Research ArticlePHYSICS

Observation of symmetry-protected topological band with ultracold fermions

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Science Advances  23 Feb 2018:
Vol. 4, no. 2, eaao4748
DOI: 10.1126/sciadv.aao4748

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.

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 (46) and symmetry-protected topological (SPT) phases (710), which exhibit long- and short-range quantum entanglement, respectively. Unlike topological orders, which are robust against any local perturbations, an 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 1D Su-Schrieffer-Heeger (SSH) model for fermions (11) and the 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 at the boundary and are characterized by a ℤ2 invariant (13, 14). Topological insulators also inspired generalizations to 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 these phases have been observed in the experiments. In comparison with solid-state materials, ultracold atoms are extremely clean systems, which can provide controllable platforms in exploring new SPT phases, with many theoretical schemes having been proposed (1618). To date, some of the novel topological features of the SSH model have been probed in experiments with ultracold bosons (1921), whereas the realization of the SPT phase, which can only exist for fermions in the noninteracting limit, has not been reported. Here, we propose and realize, for the first time, a 1D optical Raman lattice of fermions, which hosts a new SPT phase that is beyond particle-hole or chiral symmetry protection, as usually required in Altland-Zirnbauer (AZ) classification of 1D states (15, 22). We measure the topology via Bloch states at symmetric momenta (23), and the quench dynamics and unveil two fundamental types of spin-relaxation mechanisms related to band topology in this SPT system.

RESULTS AND DISCUSSION

SPT 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 laterEmbedded Image(1)where Embedded Image is the kinetic energy for motion in the x direction, Embedded Image is a two-by-two unit matrix, σx,y,z are Pauli matrices in spin space, Embedded Image denote the 1D optical lattice potentials for spin | ↑, ↓ 〉 states, with V↑,↓ being the lattice depths and k0 = π/a (a is the lattice constant), the Raman lattice potential Embedded Image(x) = M0 cos(k0x) with amplitude M0, and δ is the two-photon detuning of Raman coupling.

Fig. 1 SPT phases and band topology.

(A) Sketch of the 1D Raman lattice model. A spin-dependent lattice potential induces spin-conserved hopping, whereas the Raman potential contributes to spin-flip hopping. (B) Bulk and boundary energy spectra obtained by full-band diagonalization with a Zeeman perturbation field of strength 0.1Er and along the zz, orange circles), yy, blue triangles), and xx, green squares) direction, respectively. The in-gap states are boundary modes. The parameters are taken as V = 5Er, and M0 = 1.0Er [the same for (C) and (D)]. (C) Band structure (the second row) and spin textures of both the lowest band (the third row) and the second band (the first row) for δ = −0.2Er (left), δ = 0.7Er (middle), and δ = 1.5Er (right), respectively. (D) The spin polarizations at Γ and M points of the lowest band determine the topological trivial (ν = 0) and nontrivial (ν = 1) regimes.

The Hamiltonian (Eq. 1) supports novel topological phases protected by symmetries. With the lattice potential Embedded Image, the spin-conserved hopping (t↑,↓) is induced, whereas the Raman potential Embedded Image(x) accounts for the hopping (tso) that flips the atomic spin (Fig. 1A). The Bloch Hamiltonian for the lowest s-band physics can be obtained by Embedded Image (see the Supplementary Materials), where we take Embedded Image for a generic study, qx as the Bloch quasi-momentum, and the hopping coefficients t0/1 = (t ± t)/2. The Hamiltonian Embedded Image with mx = 0 exhibits a chiral symmetry defined by σxEmbedded Imageσx = −Embedded Image if the lattice is spin-independent Embedded Image, which gives t = t, and belongs to the AIII class according to the AZ classification (16, 22). The topology of this AIII class phase is quantified by an integer winding number (ℤ) in the single-particle regime, pictorially characterized by the circles that the 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 ℤ to ℤ4 (16), which is closely related to the Fidkowski-Kitaev ℤ8 classification of 1D interacting Majorana chains of the BDI (chiral orthogonal) class (24).

