Observation of symmetry-protected topological band with ultracold fermions

TL;DR

This paper reports the experimental observation of a symmetry-protected topological band in ultracold fermions within a one-dimensional optical lattice, highlighting the role of symmetry in topological phases and their dynamics.

Contribution

It demonstrates the realization of a noninteracting SPT band protected by magnetic and chiral symmetries in ultracold atoms, and explores quench dynamics between topologically distinct regimes.

Findings

Observation of a symmetry-protected topological band in ultracold atoms

Distinct spin dynamics observed in quenches to trivial and nontrivial regimes

Topological properties confirmed via Bloch states at symmetric momenta

Abstract

Symmetry plays a fundamental role in understanding complex quantum matter, particularly in classifying topological quantum phases, which have attracted great interests in the recent decade. An outstanding example is the time-reversal invariant topological insulator, a symmetry-protected topological (SPT) phase in the symplectic class of the Altland-Zirnbauer classification. We report the observation for ultracold atoms of a noninteracting SPT band in a one-dimensional optical lattice and study quench dynamics between topologically distinct regimes. The observed SPT band can be protected by a magnetic group and a nonlocal chiral symmetry, with the band topology being measured via Bloch states at symmetric momenta. The topology also resides in far-from-equilibrium spin dynamics, which are predicted and observed in experiment to exhibit qualitatively distinct behaviors in quenching to trivial and nontrivial regimes, revealing two fundamental types of spin-relaxation dynamics related to bulk topology. This work opens the way to expanding the scope of SPT physics with ultracold atoms and studying nonequilibrium quantum dynamics in these exotic systems.

Equations

$$H=\bigr{[}\frac{p_{x}^{2}}{2m}+\frac{V^{\rm latt}_{\uparrow}(x)+V^{\rm latt}_{\downarrow}(x)}{2}\bigr{]}\otimes{\bf\hat{1}}+\bigr{[}\frac{\delta}{2}+\frac{V^{\rm latt}_{\uparrow}(x)-V^{\rm latt}_{\downarrow}(x)}{2}\bigr{]}\sigma_{z}+{\cal M}(x)\sigma_{x}.$$

$$\dot{\rho }=-\frac{i}{\hslash }[H^{\prime },\rho ]+\gamma \left(L\rho L^{†}-\frac{1}{2}\{L^{†}L,\rho \}\right),$$

$${\Phi }_{\alpha \uparrow }(q_x)=\frac{1}{\sqrt{L}}{e}^{i(q+2k_0\alpha )x},{\Phi }_{\beta \downarrow }(q_x)=\frac{1}{\sqrt{L}}{e}^{i(q+2k_0\beta +k_0)x}.$$

$$|{\Psi }_n(q_x)⟩=\sum _{\alpha }{A}_{\alpha }^{(n)}{\Phi }_{\alpha \uparrow }(q_x)|\uparrow ⟩+\sum _{\beta }{B}_{\beta }^{(n)}{\Phi }_{\beta \downarrow }(q_x)|\downarrow ⟩,$$

$${\bar{\sigma }}_z^{(n)}(q_x)$$

$$H_{\mathrm{TB}}=\sum _{⟨i,j⟩,\sigma }(t_{\uparrow }^{ij}{c}_{i\uparrow }^{†}{c}_{j\uparrow }+t_{\downarrow }^{ij}{c}_{i\downarrow }^{†}{c}_{j\downarrow })+\sum _{⟨i,j⟩}(t_{\mathrm{so}}^{ij}{c}_{j\uparrow }^{†}{c}_{j+1\downarrow }+h.c.)+\sum _i{m}_z(n_{i\uparrow }-n_{i\downarrow }),$$

$$t_{\sigma }^{ij}=\int dx{\varphi }_{s\sigma }^{(i)}(x)\left[\frac{{\hslash }^2{k}_x^2}{2m}+V_{\sigma }^{\mathrm{latt}}(x)\right]{\varphi }_{s\sigma }^{(j)}(x),$$

$$t_{\mathrm{so}}^{ij}=\int dx{\varphi }_{s\uparrow }^{(i)}(x)\mathcal{M}(x){\varphi }_{s\downarrow }^{(j)}(x),$$

$$\Delta m_z=\int dx{\varphi }_{s\uparrow }^{(0)}(x)\left[\frac{{\hslash }^2{k}_x^2}{2m}+V_{\uparrow }{\cos }^2(k_{0}x)\right]{\varphi }_{s\uparrow }^{(0)}(x)-\int dx{\varphi }_{s\downarrow }^{(0)}(x)\left[\frac{{\hslash }^2{k}_x^2}{2m}+V_{\downarrow }{\cos }^2(k_{0}x)\right]{\varphi }_{s\downarrow }^{(0)}(x),$$

