text{R0}}\approx 3.4E_{\rm{r}}$ to realize
Raman pulse, which is enough to rotate $\left\langle\sigma_{z}\right\rangle$
to $\left\langle\sigma_{x,y}\right\rangle$ with $\rho\approx 0.97$. On the
other hand, intuitively, the duration time of $\pi/2$ Raman pulse
$t_{\text{R}}=10\mu s$ is much smaller than characteristic time of Raman
lattices $t_{\text{c}}=h/E_{\rm{r}}\approx 270\mu s$, and the atoms could
hardly evolve within the time $t_{\text{R}}$.

##  .3 The sensitivities of the Bloch-state tomography

The sensitivities of the Bloch-state tomography are determined by the
parameters associated with the Raman pulse, including the strength
$\Omega_{\rm{R0}}$ and the detuning $\delta_{\rm{R}}$ of the Raman pulse, as
well as the relative phase between the Raman lattice and the Raman pulse
$\Delta\varphi$. We estimate the effects of $\Omega_{\rm R0}$,
$\delta_{\rm{R}}$ and $\Delta\varphi$ when measuring
$\langle\sigma_{x,y}(\bm{q})\rangle$ via calculating the correlation
coefficient $\rho$ between $P(\bm{q},t_{\rm{R}})$ and
$\langle\sigma_{x,y}(\bm{q})\rangle$ for the lowest band, respectively.
Figure. S2 shows an example for $\langle\sigma_{y}(\bm{q})\rangle$. $\rho$
increases from $\sim 0$ to 1 as $\Omega_{\rm R0}$ is tuned from $\sim 0$ to
7$E_{\rm{r}}$ (Fig. S2(a)). As long as $\Omega_{\rm R0}$ is strong enough,
$P(\bm{q},t_{\rm{R}})$ basically does not change with $\Omega_{\rm R0}$. As
$\Delta\varphi$ tunes, $\rho$ oscillates with the amplitude of 1 (Fig. S2(b)).
$P$ is insensitive to changes in $\Delta\varphi$ close to $\pi/2$. $P$ changes
only slightly when $\delta_{\rm R}$ is adjusted from $-2E_{\rm r}$ to $2E_{\rm
r}$ (Fig. S2(c)). In short, $P$ is rather insensitive to the fluctuation of
the parameters, which is close to the selected parameters by this experiment,
i.e., $\Omega_{\rm{R0}}\approx 3.4E_{\rm{r}}$, $\delta_{\rm{R}}=0$ and
$\Delta\varphi=\pi/2$.

In experiment, we also measure the standard deviation of these three
parameters. The standard deviation of $\Omega_{\rm R0}$, $\delta_{\rm{R}}$ and
$\Delta\varphi$ are respectively $\pm 0.046E_{\rm r}$, $\pm 0.01\pi$ and $\pm
0.027E_{\rm r}$ (gray shaded regions in Fig. S2). For all three cases,
$\rho>0.97$ within the standard deviation. Therefore, the Raman pulse is very
stable. In other words, the Bloch-state tomography is insensitive to the
perturbation of the present parameters.

Figure S3: The expectation value of three Pauli matrices
$\langle\sigma_{x,y,z}\rangle$ in the lowest band and the relative phase
between the eigenfunction $u_{\uparrow,\downarrow}$ under two-band tight-
binding approximation of the Hamiltonian Eq.(1) (a) and the multiple bands
determined by Eq. (1) (b). The yellow solid curves are determined by
$\langle\sigma_{x,y,z}\rangle=0$. The intersections of the yellow curves in
$\langle\sigma_{x,y}\rangle$ locate the phase vortices, which are marked by
red squares. Here, we emphasize that the normalization for $P_{x,y,z}$ does
not change the relative phase between the eigenfunction
$u_{\uparrow,\downarrow}$ in the FBZ. Parameters:
$(V_{0},\Omega_{0},\delta)=(4,1,-0.2)E_{\rm{r}}$; $t_{0}\approx 0.09\propto
V_{0}$, $t_{\rm{so}}\approx-0.05\propto\Omega_{0}$.