The striking result is that a new topological phase can be achieved beyond the chiral symmetry protection when the lattice depths Embedded Image and t↑,↓ are spin-dependent, in which the regime with 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 Embedded Image and Embedded Image, respectively, whereas only a nonzero mx term splits out the degeneracy. The existence of midgap or degenerate end states manifests new SPT phases, as shown below. We first consider the case with Embedded Image. One can verify that Embedded Image satisfies a magnetic group symmetry defined as the product of time-reversal and mirror symmetries (see the Supplementary Materials), giving Mx = σzKRx with Embedded Image, where K is the complex conjugate, and Rx denotes the spatial reflection along the x axis. Besides, the Hamiltonian also satisfies a nonlocal chiral symmetry defined as Embedded Image, where Tx(k0) is a k0-momentum translation, yielding Embedded Image. The midgap end states at left-hand (|ψL〉) and right-hand (|ψR〉) boundaries are protected by the both symmetries under a novel mechanism. First, by definition, both Mx and Embedded Image transform one of |ψL/R〉 to the other. Moreover, their commutation and anticommutation relations with Embedded Image imply that the magnetic group symmetry (nonlocal chiral symmetry) connects the two end states with identical (opposite) energy, which forces the two end states |ψL/R〉 to be of zero energy (that is, midgap states in the presence of both symmetries), which is the case for Embedded Image along the y axis. The mz term keeps the Mx symmetry, hence protects the end-state degeneracy, with the degenerate end modes existing as long as the gap is not closed and is in the topological regime for |mz| < 2t0 (see the Supplementary Materials), whereas the mx term breaks the Mx symmetry and splits out the degeneracy, leading to the end-state spectra in Fig. 1B. Note that the identity matrix t1 term in the Hamiltonian does not affect the spin winding of the Bloch states. Thus, the topological invariant of the present system is still characterized by an integer winding number Embedded Image when the Mx symmetry preserves (without mx term), whereas the nonlocal chiral symmetry Embedded Image further protects the end states to be zero-energy modes. In the experiment, a nonzero mz term can be easily engineered by manipulating the two-photon detuning δ of the Raman coupling. In this case, the magnetic group ensures that Bloch states at the two symmetric momenta {Λj} = {Γ(qx = 0), M(qx = π/a)} are eigenstates of σz, having the spin polarization Pj) = ±1. The winding number of the lowest band can be further obtained from a spin index through Embedded Image, with the spin index ν being introduced as (−1)ν = Πj sgn[Pj)], similar to the study of Liu et al. (23). The topologically nontrivial (trivial) phase corresponds to ν = 1(0). Figure 1C shows the spin texture with different parameter conditions, from which the ℤ2 index and then the winding number can be read out directly, and Fig. 1D presents the phase diagram versus δ.

Quench dynamics

We examine the spin dynamics after a quench from one phase to another. The quench is performed by suddenly ramping the two-photon detuning from δi to δf, with δi and δf corresponding to two topologically distinct regimes. The quantum dynamics are captured by the time-dependent density matrix ρ(t), which satisfies the Lindblad master equation (25, 26)Embedded Image(2)where γ denotes the noise-induced decay rate, and L is the Lindblad operator characterizing the effects of the environment. For a real ultracold atom system, we consider an external trapping potential for the study and the Hamiltonian H′ = H + Vtrap, with Embedded Image and ωx being the trapping frequency. The sharp contrast between the topological and trivial phases, which we assume for this study to be away from the critical point of phase transition, is that the spin of the Bloch states in the FBZ winds over all directions in the y-z plane in the former case, whereas it 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 first consider the case without dissipation (γ = 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 qx due to nonequilibrium population after the quench and the intraband transition induced by the trapping potential between Bloch states of different momenta. The numerical simulation of the spin dynamics at Γ and M points is shown in Fig. 2 (A and B) for a quench to topological (δf = 0.7Er) and trivial (δf = 2.0Er) 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 Γ and M points. However, the spin dynamics for quench to topological phase also exhibit 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 (see the Supplementary Materials). 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 that is away from critical point, 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.

Fig. 2 Quench dynamics of SPT bands.