$$t_{\sigma }=-\int dx{\varphi }_{s\sigma }^{(0)}(x)\left[\frac{{\hslash }^2{k}_x^2}{2m}+V_{\sigma }{\cos }^2(k_{0}x)\right]{\varphi }_{s\sigma }^{(0)}(x-a),$$

$$t_{\mathrm{so}}=M_0\int dx{\varphi }_{s\uparrow }^{(0)}(x)\cos (k_{0}x){\varphi }_{s\downarrow }^{(0)}(x-a).$$

$$H_{\mathrm{TB}}=\sum _{q_x}{\Psi }_{q_x}^{†}\mathcal{H}(q_x){\Psi }_{q_x},$$

$$\mathcal{H}(q_x)=-2t_1\cos (q_{x}a)\otimes 1+[m_z-2t_0\cos (q_{x}a)]{\sigma }_z-2t_{\mathrm{so}}\sin (q_{x}a){\sigma }_y.$$

$$\mathcal{H}(q_x)=-2t_0\cos (q_{x}a){\sigma }_z-2t_{\mathrm{so}}\sin (q_{x}a){\sigma }_y-2t_1\cos (q_{x}a)\otimes 1+\vec{m}\cdot \vec{\sigma }.$$

$$M_{0x}\sin (k_{0}x){\sigma }_x,\mathrm{and}{M}_{0y}\sin (k_{0}x){\sigma }_y,$$

$$m_x=M_{0x}\int dx{\varphi }_{s\uparrow }^{(0)}(x)\sin (k_{0}x){\varphi }_{s\downarrow }^{(0)}(x),m_y=M_{0y}\int dx{\varphi }_{s\uparrow }^{(0)}(x)\sin (k_{0}x){\varphi }_{s\downarrow }^{(0)}(x).$$

$$\mathcal{S}={\sigma }_x,$$

$$M_x=\mathcal{T}{\mathcal{M}}_x={\sigma }_{z}K\otimes R_x,$$

$$\mathcal{S}={\sigma }_z\otimes T_x(k_0)\otimes R_x,$$

$$H_{\mathrm{TB}}=\sum _j(\mathcal{A}{c}_j^{†}{c}_{j+1}+{\mathcal{A}}^{†}{c}_{j+1}^{†}{c}_j),$$

$$\mathcal{A}=-t_0{\sigma }_z+i{t}_{\mathrm{so}}{\sigma }_y-t_{1}1.$$

$$|{\psi }_{L/R}(x_j)⟩=\frac{1}{\sqrt{\mathcal{L}}}{\lambda }^{x_j/a}|{\phi }_{L/R}⟩,$$

$$(\lambda \mathcal{A}+{\lambda }^{-1}{\mathcal{A}}^{†})|{\phi }_{L/R}⟩=E|{\phi }_{L/R}⟩,$$

$$E_{\pm }(\lambda )=-t_1(\lambda +{\lambda }^{-1})\pm \sqrt{t_0^2(\lambda +{\lambda }^{-1}{)}^2-t_{\mathrm{so}}^2(\lambda -{\lambda }^{-1}{)}^2}.$$

$${\lambda }_{L/R}^{\pm }=\mp i{\lambda }_{L/R},$$

$${\lambda }_{\mathrm{L}}={\left(\frac{\sqrt{t_0^2-t_1^2}-t_{\mathrm{so}}}{\sqrt{t_0^2-t_1^2}+t_{\mathrm{so}}}\right)}^{1/2},{\lambda }_{\mathrm{R}}=1/{\lambda }_{\mathrm{L}}.$$

$$|{\phi }_{\mathrm{L}}⟩=\frac{1}{\sqrt{1+t_2}}(\genfrac{}{}{0ex}{}{1}{-\sqrt{t_2}}),$$

$$|{\phi }_{\mathrm{R}}⟩=\frac{1}{\sqrt{1+t_2}}(\genfrac{}{}{0ex}{}{1}{\sqrt{t_2}}).$$

$$|{\psi }_{\mathrm{L}}(x_j)⟩=\frac{1}{\sqrt{\mathcal{L}}}{\lambda }_{\mathrm{L}}^{x_j/a}\left(e^{i\pi x_j/2a}-e^{-i\pi x_j/2a}\right)|{\phi }_{\mathrm{L}}⟩$$

$$|{\psi }_{\mathrm{R}}(x_j)⟩=\frac{1}{\sqrt{\mathcal{L}}}{\lambda }_{\mathrm{R}}^{(x_j-L)/a}\left[e^{i\pi (x_j-L)/2a}-e^{-i\pi (x_j-L)/2a}\right]|{\phi }_{\mathrm{R}}⟩.$$