##  .4 The origin of the phase vortices deviating from the high-symmetry
points

The asymmetry in the phase vortices in Fig.3 stems mainly from the effects of
the higher bands, since the eigenfunction of the lowest band is modified by
the higher bands. We calculate the expectation values of three Pauli matrices
by both the two-bands tight-binding approximation (TBA) of the Eq. (1) (see
below) and the Eq. (1) with multiple-bands. For simplicity,
$\langle\sigma_{x,y,z}\rangle$ here refer to the results evaluated in the
lowest band. (a) The Bloch Hamiltonian within TBA is $\mathcal{H}_{\rm
TBA}=\bm{h}\cdot\bm{\sigma}=h_{x}\sigma_{x}+h_{y}\sigma_{y}+h_{z}\sigma_{z}=2t_{\rm
so}\sin{q_{y}}\sigma_{x}+2t_{\rm so}\sin{q_{x}}\sigma_{y}+(\delta/2-2t_{\rm
so}(\cos q_{x}+\cos q_{y}))\sigma_{z}$, which gives
$\langle\sigma_{x,y,z}\rangle=-h_{x,y,z}/|\bm{h}|$. (b)
$\langle\sigma_{x,y,z}\rangle$ of the Hamiltonian Eq.(1) with the multiple-
bands are obtained by the formulas
$\langle\sigma_{x,y,z}\rangle=P_{x,y,z}/\sqrt{P_{x}^{2}+P_{y}^{2}+P_{z}^{2}}$
and $P_{x,y,z}=\langle\Psi(\bm{q})|\sigma_{x,y,z}|\Psi(\bm{q})\rangle$, where
$|\Psi(\bm{q})\rangle$ is the eigenfunction of the Hamiltonian Eq.(1). Here,
the eigenfunction of the lowest band is modified by the coupling between the
lowest band and the higher bands. We normalize $P_{x,y,z}$ for each band in
order to calculate the eigenfunctions, the Berry curvature and the quantum
metric tensor of the lowest bands. Basically, such normalization is reasonable
for the present parameters and hardly changes the symmetry of $P$ (see Fig. S2
and Fig. S3 for $\langle\sigma_{y}\rangle$).

In Fig. S3, we plot the expectation values of the three Pauli matrices
$\langle\sigma_{x,y,z}\rangle$ and the relative phase between the
eigenfunction $u_{\uparrow,\downarrow}$ with the lowest band for the above two
cases. The relative phase between the eigenfunction is obtained by
$\langle\sigma_{x}\rangle$ and $\langle\sigma_{y}\rangle$, i.e.,
$\phi_{\bm{q}}=\arg(\left\langle\sigma_{x}\right\rangle+i\left\langle\sigma_{y}\right\rangle)\in[0,2\pi)$.
$\langle\sigma_{z}\rangle$ ($\langle\sigma_{x,y}\rangle$) is symmetric along
the $q_{x}$ and $q_{y}$ ($q_{x,y}$) directions within the TBA of Eq. (1),
being consistent with the multiple-band results determined by Eq. (1).
However, there also exist some subtle differences. Figure S3(a) shows two
straight lines in $\langle\sigma_{x,y}\rangle=0$ along the $q_{x,y}$
directions, whereas these two straight lines are bent due to the effects of
the higher bands, shown in Fig. S3(b). The phase vortices are determined by
$(\langle\sigma_{x}\rangle,\langle\sigma_{y}\rangle,\langle\sigma_{z}\rangle)=(0,0,\pm
1)$, whose locations are indicated by the intersections of the yellow straight
or bent lines in the FBZ. We then see that, due to the effects of the higher
bands, the position of the phase vortices deviates from the high-symmetry
points.