The numerical simulation is performed in the half-filling condition (one atom per site) for quench process from trivial to topological regimes, with δi = −0.2Er and δf = 0.7Er in (A) and (C), and from topological to trivial regimes, with δi = 0.7Er and δf = 2.0Er in (B) and (D). In all the cases, the trapping frequency reads ωx = (2π)300 Hz, and the optical Raman lattice potentials V = 2.5Er, V = 5Er, and M0 = 1.0Er. For comparison, we plot the dashed line in (A) for an oscillation of frequency Embedded Image, with ΔE = 0.07Er characterizing a detuning of magnitude between the bandwidth and the energy difference of Γ and M points. (A and B) Quench dynamics for spin polarizations at Γ and M points at zero temperature and without dissipation. (C) Quench spin dynamics from the trivial to the topological regime at the temperature T = 100 nK, with the noise-induced dephasing rate γ = 0.03. (D) Quench spin dynamics from topological to trivial at T = 100 nK, with dissipation rate taken as γ = 0 and 0.05, respectively.

We further show in Fig. 2 (C and D) the quench dynamics with dissipation, as captured by the nonunitary part of Eq. 2. For a quench to the trivial phase, because the spin dynamics are dominated by the interband oscillations, the dissipation mainly results from the decay of the excited bands. In comparison, for the 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 Embedded Image (for decay) and Embedded Image (for dephasing), respectively, and for qualitative modification of dynamical behaviors, the latter effect is typically much weaker than the former. Here, |n〉 denotes the eigenstates of the postquench Hamiltonian Hf) + Vtrap. The numerical simulation shows that the quench dynamics from trivial to topological regime resemble 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), whereas the dynamics in the other way round render a pure decay from initial state to the final equilibrium phase (Fig. 2D). With these results, we draw the conclusion that the trapping potential (dissipation) has dominant effects on the spin dynamics after the quench due to the topological (trivial) spin textures of the final SPT band, as observed in the present experiment.

Optical Raman lattices for ytterbium atoms

To realize the Hamiltonian (1) in the experiment, we use a 1D optical lattice dressed by a periodic Raman coupling potential, making up a so-called optical Raman lattice, as described in Fig. 3A (16, 27). The optical Raman lattice is generated by the use of the intercombination λ0 = 2π/k0 = 556 nm transition of 173Yb atoms, blue-detuned by ~1 GHz from the Embedded Image transition. The lattice potential produced by counter-propagating lights, being linearly polarized along the z direction, forms a spin-dependent potential Embedded Image, where Embedded Image, with Embedded Image and Embedded Image (28). The effective Rabi frequencies Embedded Image and the single-photon detunings Embedded Image are determined from all relevant transitions to the excited Embedded Image states in the 3P1 manifold (Fig. 3B). With the quantized axis set along the z direction, a standing-wave lattice light of frequency ω0 = k0c (c is the speed of light), denoted as Embedded Image, where Ē1x is the amplitude of the light field and ez is the unit vector in the z direction, induces a π transition between the ground and the excited states, thus resulting in the relation V/V = 3.84 (Fig. 3A).

Fig. 3 A sketch of the experimental setup.

(A) The experimental setup consists of a 1D optical lattice with lattice along the x direction and a perpendicular Raman beam generating a periodic Raman potential ℳ(x) = M0 cos(k0x). A circularly polarized beam is added along the quantized axis in the z direction, introducing the spin-dependent ac Stark shift. (B) Both the lattice and the Raman beams are near blue-detuned from the Embedded Image intercombination transition and induce the Raman transition between the Embedded Image and Embedded Image hyperfine states of the 1S0 ground manifold. (C) TOF image of mF = 5/2 atoms without pulsing optical Raman lattice beams. The direction of the gravity is along the z direction. (D) When the optical Raman lattice is briefly switched on in such a way that the mF = 1/2 atoms are coupled out, two mF = 1/2 clouds are relatively shifted by ~2k0 along the x direction.

Another circularly polarized light of frequency ω1, denoted as Embedded Image, is applied along the z direction and induces the Raman coupling between | ↑ 〉 and | ↓ 〉 states with the Rabi frequency Embedded Image. Both the lattice and Raman lights are generated by a single laser source and controlled by phase-locked acousto-optical modulators, making the frequency difference δω = ω0 − ω1 tunable. The Raman coupling potential Embedded Image(x, z) takes the form Embedded Image, where Embedded Image. Because we focus on the motion along the x direction, which is uncoupled to the other two directions, we neglect the irrelevant phase Embedded Image and write the Raman potential as Embedded Image(x) = M0 cos(k0x). To separate out spin-1/2 space from other hyperfine states of the ground manifold, an additional circularly polarized 556-nm light, Embedded Image (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 Embedded Image adds an additional energy shift, δ0, 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 mz = (δ − δ0)/2, where δ0 is the on-site energy difference in the spin-dependent lattice.