Full text

{affiliations}

Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

International Center for Quantum Materials, School of Physics, Peking University, Beijing, China

Collaborative Innovation Center of Quantum Matter, Beijing 100871, China

Introduction

The discovery of the quantum Hall effect in a two-dimensional (2D) electron gas [1, 2] brought about a new fundamental concept, topological quantum phase whose characterization is beyond Landau symmetry-breaking theory, to condensed-matter physics [3]. Recent extensive studies have generated two broad categories of topological matter, the topologically ordered phases [4, 5, 6] and symmetry-protected topological (SPT) phases[7, 8, S3, 10], which exhibit long-range and short-range quantum entanglement, respectively. Unlike topological orders which are robust against any local perturbations, a SPT phase with a bulk gap has gapless or degenerate boundary modes that are only robust against local perturbations respecting the given symmetries. The earliest examples of SPT phases include the one-dimensional (1D) Su-Schrieffer-Heeger (SSH) model for fermions [11] and spin-$1$ antiferromagnetic chain for bosons [12]. The search for new SPT phases has been greatly revived in the recent decade due to the groundbreaking discovery of time-reversal invariant topological insulators in 2D and 3D materials, which exhibit symmetry-protected helical edge or surface modes in the boundary and are characterized by a $\mathbb{Z}_{2}$ invariant [13, 14]. The topological insulators also inspired generalizations to the SPT phases arising from spatial symmetries, in unconventional superconductors and superfluids [15].

Despite the broad classes of SPT phases predicted in theory, only a small portion of such phases have been observed in experiment. In comparison with solid state materials, the ultracold atoms are extremely clean systems which can provide controllable platforms in exploring new SPT phases, with many theoretical schemes having been proposed [S1, 18, 17]. To date, some novel topological features of the SSH model have been probed in experiments with ultracold bosons [19, 20], while the realization of a SPT phase, which has to be in a fermionic system for a noninteracting phase, is not available. Here we propose and realize for the first time a new SPT phase for ultracold fermions which is beyond particle-hole or chiral symmetry protection as usually required in Altland-Zirnbauer (AZ) classification of 1D states [S2, 15]. We measure the topology via Bloch states at symmetric momenta [22] and the quench dynamics, and unveil a deep topology-dependent mechanism for the spin relaxation in this SPT phase.

Symmetry-protected topological phases

We start with the Hamiltonian for spin-$1/2$ ultracold fermions trapped in a 1D optical Raman lattice potential (Fig. 1a), whose realization in experiment will be presented later:

[TABLE]

Here $p^{2}_{x}/2m$ is the kinetic energy for motion in $x$-direction, ${\bf\hat{1}}$ is a two-by-two unit matrix, $\sigma_{x,y,z}$ are Pauli matrices in spin space, $V^{\rm latt}_{\uparrow,\downarrow}(x)=V_{\uparrow,\downarrow}\cos^{2}(k_{0}x)$ denote the 1D optical lattice potentials for spin $|\uparrow,\downarrow\rangle$ states, respectively, with $V_{\uparrow,\downarrow}$ being the lattice depths and $k_{0}=\pi/a$ ($a$ is the lattice constant), the Raman lattice potential ${\cal M}(x)=M_{0}\cos(k_{0}x)$ with amplitude $M_{0}$, and $\delta$ denotes the two-photon detuning of Raman coupling.

