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
  • 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.

  • 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.

  • 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.

  • 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.

  • 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).

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.

  • Supplementary Materials

    This PDF file includes:

    • 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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