Symmetry in an optical Raman lattice

To investigate a new SPT band in the experiment, it is crucial to test that the system has a mirror-symmetric optical Raman lattice and that a Zeeman term Embedded Image is parallel to the mirror plane, leading to a magnetic group symmetry. In the experiment, the two-photon detuning δ sets a nonzero mz, but a finite mx term may be generated when the Raman potential Embedded Image(x) breaks a mirror symmetry of the lattice and thus induces on-site spin flipping (see the Supplementary Materials). When the Raman beam E1z is not perfectly along the z direction with the wave vector Embedded Image, the Raman potential has an additional phase factor Embedded Image, therefore breaking a mirror symmetry unless k⊥,x is close to zero.

We begin with spin-polarized mF = 5/2 fermions (Fig. 3C), followed by a brief pulse of the optical Raman lattice beams, and then examine the wave number k⊥,x of the Raman beam E1z. After the time-of-flight (TOF) expansion, as described in Fig. 3D, mF = 5/2 atoms are shifted downward (in the −Embedded Image direction), whereas the mF = 3/2 atoms are shifted upward (in the +Embedded Image direction), which manifests spin-orbit coupling induced by the Raman transition (2834). However, along the x direction, mF = 3/2 atoms absorb the momentum shift of ±1/2k0, resulting to two small mF = 3/2 clouds after TOF. In the experiment, we spin-orbit–couple multiple hyperfine states mF = 5/2, 3/2, 1/2 in such a way that two small daughter clouds of mF = 1/2 atoms are well separated from the mF = 5/2 atoms. Finally, we assess the left-right symmetry of the atom diffraction and determine the angle θ between the lattice and the Raman beam to be θ = 89.3 ± 1.5 or the wave number k⊥,x = 0.01k0, which ensures negligible mx term in the Hamiltonian (Eq. 1) and, therefore, a magnetic group symmetry.

Observation of the SPT band

We now observe an SPT band for ultracold fermions by loading an equal mixture of | ↑ 〉 and | ↓ 〉 atoms, prepared at T/TF = 0.4 in a crossed optical dipole trap (ODT), where TF 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 a 10-ms exponential ramp to the final value. During the ramp, the crossed ODT power is increased to compensate the antitrapping effect arising from the blue-detuned Raman potential. Consequently, the temperature is increased to T = 125(20) nK when all optical potentials with V (V) = 1.1Er (4.2Er) and Raman coupling M0 = 1.88 Er, where Embedded Image and m is the mass of an ytterbium atom, are turned on. In the experiment, the spin-resolved 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 bands is the nontrivial spin texture within the FBZ. Figure 4A shows measured spin textures P(qx) = (n(qx) − n(qx))/(n(qx) + n(qx)) within the FBZ reconstructed from the integrated momentum distribution along the Embedded Image direction for different values of the two-photon detuning. Because the two-photon detuning δ is scanned from the negative to positive value, the overall spin polarization of the lowest s-band gradually changes from | ↑ 〉 (red) to | ↓ 〉 (blue). However, when the Raman transition resonantly couples the two lowest s-bands corresponding to | ↑ 〉 and | ↓ 〉, respectively, the spin polarization changes from one to the other within the FBZ, as shown in Fig. 4B. Figure 4C also visualizes the coexistence of different spin domains within the FBZ in the topological regime. Finally, we use the connection between the spin index and the P(Γ, M) and determine (−1)ν = ∏i = M,Γsgn[Pi)] by measuring the averaged spin polarization Embedded Image that takes into account the optical resolution of the imaging system (23). Figure 4D shows the emergence of SPT bands (ν = 1) out of trivial bands (ν = 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).

Fig. 4 In-equilibrium spin textures and their ℤ2 invariant across the topological transition.