The Hamiltonian (1) realizes novel topological phases protected by symmetries. With the lattice potential $V^{\rm latt}_{\uparrow,\downarrow}(x)$ the spin-conserved hopping ($t_{\uparrow,\downarrow}$) is induced, while the Raman potential ${\cal M}(x)$ accounts for the hopping ($t_{\rm so}$) that flips atomic spin (Fig. 1a). The Bloch Hamiltonian for the lowest $s$-band physics can be obtained by [23] ${\cal H}(\vec{m})=-2t_{0}\cos q_{x}a\sigma_{z}+2t_{\rm so}\sin q_{x}a\sigma_{y}+\vec{m}\cdot\sigma+2t_{1}\cos q_{x}a{\bf\hat{1}}$, where we take $\vec{m}=(m_{x},m_{y},m_{z})$ for a generic study, $q_{x}$ is Bloch quasi-momentum, and the hopping coefficients $t_{0/1}=(t_{\uparrow}\pm t_{\downarrow})/2$. It is known that the Hamiltonian ${\cal H}$ with $m_{x}=0$ exhibits a chiral symmetry defined by $\sigma_{x}{\cal H}\sigma_{x}=-{\cal H}$ in the spin-independent lattice regime $V_{\uparrow}^{\rm latt}=V_{\downarrow}^{\rm latt}$, which gives $t_{\uparrow}=t_{\downarrow}$, and belongs to the AIII class according to the AZ classification [S2, S1]. The topology of such AIII class phase is quantified by an integer winding number ($\mathbb{Z}$) in the single particle regime, pictorially characterized by the circles that spin of Bloch states winds over the first brillouin zone (FBZ), and determines the number of midgap end states due to bulk-boundary correspondence. Furthermore, in the interacting regime, it was shown that the topology of the 1D AIII class topological insulator is reduced from $\mathbb{Z}$ to $\mathbb{Z}_{4}$ [S1], which is closely related to the Fidkowski-Kitaev $\mathbb{Z}_{8}$-classification of 1D interacting Majorana chains of BDI class [24]. The AIII class topological insulator has not been studied previously in experiment.

The striking result is that a new topological phase can be achieved beyond the chiral symmetry protection when the lattice $V^{\rm latt}_{\uparrow,\downarrow}$ and $t_{\uparrow,\downarrow}$ are spin-dependent, in which regime no chiral symmetry preserves and the bulk energy spectrum is asymmetric. From the numerical results in Fig. 1b one finds that the midgap and degenerate end states are obtained for $\vec{m}=m_{y}\hat{e}_{y}$ and $\vec{m}=m_{z}\hat{e}_{z}$, respectively, while only a nonzero $m_{x}$ term splits out the degeneracy. The existence of midgap or degenerate end states manifests new SPT phases, as shown below. We consider first the case with $\vec{m}=0$. One can verify that ${\cal H}(\vec{m}=0)$ satisfies a magnetic group symmetry defined as the product of time-reversal and mirror symmetries [23], giving $M_{x}=\sigma_{z}K\otimes R_{x}$ with $M_{x}{\cal H}(q_{x})M_{x}^{-1}={\cal H}(-q_{x})$, where $K$ is complex conjugate and $R_{x}$ denotes the spatial reflection along $x$ axis. Besides, the Hamiltonian also satisfies a nonlocal chiral symmetry defined as ${\cal S}=\sigma_{z}\otimes T_{x}(k_{0})\otimes R_{x}$, where $T_{x}(k_{0})$ is a $k_{0}$-momentum translation, giving that ${\cal S}{\cal H}(q_{x}){\cal S}^{-1}=-{\cal H}(q_{x})$. The midgap end states at left ($|\psi_{L}\rangle$) and right ($|\psi_{R}\rangle$) hand boundaries are protected by the both symmetries under a novel mechanism. First, by definition both $M_{x}$ and ${\cal S}$ transform one of $|\psi_{L/R}\rangle$ to the other. Moreover, their commutation and anti-commutation relations with ${\cal H}$ imply that the magnetic group (nonlocal chiral) symmetry connects two states with identical (opposite) energy. For this $|\psi_{L/R}\rangle$ have to be zero energy, i.e. midgap states in the presence of both symmetries, which is the case for $\vec{m}$ along $y$ axis. The $m_{z}$ term keeps $M_{x}$ symmetry, hence protects the end state degeneracy, while $m_{x}$ term breaks $M_{x}$ symmetry and splits out the degeneracy, leading to the end state spectra in Fig. 1b. In the experiment a nonzero $m_{z}$ term can be easily engineered by manipulating the two-photon detuning $\delta$ of the Raman coupling. In this regime the magnetic group ensures that Bloch states at the two symmetric momenta $\{\Lambda_{j}\}=\{\Gamma(q_{x}=0),M(q_{x}=\pi/a)\}$ are eigenstates of $\sigma_{z}$, having spin polarization $P(\Lambda_{j})=\pm 1$, with which a $\mathbb{Z}_{2}$ invariant [22] $(-1)^{\nu}=\Pi_{j}{\rm sgn}[P(\Lambda_{j})]$ can be introduced to characterize the band topology. The topologically nontrivial (trivial) phase corresponds to $\nu=1$ (0). Fig. 1c shows the spin texture with different parameter conditions, from which the $\mathbb{Z}_{2}$ invariant can be read out directly, and Fig. 1d presents the phase diagram versus $\delta$.