##  .5 The experimental protocol

We measure the expectation values of three Pauli matrices
$\langle\sigma_{x,y,z}\rangle$ by applying a Raman pulse to the eigen-state of
Raman lattices. We divide the experimental protocol into three steps:

Figure S4: (a) The expectation value of Pauli matrices
$\langle\sigma_{y,z}\rangle^{s1,s2}$ and the ratio of the atoms $p^{s1,s2}$ in
the FBZ. $\langle\sigma_{y,z}\rangle_{\rm T}$ and the calculated spin textures
$P_{\text{m},\langle\sigma_{y,z}\rangle}$ is obtained by Eq. S9 and Eq. S8,
respectively. (b) The lowest two bands $s1$ and $s2$. Parameters:
$(V_{0},\Omega_{0},\delta)=(4,1,-0.2)E_{\rm{r}},t_{\rm{R}}=10\mu
s,\delta_{\rm{R}}=0,\Omega_{\rm{R0}}\approx 3.4E_{\rm{r}}$.

_i. The preparation of the state_ –The atoms is cooled by evaporative cooling.
Whereafter, the atoms are loaded adiabatically into Raman lattices in 100ms by
ramping the intensity of $\bm{E}_{x,y}$ with the temperature $\sim
100\text{nK}$, which is controlled by the power of $\rm{RF}_{1,3}$. Meanwhile,
the phase and frequency of $\rm{RF}_{1,3}$ are fixed. And $\rm{AOM}_{3,4}$ and
$\rm{RF}_{2,4}$ are turned off. Then, the atoms occupy each momentum point in
the Brillouin zone; most of them are in the lowest band and a small part of
them are in the higher band.

_ii. The realization of the Raman pulse_ –The Raman pulse is realized by
simultaneously performing 4 manipulations on the beams $\bm{E}_{x,y}$: 1) The
reflected beams of $\bm{E}_{x,y}$ are shut down by turning on
$\rm{AOM}_{3,4}$. The ratio of beams $\bm{E}_{x,y}$ reflected back the atoms
$\gamma$ is about 10$\%$, which is evaluated by calibrating the optical
lattice depth with $\rm{AOM}_{3,4}$ turned on. The reason for $\gamma>0$ is
that the diffraction efficiency of AOM can not reach 100$\%$. 2) The strength
of the Raman pulse is set as $\Omega_{\text{R0}}\approx 3.4E_{\rm{r}}$ by
tuning the intensity of $\bm{E}_{x,y}$. 3) The relative phase $\Delta\varphi$
is adjusted by tuning the phase of $\bm{E}_{x,y}$. $\Delta\varphi$ is
determined by matching the experimental measurements and numerical
calculations [In fact, the absolute $\Delta\varphi$ is not necessary. In
experiment, only a difference of $\Delta\varphi=\pi/2$ is necessary for
measuring $\langle\sigma_{x}\rangle$ and $\langle\sigma_{y}\rangle$. In order
to compare the experiment and theory at the same $\Delta\varphi$, we make the
phase of the experimental data consistent with the numerical calculation by
shifting a offset of the experimental $\Delta\varphi$.], as depicted in
Fig.1(e). And the stability of $\Delta\varphi$ is assessed by comparing the
spin polarization $P_{\rm{m}}$ in the presence and absence of the Raman pulse.
The experimental measurements show that the uncertainty of $P_{\rm{m}}$ are
both 0.03 for 100 photos regardless of the presence of the Raman pulse, which
indicates that the uncertainty of $\Delta\varphi$ does not disturb the
experimental observations. 4) The detuning of Raman pulse $\delta_{\rm{R}}$ is
set as 0 by adjusting the frequency of $\bm{E}_{x,y}$. Meanwhile, ratio-
frequency signals are switched from $\rm{RF}_{1,3}$ to $\rm{RF}_{2,4}$ by
$\rm{Switch}_{1,2}$, and the switching speed is about 200ns, which is much
less than $t_{\text{R}}=10\mu$s. The intensity, phase and frequency of
$\bm{E}_{x,y}$ are exactly controlled by the power, phase and frequency of
$\rm{RF}_{1,2,3,4}$ input to $\rm{AOM}_{1,2}$, respectively.