(A) Experimental measurement of spin polarization P(qx) = (n(qx) − n(qx))/(n(qx) + n(qx)) within FBZ as a function of two-photon detuning δ. For all measurements, the lattice depth and the Raman coupling strength are set to V = 1.1(1)Er, V = 4.2(4)Er, and M0 = 1.88(10)Er. Numerical calculation at T = 0 nK is shown for comparison. (B) The spin polarization is obtained at the symmetry points Γ and M. The inset figure gives numerical calculation based on experimental parameters. (C) Coexistence of | ↑ 〉 and | ↓ 〉 within the FBZ is visualized by measuring the width of the | ↑ 〉 domain of the spin texture normalized by 2ℏk0 as a function of mz. (D) The measured ℤ2 invariant ν reveals the SPT band, which is consistent with the predicted region (shaded region) for the current experiment parameters. The vertical dashed line indicates the predicted phase transition point at T = 0 nK. (E) A topological band, adiabatically prepared at δ = 0.8(3)Er initially, is converted into a trivial one at δ = −2.1(3)Er and restored back to the original SPT band. The dashed line of the inset serves as a visual guide.

Adiabatic control of the band topology

In a second set of experiments, we demonstrate the adiabatic preparation of the topological band starting from the trivial regime and vice versa. In the experiment, a topological band is adiabatically prepared at δ = 0.8(3)Er, as described in the previous experiment, and subsequently, the two-photon detuning δ is linearly ramped to the final value of δ = −2.1(3)Er, where the trivial band is expected in equilibrium (Fig. 4A). We first 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 than 5 ms. As a final examination of the adiabatic control, we show that the SPT band can be converted into the trivial one and restored back to the topological one, as shown in Fig. 4E. The spin texture and its invariant measured at each time (0, 6, and 12 ms) are consistent with the equilibrium case.

Spin relaxation dynamics after quench

Having known the 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 the dynamic control of the Hamiltonian, we particularly consider a bidirectional quench between the topological regime [δ = 0.8(3)Er] and the polarized spin-| ↑ 〉 trivial regime [δ = −2.1(3)Er] with the trap frequency ωx = 2π × 241(20) Hz and monitor the spin dynamics under the postquench Hamiltonian. Figure 5 shows experimental measurements of spin polarization at symmetric points after bidirectional quench, which reflects the topological nature of the postquench Hamiltonian. If the system is quenched to the topological regime with spin-orbit fields (Fig. 5A), then the qx-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 band where the spin is averagely polarized to the z direction and the dominant spin dynamics show monotonous relaxation (Fig. 5B). It shows that the collective oscillation exhibits a frequency larger than the trapping frequency. This is because the trapping-induced intraband transition can experience an energy detuning between Bloch states at different momenta, which enhances the collective oscillation frequency (see the Supplementary Materials). The observed topology-dependent spin dynamics show a quite very good agreement with numerical calculations taking into account the trap geometry, the finite temperature effect, and the dissipation process (Fig. 5, C and D). We emphasize that in the simulation of Fig. 5C, a vanishing decay coefficient has been chosen, while the simulation matches well with the experimental observations. This indicates that the spin dynamics in the quench to the topological regime are dominated by the trapping-induced intraband scatterings. We note that, although in general, the quench spin dynamics studied here may not be able to distinguish the phase close to the critical point of phase transition, the observed spin relaxation dynamics reflect the topological property of the band away from critical point after quench.

Fig. 5 Far-from-equilibrium spin dynamics after quench to SPT and trivial bands.

(A and B) Measured spin dynamics after a quench from trivial to topological (A) or from topological to trivial (B). The parameters are V = 1.1(1)Er, V = 4.2(4)Er, and M0 = 1.88(10)Er. The two-photon detuning is set to be δ = −2.1(3)Er and 0.8(3)Er for the trivial and topological regime, respectively. The shaded region serves as a visual guide. (C and D) Numerical calculations of the quench dynamics. Parameters all take the same values as in (A) and (B), except δ = −1.5Er (δ = 0.8Er) for the trivial (topological) regime. The temperature is set at T = 150 nK. The decay coefficient is taken as γ = 0 for (C) and γ = 0.08 for (D).