Quench dynamics

We examine the spin dynamics after a quench from one SPT phase to another. The quench is performed by ramping suddenly the two-photon detuning from $\delta_{i}$ to $\delta_{f}$, with $\delta_{i}$ and $\delta_{f}$ corresponding to two topologically distinct regimes. The quantum dynamics is captured by the time-dependent density matrix $\rho(t)$, which satisfies the Lindblad master equation [25, 26]

[TABLE]

where $\gamma$ denotes the noise-induced decay rate and $L$ is the Lindblad operator characterizing effects of the environment. For a real ultracold atom system, we consider an external trapping potential for the study and the Hamiltonian $H^{\prime}=H+V_{\rm trap}$, with $V_{\rm trap}(x)=\frac{1}{2}m\omega_{x}^{2}x^{2}$ and $\omega_{x}$ being the trapping frequency. The sharp contrast between topological and trivial phases is that the spin of Bloch states over FBZ winds over all direction in $y$-$z$ plane in the former case, while is polarized to $+z$ (or $-z$) direction in the latter (Fig. 1c). This feature accounts for different fundamental mechanisms governing the quench spin dynamics, as given below.

We consider first the case without dissipation ($\gamma=0$). In this regime the time-evolution is unitary, and the spin dynamics are governed by the following two quantum processes, namely, the interband transition at each quasi momentum $q_{x}$ due to nonequilibrium population after quench, and the intraband transition induced by trapping potential between Bloch states of different momenta. The numerical simulation of the spin dynamics at $\Gamma$ and $M$ points are shown in Fig. 2a and Fig. 2b for quench to topological ($\delta_{f}=0.7E_{\rm r}$) and trivial ($\delta_{f}=2.0E_{\rm r}$) phases, respectively. In both cases, the spin dynamics exhibit a fast oscillation due to interband transitions, with the oscillation frequency determined by the local band gap at $\Gamma$ and $M$ points. However, the spin dynamics for quench to topological phase exhibit also a slower spin-wave like collective oscillation associated with a fast decay in the very beginning stage, which is absent in the quench to trivial phase. This novel behavior is a consequence of the topological spin texture or spin-orbit fields in momentum space [23]. In the topological regime, the intraband transitions induced by trapping potential randomize the fast oscillation caused by interband transitions due to the strongly momentum-dependent spin-orbit fields, leading to a fast decay in the very beginning and leaving only a collective oscillation governed by the trapping potential. Instead, in the trivial regime, all the spin states and spin-orbit fields of FBZ approximately point to the $+z$ (or $-z$) direction. Thus the trapping-induced intraband transitions have negligible effect on the spin oscillation.

We further show in Fig. 2c,d the quench dynamics with dissipation, as captured by the non-unitary part of Eq. (2). For quench to the trivial phase, since the spin dynamics are dominated by the interband oscillations, the dissipation mainly results from the decay of the excited bands. In comparison, for quench to the topological phase, the dissipation mainly describes a spin dephasing of the collective oscillation induced by trapping. The Lindblad operator $L$ then takes the form $L=(\widetilde{\sigma}_{x}+i\widetilde{\sigma}_{y})/2$ (for decay) and $L=\widetilde{\sigma}_{z}$ (for dephasing), respectively, and the latter effect is typically much weaker than the former [26]. Here $\widetilde{\sigma}_{x,y,z}$ denote the Pauli matrices in the eigenbasis of the post-quench Hamiltonian $H(\delta_{f})+V_{\rm trap}$. The numerical simulation shows that the quench dynamics from trivial to topological regime resembles the one without dissipation except for a small decay in the amplitude of collective oscillation, followed by a quick interband balance in the very beginning (Fig. 2c), while the dynamics in the other way round renders a pure decay from initial state to the final equilibrium phase (Fig. 2d). With these results we draw a novel conclusion that the dissipation and trapping potential have dominant (weak) effects on the spin dynamics with quench to trivial (topological) and topological (trivial) phases, respectively, due to the topological and trivial spin textures of the final SPT phase, as observed in the present experiment.