_iii. The detection of the spin polarization_ –After the Raman pulse is
applied to the atoms for a certain time $t_{\text{R}}$, all the lasers and the
bias field are switched off in less than $1\mu$s. Then, the Stern-Gerlach
magnetic field is turned on so that the atom cloud of $|\uparrow\rangle$ and
$|\downarrow\rangle$ are separated. After the atoms expand freely for 25ms, we
take photos of the atoms along $\hat{z}$ direction to obtain the atomic
distribution in $|\uparrow\rangle$ and $|\downarrow\rangle$. Thus, the atom
numbers $n_{\uparrow}(\bm{q})$ and $n_{\downarrow}(\bm{q})$ are obtained.

Figure S5: The spin polarization as a function of the duration time
$t_{\rm{R}}$ of Raman pulse with $T=100$nK. The dots with error bars (solid
curve) is experimental data (the fitting using damped sine function). The spin
textures are from the original experimental measurements and numerical
calculations at $t_{\rm{R}}=20\mu s$. Parameters:
$(V_{0},~{}\Omega_{0},~{}\delta)=(4.0,1.0,-0.2)E_{\rm{r}}$,
$\Delta\varphi=\pi,~{}\Omega_{\rm{R0}}\approx 3.4E_{\rm{r}}$.

##  .6 Data analysis

We first analysis the experimental protocol measuring the spin textures.

i. The atoms are loaded into Raman lattices in 100ms, which can be regarded as
an adiabatic process and then induces the populations of the atom in
accordance with the Bose distribution. The ratio of the atoms populated to the
momentum point $\bm{q}$ of the $j$th band $p^{j}(\bm{q})\in[0,1]$ is
calculated numerically by the Bose distribution with the input parameters,
including atomic temperature $T$, atomic density $n_{d}$ and the band
structures, as shown in Fig. S4. Here, $T$ and $n_{d}$ are measured
experimentally, and the band structures are obtained by diagonalizing the
Hamiltonian Eq.(1) in the main text. The atoms are almost entirely distributed
in the two lowest $s$ bands when the atomic temperature is $T=100$nK and
atomic density is $n_{d}=3\times 10^{18}m^{-3}$. About 67% (33%) atoms
populate to the $s1~{}(s2)$ band for $\delta=-0.2E_{\rm{r}}$.

ii. After applying the Raman pulse, the spin polarization evolves according to
by Eq. (3) in the main text. Numerically, the measured spin textures after
applying a Raman pulse can be decomposed into the sum of the individual bands,
i.e.,

$\displaystyle P_{\rm
m}(\bm{q})=\frac{n_{\uparrow}(\bm{q})-n_{\downarrow}(\bm{q})}{n_{\uparrow}(\bm{q})+n_{\downarrow}(\bm{q})}=\frac{\sum_{j=1}^{j=N}p^{j}(\bm{q})[A_{\uparrow}^{j}(\bm{q})-A_{\downarrow}^{j}(\bm{q})]}{\sum_{j=1}^{j=N}p^{j}(\bm{q})[A_{\uparrow}^{j}(\bm{q})+A_{\downarrow}^{j}(\bm{q})]},$
(S8)

where $A_{\uparrow}^{j}=|\widetilde{\Psi}_{\uparrow}^{j}(\bm{q})|^{2}$
($A_{\downarrow}^{j}=|\widetilde{\Psi}_{\downarrow}^{j}(\bm{q})|^{2}$) is
determined by the evolved state
$|\widetilde{\Psi}_{\uparrow}^{j}(\bm{q})\rangle$
($|\widetilde{\Psi}_{\downarrow}^{j}(\bm{q})\rangle$) of $\mid\uparrow\rangle$
($\mid\downarrow\rangle$) in the $j$th band. Here, we have assumed that
$p^{j}(\bm{q})$ does not change over time with $t_{\rm{R}}\in[0,10)\mu s$,
which is reasonable according to the following numerical evidence and
experimental data.