CONCLUSION

The novel SPT band we uncovered here highlights the great capabilities to realize and explore new topological physics with ultracold atoms. In particular, the study may open the way to observing all SPT phases of 1D in the AZ classification (9), including the chiral topological state of AIII class, which can be readily achieved by considering a spin-independent, rather than spin-dependent, optical lattice in Eq. 1 (16), the BDI 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 generally causes difficulty in engineering the symmetries, the full steerability of ultracold atoms can enable a precise study of broad classes of SPT phases, of which probing the reduction of topological classification of SPT phases from single particle to interacting regimes is a most fascinating topic has been extensively discussed in theory (9, 24, 38, 39). This important issue is very difficult to investigate in solid-state experiments because of the great challenges of realization but it is promising for ultracold atoms based on the current study with properly engineered interactions (16, 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 because 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 these 2D Dirac semimetals. Finally, the nonequilibrium quench dynamics unveiled in this work show novel spin relaxation dynamics related to bulk topology. 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.

MATERIALS AND METHODS

Experimental procedure

A degenerate equal mixture of N↑,↓ = 5 × 103 173Yb atoms at T/TF = 0.4 was prepared after forced-evaporative cooling in an ODT (28, 47), where TF is the Fermi temperature of the trapped fermions with a trapping frequency of Embedded Image. A quantized axis was fixed by the bias magnetic field of 5 G applied along the Embedded Image direction. Before the two-photon Raman transition that couples | ↑ 〉 and | ↓ 〉 states was switched on, the intensity of the σ-polarized light (called a lift beam) was linearly ramped up within 3 ms to lift the degeneracy of the ground levels. Simultaneously, we compensated the antitrapping effect of the blue-detuned light by increasing the depth of the dipole trap. A pair of Raman and lattice beams, with fixed two-photon detuning δ, was then exponentially ramped to the final value with the time constant of τ = 7 ms, by which fermions were adiabatically loaded into the optical Raman lattice. Finally, the two-photon detuning δ (or equivalently mz) was changed on demand, depending on the experiments. In the measurement of the topological invariant, the fermions were directly loaded into the Raman lattice with the fixed value of mz. In the measurement of quench dynamics, two-photon detuning was suddenly switched from the initial (δ = 0.8Er, corresponding to the topological phase) to the final value (δ = −2.1Er, corresponding to the trivial phase) and vice versa (see fig. S5).

To obtain a TOF image sensitive to the spin state, we applied 556-nm blast pulses resonant to 1S0 → 3P1(F′ = 7/2) transition before the 6-ms TOF expansion. Using a 399-nm light resonant to 1S0 → 3P1(F′ = 7/2) transition with a blast pulse, three absorption images were obtained, providing a TOF atomic distribution Embedded Image after killing mF-state atoms for 200 to 400 μs. Finally, an atomic distribution of mF = 5/2 and mF = 3/2 atoms was extracted from Embedded Image and Embedded Image, respectively. We integrated the whole atomic cloud along the kz direction and obtained the atomic profile in the kx momentum domain, as described in fig. S6.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/2/eaao4748/DC1

Supplementary Text

fig. S1. Total spin polarizations of the lowest two bands at Γ and M points.

fig. S2. Energy spectra of 1D Raman lattice Hamiltonian H, given by Eq. 1 in the main text, with open boundary condition, showing the existence of midgap end states.

fig. S3. Quench dynamics under various conditions.

fig. S4. Electronic hyperfine level of 173Yb atoms and Raman coupling.

fig. S5. Experimental sequences.

fig. S6. Spin-sensitive TOF images.

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REFERENCES AND NOTES

Acknowledgments: We thank T.-L. Ho, Q. Zhou, K. T. Law, S. Z. Zhang, and C.-K. Chan for the useful discussion. We also thank Y. Zou, W. Huang, and G. Cai for the experimental assistance. Funding: G.-B.J. acknowledges the support from the Hong Kong Research Grants Council and the Croucher Foundation through ECS26300014, GRF16300215, GRF16311516, GRF16305317, and the Croucher Innovation grants. X.-J.L. acknowledges the support from the National Key R&D Program of China (under grant 2016YFA030160X), the NSFC (nos. 11574008 and 11761161003), and the Thousand-Young-Talent Program of China. Author contributions: B.S., C.H., E.H., and S.Z. carried out the experiment, data analysis, and the numerical calculation. L.Z. and T.F.J.P. performed the theoretical modeling and the numerical calculations. All authors discussed the results and commented on the manuscript. G.-B.J. and X.-J.L. supervised the research. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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