Experimental Result

Optical Raman lattices for ytterbium atoms

To realize the Hamiltonian (1) in experiment, we utilize a 1D optical lattice dressed by a periodic Raman coupling potential, making up a so-called optical Raman lattice [S1, 27]. The optical Raman lattice is generated by the use of the intercombination $\lambda_{0}=2\pi/k_{0}=$ 556 nm transition of 173Yb atoms, blue-detuned by $\sim$1 GHz from the 1S${}_{0}(F=\frac{5}{2})\leftrightarrow{}^{3}$P${}_{1}(F^{\prime}=\frac{7}{2})$ transition. The lattice potential produced by counter-propagating lights, being linearly-polarized along the $z$ direction, forms a spin-dependent potential $V_{\sigma}^{\rm latt}(x)=V_{\sigma}\cos^{2}(k_{0}x)$, where $V_{\sigma=\{\uparrow,\downarrow\}}=\sum_{F^{\prime}}\hbar\frac{|\Omega^{1x}_{\sigma,F^{\prime}}|^{2}}{4\Delta_{F^{\prime}}}$ with $|\uparrow\rangle=|m_{F}=\frac{5}{2}\rangle$ and $|\downarrow\rangle=|m_{F}=\frac{3}{2}\rangle$ [28]. The effective Rabi frequencies $\Omega^{1x}_{\sigma,F^{\prime}}$ and the single-photon detunings $\Delta_{F^{\prime}}$ are determined from all relevant transitions to the excited $F^{\prime}=(\frac{7}{2},\frac{5}{2},\frac{3}{2})$ states in the 3P1 manifold [23]. With the quantized axis set along the $z$ direction, a standing-wave lattice light of frequency $\omega_{0}=k_{0}c$ ($c$ is the speed of light), denoted as $\textbf{E}_{1x}\propto\textbf{e}_{z}2\overline{E}_{1x}e^{-i\omega_{0}t}\text{cos}(k_{0}x)$ where $\overline{E}_{1x}$ is the amplitude of the light field and $\textbf{e}_{z}$ is the unit vector in $z$-direction, induces a $\pi$-transition between the ground and the excited states, thus resulting in the relation $V_{\downarrow}/V_{\uparrow}=3.84$.

Another circularly polarized light of frequency $\omega_{1}$, denoted as $\textbf{E}_{1z}\propto(\textbf{e}_{x}+i\textbf{e}_{y})\overline{E}_{1z}e^{i(k_{0}z-\omega_{1}t)}$, is applied along the $z$ direction, and induces the Raman coupling between $|\uparrow\rangle$ and $|\downarrow\rangle$ states with the Rabi frequency $\Omega^{1z}_{\downarrow,F^{\prime}}$. Both the lattice and Raman lights are generated by a single laser source and controlled by phase-locked acousto-optical modulators (AOMs), making the frequency difference $\delta\omega=\omega_{0}-\omega_{1}$ tunable. The Raman coupling potential $\mathcal{M}(x,z)$ takes the form $\mathcal{M}(x,z)\propto M_{0}e^{ik_{0}z}(e^{ik_{0}x}+e^{-ik_{0}x})$, where $M_{0}=\sum_{F^{\prime}}\Omega^{1x}_{\uparrow,F^{\prime}}\Omega^{1z}_{\downarrow,F^{\prime}}/2\Delta_{F^{\prime}}$. Since we focus on the motion along the $x$ direction, which is uncoupled to the other two directions, we neglect the irrelevant phase $e^{ik_{0}z}$ and write the Raman potential as $\mathcal{M}(x)=M_{0}\cos(k_{0}x)$. To separate out spin-$1/2$ space from other hyperfine states of the ground manifold, an additional circularly-polarized 556 nm light, $\textbf{E}_{2z}\propto(\textbf{e}_{x}-i\textbf{e}_{y})\overline{E}_{2z}e^{ik_{0}z}$ (called as a lift beam), is applied along the $z$ direction to induce spin-dependent AC Stark shift [28]. Note that the spin-dependent lattice potential $V_{\uparrow,\downarrow}^{\rm latt}(x)$ adds an additional energy shift to the Bloch bands corresponding to two spin states, which can be readily compensated by the two-photon detuning. Finally a Zeeman term is given as $m_{z}=({\delta-\delta_{o}})/{2}$, where $\delta_{0}$ is the on-site energy difference in the spin-dependent lattice [23].

Symmetry in an optical Raman lattice

To investigate a new SPT phase in experiment, it is crucial to test that the system has a mirror symmetric optical Raman lattice and a Zeeman term $\vec{m}$ is parallel to the mirror plane, leading to a magnetic group symmetry. In the experiment, the two-photon detuning $\delta$ sets a nonzero $m_{z}$, but a finite $m_{x}$ term may be generated when the Raman potential $\mathcal{M}(x)$ breaks a mirror symmetry of the lattice and thus induces on-site spin-flipping (see a supplementary note). When the Raman beam $\textbf{E}_{1z}$ is not perfectly along the $z$ direction with the wave vector $\hat{k}=k_{0}\textbf{e}_{z}+k_{\perp,x}\textbf{e}_{x}+k_{\perp,y}\textbf{e}_{y}$, the Raman potential has an additional phase factor $\mathcal{M}(x)=M_{0}\cos(k_{0}x)e^{ik_{\perp,x}x}$, therefore breaking a mirror symmetry unless $k_{\perp,x}$ is close to zero.