For the numerical evidence, we can calculate the expectation values of three
Pauli matrices with non-zero temperature by the sum of
$\langle\sigma_{x,y,z}\rangle^{j}$ with the ratio $p^{j}(\bm{q})$, i.e.,

$\langle\sigma_{x,y,z}(\bm{q})\rangle_{\rm
T}=\sum_{j=1}^{j=N}p^{j}(\bm{q})\langle\sigma_{x,y,z}(\bm{q})\rangle^{j}.$
(S9)

From Eq. (S9), $\langle\sigma_{x,y,z}(\bm{q})\rangle_{\rm T}$ contains the
corrections of higher-band population. To validate our assumptions, we need to
verify that the expectation value of Pauli matrices measured using Raman
pulses is consistent with directly calculated using the eigenstate, i.e.,
$P_{\text{m},i}\simeq\langle\sigma_{i}\rangle_{\rm T}~{}(i=x,y,z)$. Hence, we
calculate numerically Eq. (S9) and Eq. (S8) with the atomic temperature
$T=100$nK and atomic density $n_{d}=3\times 10^{18}m^{-3}$, as shown in Fig.
S4. For individual bands, the spin textures
$\langle\sigma_{y,z}\rangle^{s1,s2}$ have high contrast. However, about 33% of
the atoms populate the s2 band, which reduces the contrast of the spin
textures. The correlation coefficient $\rho$ between
$P_{\text{m},\langle\sigma_{i}\rangle}$ and $\langle\sigma_{i}\rangle_{\rm T}$
is more than 0.9 for a $\pi/2$ Raman pulse of 10$\mu$s, which indicates
$P_{\text{m},i}\simeq\langle\sigma_{i}\rangle_{\rm T}$.

For the experimental data, we compare the band gap and the Rabi frequency of
the Raman pulse to validate our assumptions. The band gap is from a few
hundred hertz to a few kilohertz depending on which quasimomentum point the
atom is located, which is much less (more) than the Rabi frequency of Raman
pulse 25kHz (See Eq. (3) in the main text). Hence, the atoms moving to other
bands should not be observed experimentally. To verify it, we measure
$P_{\rm{tot}}=(N_{\uparrow}-N_{\downarrow})/(N_{\uparrow}+N_{\downarrow})$
with the atom number of $|\uparrow,\downarrow\rangle$
$N_{\uparrow,\downarrow}$, as shown in Fig. S5. The period of
$P_{\rm{tot}}(t_{\rm{R}})$ is about 35$\mu s$, which is determined by the
strength $\Omega_{\rm{R0}}$ and has nothing to do with the gap. Further, the
spin texture at $t_{\rm{R}}=20\mu s$ is consistent with the numerical
calculations, implying the atoms moving to other bands can be ignored when
$t_{\rm{R}}=10\mu s$.

Figure S6:  The measured spin textures and the revised spin textures in the
lowest band. Parameters: $(V_{0},\Omega_{0})=(4,1)E_{\rm{r}}$.

