Topological optical Raman superlattices

Jia-Hui Zhang, Bei-Bei Wang, Feng Mei, Jie Ma, Liantuan Xiao, and Suotang Jia

Phys. Rev. A 105, 033310 – Published 16 March 2022

Abstract

Topological phases of ultracold atoms recently have been intensively studied in both optical superlattices and Raman lattices. However, the topological features induced by the interplay between such two lattices remain largely unexplored. Here we present an optical Raman superlattice system that incorporates an optical superlattice and a Raman superlattice. The Raman superlattice presented here supports tunable dimerized spin-orbit couplings and staggered on-site spin flips. We find that such a system respects a spin-rotation symmetry and has much richer topological properties. Specifically, we show that various topological phases could emerge in the optical Raman superlattice, such as four different chiral topological insulator phases and two different quantum spin Hall insulator phases, identified by spin winding and spin Chern numbers, respectively. We also demonstrate that the spin-dependent topological invariants could be directly measured by quench dynamics.

Received 28 November 2021
Accepted 2 March 2022

DOI:https://doi.org/10.1103/PhysRevA.105.033310

Physics Subject Headings (PhySH)

Cold gases in optical lattices Spin-orbit coupling Topological phases of matter

Ultracold gases

Atomic, Molecular & Optical

Authors & Affiliations

Jia-Hui Zhang , Bei-Bei Wang, Feng Mei ^*, Jie Ma, Liantuan Xiao, and Suotang Jia

State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China and Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China

^*meifeng@sxu.edu.cn

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Vol. 105, Iss. 3 — March 2022

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Images

Figure 1
(a) Schematic illustration of the double- $Λ$ configuration for creating a Raman superlattice potential. Each Raman coupling is induced by one standing-wave laser with Rabi frequency $Ω_{1 x}$ or $Ω_{2 x}$ and one plane-wave laser with Rabi frequency $Ω_{0}$ . $Δ_{1, 2}$ are the detunings from the auxiliary excited states ${| e 〉}_{1, 2}$ . (b) Implemented lattice model with dimerized spin-orbit couplings and staggered on-site spin flips. Specifically, each unit cell has two sublattice sites $a$ and $b$ ; the intra- and intercell spin-conserved hoppings are $J_{1}$ and $J_{2}$ , the intra- and intercell spin-flip hoppings are $K_{1}$ and $K_{2}$ , and the on-site spin-flip strengths for the sublattices $a$ and $b$ are $\pm M$ .
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Figure 2
Topological phase diagram and spin winding number values in the parameter space of $K_{1} / J_{0}$ and $K_{2} / J_{0}$ for (a) $J_{1} = 2 J_{0}$ and $J_{2} = J_{0}$ , (b) $J_{1} = J_{2} = J_{0}$ , and (c) $J_{2} = 2 J_{0}$ and $J_{1} = J_{0}$ . The transitions between different topological phases are determined by the gap-closing conditions $E_{+} = 0$ (red solid line, $A_{+} = \pm B_{+}$ ) and $E_{-} = 0$ (blue dashed line, $A_{-} = \pm B_{-}$ ). The energy spectra with open boundary conditions are shown in (d) $K_{1} = 1.5 J_{0}$ and $K_{2} = 4 J_{0}$ , (e) $K_{1} = - 3 J_{0}$ and $K_{2} = 3 J_{0}$ , (f) $K_{1} = 3 J_{0}$ and $K_{2} = 1 J_{0}$ , and (g) $K_{1} = 4 J_{0}$ and $K_{2} = - 3 J_{0}$ , corresponding to the four topological phases shown in panel (c), manifesting the bulk-edge correspondence. Here $J_{0}$ is used as the energy unit.
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Figure 3
Time-averaged spin polarizations $\bar{〈 s_{x \pm, y \pm} 〉}$ as a function of $k_{x}$ for (a), (b) $K_{1} = 4 J_{0}$ and $K_{2} = - 4 J_{0}$ , (c), (d) $K_{1} = 3 J_{0}$ and $K_{2} = J_{0}$ , (e), (f) $K_{1} = - 3 J_{0}$ and $K_{2} = 4 J_{0}$ , and (g), (h) $K_{1} = 1.5 J_{0}$ and $K_{2} = 4 J_{0}$ . The sign $+$ ( $-$ ) denotes the region where $d_{y \pm} > 0$ ( $d_{y \pm} < 0$ ). The other parameters are $J_{1} = J_{0}, J_{2} = 2 J_{0}$ , and $T = 10 / J_{0}$ .
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Figure 4
(a) Topological phase diagram and spin Chern number values in the parameter space of $α$ and $β$ . The solid (dashed) lines $α = - 1$ and $β = - 1$ ( $α = 1$ and $β = 1$ ) are the gap-closing lines in which $E_{+} = 0$ ( $E_{-} = 0$ ). The integer values in the coordinates represent the spin Chern number values $C_{\pm}$ , while the values of the laser phase $φ$ in the coordinates reflect the positions at which the edge states of $h_{\pm}$ cross. For example, $(1, - 1)$ represents that the spin Chern number values are $(C_{+} = 1, C_{-} = - 1)$ ; $(0, π)$ represents that the edge states of $h_{+}$ and $h_{-}$ in this region of parameter space respectively cross at $φ = 0$ and $π$ [see panels (g)–(k)]. The bulk energy spectra in the synthetic first Brillouin zone for (b) $α = 1$ and $β = 0.5$ , (c) $α = - 1$ and $β = 0.5$ , (d) $α = 0.5$ and $β = - 1$ , (e) $α = 0.5$ and $β = - 1$ , and (f) $α = 1$ and $β = 1$ . With an open boundary condition along the genuine dimension, the corresponding energy spectra in the five regions of parameter space [labeled A–E in panel (a)] are respectively plotted in panels (g)–(k) and the spectra for the flat band case are shown in panel (l). The specified parameters are (g) $α = 0.5$ and $β = 0.5$ , (h) $α = 1.5$ and $β = 0.5$ , (i) $α = - 1.5$ and $β = 0.5$ , (j) $α = 0.5$ and $β = 1.5$ , (k) $α = 0.5$ and $β = - 1.5$ , and (l) $α = 1$ and $β = 1$ . The red solid and black dashed lines respectively denote the left and right in-gap topological edge states. The other parameters are $t_{1} = 0.8 t_{0}, t_{2} = 0.3 t_{0}$ , and $δ = - t_{0}$ . Here $t_{0}$ is used as the energy unit.
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Figure 5
(a)–(c) Time-averaged spin polarizations $\bar{〈 s_{x \pm, y \pm, z \pm} 〉}$ as a function of $k_{x}$ and $φ$ . The black dashed lines denote the BISs that divide the synthetic Brillouin zone into two regions, i,e., $V_{+}$ with $d_{x \pm} > 0$ and $V_{-}$ with $d_{x \pm} < 0$ . The spin textures for the dynamic fields ${\vec{g}}_{\pm}$ on the BISs are given in panel (d). The parameters are chosen as $t_{1} = 0.8 t_{0}, t_{2} = 0.3 t_{0}, δ = - t_{0}, α = 4, β = 0.3$ , and $T = 10 / t_{0}$ .
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