We begin with spin-polarized $m_{F}=5/2$ fermions followed by a brief pulse of the optical Raman lattice beams, and then examine the wave number $k_{\perp,x}$ of the Raman beam $\textbf{E}_{1z}$. After the time-of-flight expansion, $m_{F}=5/2$ atoms are shifted downward (in $-\hat{z}$ direction) while $m_{F}=3/2$ atoms upward (in $+\hat{z}$ direction) which manifests spin-orbit coupling induced by the Raman transition [29, 30, 31, 32, 28, 33, 34]. Along the $x$ direction, however, $m_{F}=3/2$ atoms absorb the momentum shift of $\pm 1/2\hbar k_{0}$ resulting two small $m_{F}=3/2$ clouds after time-of-flight. In the experiment, we spin-orbit-couple multiple hyperfine states $m_{F}=5/2,3/2,1/2$ in such a way that two small daughter clouds of $m_{F}=1/2$ atoms are well separated from the $m_{F}=5/2$ atoms. Finally we assess the left-right symmetry of the atom diffraction and determine the angle $\theta$ between the lattice and the Raman beam to be $\theta=$89.3$\pm$1.5 or the wavenumber $k_{\perp,x}=0.01k_{0}$, which ensures negligible $m_{x}$ term in the Hamiltonian (1) and therefore a magnetic group symmetry.

Observation of the symmetry-protected topological phase

We now observe a SPT phase for ultracold fermions by loading a equal mixture of $|\uparrow\rangle$ and $|\downarrow\rangle$ atoms, prepared at $T/T_{F}=$ 0.4 in a crossed optical dipole trap (ODT) where $T_{F}$ is the Fermi temperature of the system [28], into the optical Raman lattice. An optical AC Stark shift, separating out an effective spin-1/2 subspace from other hyperfine levels, is applied within 5 ms before the optical lattice potential and the Raman-dressing beams are adiabatically switched on with an 10 ms exponential ramp to the final value. During the ramp, the crossed ODT power is increased to compensate the anti-trapping effect arising from the blue-detuned Raman potential. Consequently the temperature is increased to $T=$ 125(20) nK when all optical potentials with $V_{\uparrow}$($V_{\downarrow}$)=1.1 $E_{r}$ (4.2 $E_{r}$) and Raman coupling $M_{0}$=1.88$E_{r}$, where $E_{r}=\hbar^{2}k_{0}^{2}/2m$ and $m$ is the mass of ytterbium atom, are turned on. In the experiment, the spin-resolved time-of-flight (TOF) imaging is taken after all the laser beams are suddenly switched off, followed by spin-sensitive blast lights [28].

The crucial feature of SPT phases is the non-trivial spin texture within the FBZ. Fig. 4a shows measured spin textures $P(q_{x})=(n_{\uparrow}(q_{x})-n_{\downarrow}(q_{x}))/(n_{\uparrow}(q_{x})+n_{\downarrow}(q_{x}))$ within the FBZ reconstructed from the integrated momentum distribution along the $\hat{x}$-directional for different values of the two-photon detuning. As the two-photon detuning $\delta$ is scanned from the negative to positive value, the overall spin polarization of the lowest $s$-band gradually changes from $|\uparrow\rangle$ (red) to $|\downarrow\rangle$ (blue). However when the Raman transition resonantly couples the two lowest $s$-band corresponding to $|\uparrow\rangle$ and $|\downarrow\rangle$ respectively, the spin polarization changes from one to the other within the FBZ as shown in Fig. 4b. Finally we use the connection between the $\mathbb{Z}_{2}$ invariant and the $P(\Gamma,M)$, and determine $(-1)^{\nu}=\prod_{i=M,\Gamma}sgn[P(\Lambda_{i})]$ by measuring averaged spin polarization $P(\Lambda_{i})=\frac{10}{k_{r}}\int_{\Lambda_{i}-0.05k_{r}}^{\Lambda_{i}+0.05k_{r}}P(q^{\prime})dq^{\prime}$ that takes into account the optical resolution of the imaging system [22]. Fig. 4d shows the emergence of SPT phases ($\nu=1$) out of trivial phases ($\nu=0$). The spin-dependent lattice shifts the topological regime due to the on-site energy difference, and furthermore the finite temperature effect reduces the topological regime in good agreement with the prediction (indicated by shaded region in Fig. 4d).