Now, we remove the higher band population in the measured spin textures. The
higher band population is removed to get the spin polarization of the lowest
band, that is,
$P_{0}(\bm{q})=(n_{\uparrow}^{s1}(\bm{q})-n_{\downarrow}^{s1}(\bm{q}))/(n_{\uparrow}^{s1}(\bm{q})+n_{\downarrow}^{s1}(\bm{q}))$
with atom number of the lowest band for spin up $n_{\uparrow}^{s1}(\bm{q})$
and spin down $n_{\downarrow}^{s1}(\bm{q})$. To this end, The proportion of
the atoms in the lowest band $\eta_{\uparrow,\downarrow}(T)$ needs to be
obtained. According to the evolved states
$|\widetilde{\Psi}_{\uparrow}^{j}(\bm{q})\rangle$
($|\widetilde{\Psi}_{\downarrow}^{j}(\bm{q})\rangle$) of $\mid\uparrow\rangle$
($\mid\downarrow\rangle$) in the $j$th band calculating by Eq. (S6) at
$t_{\rm{R}}=10\mu s$, $n_{\uparrow}^{j}(\bm{q})=p^{j}A_{\uparrow}^{j}$
($n_{\downarrow}^{j}(\bm{q})=p^{j}A_{\downarrow}^{j}$) with
$A_{\uparrow}^{j}=|\widetilde{\Psi}_{\uparrow}^{j}(\bm{q})|^{2}$
($A_{\downarrow}^{j}=|\widetilde{\Psi}_{\downarrow}^{j}(\bm{q})|^{2}$). We
then get the proportions
$\eta_{\uparrow,\downarrow}(T)=n_{\uparrow,\downarrow}^{s1}/n_{\uparrow,\downarrow}$,
where $n_{\uparrow,\downarrow}$ are obtained by experimental measurements.
Thus, the spin polarization of the lowest band is expressed as
$P_{0}(\bm{q})=[\eta_{\uparrow}(T)n_{\uparrow}-\eta_{\downarrow}(T)n_{\downarrow}]/[\eta_{\uparrow}(T)n_{\uparrow}+\eta_{\downarrow}(T)n_{\downarrow}]$.
Whereafter, we apply the correction to the measured data and the revised spin
textures $\langle\sigma_{x,y,z}\rangle$ are drawn in Fig.S6. Finally,
$\langle\sigma_{x,y,z}\rangle$ are displayed in the main text after
normalization
$\sqrt{\langle\sigma_{x}\rangle^{2}+\langle\sigma_{y}\rangle^{2}+\langle\sigma_{z}\rangle^{2}}=1$
and smoothed between adjacent momentum points. The smoothing is achieved by
calculating the average of the spin polarizations within a square area. For
example, the smoothing of the center point in the black square of Fig. S6 is
obtained by averaging the spin polarizations in this black square. The area of
the black square is about 0.3% of the area of the FBZ. The FBZ with around
12000 momentum points has a high resolution in momentum space and each
momentum point is about 2.6$\mu$m, given by the pixel size of the camera.

Figure S7: The amplitude of the oscillations for the $s1$ band
$\text{Amp}^{s1}$ and finite temperature $\text{Amp}_{\rm T}$ in the FBZ. The
asymmetry is from the coupling between the $s1$ band and other bands.
Parameters: $(V_{0},\Omega_{0},\delta)=(4,1,-0.2)E_{\rm{r}},t_{\rm{R}}=10\mu
s,\delta_{\rm{R}}=0,\Omega_{\rm{R0}}\approx 3.4E_{\rm{r}}$.

Besides, we should also clarify that the higher-band population directly
reduces the contrast of the spin textures and indirectly reduces the amplitude
of the oscillations. To this end, we generalize Eq. (3) in the main text to a
situation with finite temperature. For finite temperature, the amplitude of
the oscillations reads $\text{Amp}_{\rm T}=\sqrt{\langle\sigma_{z}\rangle_{\rm
T}^{2}+(\langle\sigma_{y}\rangle_{\rm
T}\cos\Delta\varphi+\langle\sigma_{x}\rangle_{\rm T}\sin\Delta\varphi)^{2}}$;
And the amplitude in $s1$ band
$\text{Amp}^{s1}=\sqrt{(\langle\sigma_{z}\rangle^{s1})^{2}+(\langle\sigma_{y}\rangle^{s1}\cos\Delta\varphi+\langle\sigma_{x}\rangle^{s1}\sin\Delta\varphi)^{2}}$.
$\text{Amp}_{\rm T}<\text{Amp}^{s1}$ since $0<p^{j}<1$. $\text{Amp}_{\rm T}$
and $\text{Amp}^{s1}$ with $\Delta\varphi=0$ are shown in Fig. S7. According
to the calculation for the amplitude of the oscillations, a decrease behavior
of the amplitude of the oscillations at finite temperature directly originates
from the reduction of the contrast of spin polarizations. However, the
reduction of the contrast of spin polarizations is directly caused by the
atoms populate in the higher bands. In other words, the atoms populate in the
higher bands indirectly reduce the amplitude of the oscillations.