Adiabatic control of the band topology

In a second set of experiment, we highlight the adiabatic preparation of the topological phase starting from the trivial regime and vice versa. In the experiment, a topological phase is adiabatically prepared at $\delta=0.8(3)E_{r}$ as described in the previous experiment and subsequently the two-photon detuning $\delta$ is linearly ramped to the final value of $\delta=-2.1(3)E_{r}$ where the trivial phase is expected in equilibrium (Fig. 4a). Here we test the adiabaticity of the sweep process by probing the spin polarization for different sweep times as shown in the inset of Fig. 4e. The value of the spin polarization at symmetric points is identical to the equilibrium case when the sweep time is longer 5 ms. As a final examination of the adiabatic control, we show that the SPT phase can be converted into the trivial phase, and restored back to the topological one as in Fig. 4e. The spin texture and its invariant measured at each time (0, 6, 12) ms is consistent with the equilibrium case.

Topology-dependent spin relaxation after quench

Having known spin textures in equilibrium and their topological invariants, we now turn our attention to the far-from-equilibrium spin dynamics after the quench between the trivial and the topological regime. Using dynamic control of the Hamiltonian, we in particular consider a bi-directional quench between the topological regime ($\delta=$0.8(3)$E_{r}$) and spin-$|\uparrow\rangle$ trivial regime ($\delta=-$2.1(3)$E_{r}$) with the trap frequency $\omega_{x}=2\pi\times 241(20)$ Hz, and moitor the spin dynamics under the post-quench Hamiltonian. Fig. 5 shows experimental measurements of spin polarization at symmetric points after bi-directional quench, which indeed reflects the topological nature of the post-quench Hamiltonian. If the system is quenched to the SPT phase with spin-orbit field (Fig. 5a), the $q_{x}$-dependent effective magnetic (spin-orbit) field leads to spin-wave-like collective dynamics with spin relaxation to the final state. This is in sharp contrast to the case that the system is quenched to the trivial phase where the spin is averagely polarized to z direction, and the dominant spin dynamics shows monotonous relaxation (Fig. 5b). It can be seen that the collective oscillation exhibits a frequency larger than the trapping frequency. This is because the trapping-induced intraband transition can have an energy detuning between Bloch states at different momenta, which enhances the collective oscillation frequency [23]. The observed topology-dependent spin dynamics shows a quite good agreement with numerical calculations taking into accounts the trap geometry, the finite temperature effect and the dissipation process (Fig. 5c,d). The spin relaxation dynamics clearly reflect the topological property of the phase after quench.

Discussion and Conclusion

The novel SPT phase we uncovered here highlights the great capabilities to realize and explore new topological states with ultracold atoms. In particular, the study may open the way to observing all SPT phases of 1D in the AZ classification [S3], including the chiral topological phase of AIII class which can be readily achieved by considering spin-independent rather than spin-dependent optical lattice in Eq. (1) [S1], the BDI class and D classes of superfluids by adding attractive interactions [35], and also the exotic phases protected by nonsymmorphic symmetries [36, 37]. Compared with the solid state materials whose complicated environment causes difficulty in general in engineering the symmetries, the full steerability of ultracold atoms can enable a precise study of broad classes of SPT phases, of which a most fascinating issue is to probe the reduction of topological classification of SPT phases from single particle to interacting regimes and has been extensively discussed in theory [S3, 24, 38, 39]. Such important issue is very difficult to be investigated in solid state experiments due to the great challenges of realization, but is promising for ultracold atoms based on the current study with properly engineered interactions [S1, 40].

Further generalization of the present study to higher dimensional systems also offers the simulation of quantum phases beyond natural conditions. For example, 2D tunable Dirac semimetals driven by spin-orbit coupling can be readily achieved by applying the present scheme to 2D systems [41], and are of particular interests since a spin-orbit coupled 2D Dirac semimetal is still not available in solid state materials [42]. By tuning attractive interactions the novel topological Fulde-Ferrell superfluid [43, 44] and Majorana zero modes protected by Chern-Simons invariant [45] have been predicted in such 2D Dirac semimetals. Finally, the nonequilibrium quench dynamics unveiled in this work show novel topology-dependent spin relaxation dynamics. Generalization of the quench study to higher dimensional systems is expected to bring about more nontrivial quantum spin dynamics [46], which may provide insights into the understanding of nonequilibrium many-particle dynamics in the exotic phases.