Optical Lattice with Torus Topology

Hwanmun Kim, Guanyu Zhu, J. V. Porto, and Mohammad Hafezi

Phys. Rev. Lett. 121, 133002 – Published 26 September 2018

Abstract

We propose an experimental scheme to construct an optical lattice where the atoms are confined to the surface of a torus. This construction can be realized with spatially shaped laser beams which could be realized with recently developed high resolution imaging techniques. We numerically study the feasibility of this proposal by calculating the tunneling strengths for atoms in the torus lattice. To illustrate the nontrivial role of topology in atomic dynamics on the torus, we study the quantized superfluid currents and fractional quantum Hall (FQH) states on such a structure. For FQH states, we numerically investigate the robustness of the topological degeneracy and propose an experimental way to detect such a degeneracy. Our scheme for torus construction can be generalized to surfaces with higher genus for exploration of richer topological physics.

Received 3 May 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.133002

Physics Subject Headings (PhySH)

Cold gases in optical lattices Quantum Hall effect Superfluidity Topological Hall effect Topological phases of matter

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Hwanmun Kim^1,2,*, Guanyu Zhu^1,†, J. V. Porto¹, and Mohammad Hafezi^1,2,3

¹Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
²Department of Physics, University of Maryland, College Park, Maryland 20742, USA
³Departments of Electrical and Computer Engineering and Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA

^*Corresponding author. hwanmun@terpmail.umd.edu
^†Corresponding author. gzhu123@umd.edu

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Issue

Vol. 121, Iss. 13 — 28 September 2018

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Images

Figure 1
(a) Schematic beam configuration for a torus surface in an optical lattice. Plane wave beams in the horizontal directions generate a rectangular lattice in the $x y$ plane. In the $z$ direction, a superlattice structure created by pairs of blue-detuned and red-detuned beams confines atoms in two layers. The - $z$ propagating blue-detuned beam has the beam shape of a square annulus. (Inset) Different laser intensities turn the interlayer tunneling on and off in different regions. To complete the torus surface, only the interlayer tunneling in the edge region is allowed. (b) Generalization of the scheme to surfaces with higher genus ( $g = 2$ , 3 shown, for example) can be achieved by puncturing more holes in the middle of the lattice.
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Figure 2
Numerically evaluated dipole potential and tunneling strengths. We consider ${Rb}^{87}$ atoms with $a_{x} ≃ a_{y} = 480 nm$ and $k_{x} = k_{z} / 2 = 2 q_{z}$ . In the unit of recoil energy $E_{r} \equiv ℏ^{2} k_{x}^{2} / 2 m (E_{r, z} \equiv ℏ^{2} k_{z}^{2} / 2 m)$ , $V_{0} = 8 E_{r}$ , $V_{E} = 60 E_{r} = 15 E_{r, z}$ , $V_{B} = 120 E_{r} = 30 E_{r, z}$ , and $V_{red} = 20 E_{r} = 5 E_{r, z}$ . (a) Dipole potentials in the $x y$ plane on the upper layer. (b) Dipole potentials in the $y z$ plane. Interlayer tunneling strengths in bulk ( $J_{z}^{bulk}$ ) and edge ( $J_{z}^{edge}$ ) are shown for comparison. (c) Numerically evaluated tunneling strengths represented as the thickness of bonds in the 3D lattice. Shown tunneling strengths range from $0.03 E_{r}$ to $0.04 E_{r}$ .
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Figure 3
(a) A scheme to generate supercurrents in two cycles. A focused, blue-detuned laser beam acts as a stirrer along each cycle, namely, cycle 1 and 2. Note that the stirrer along cycle 2 is focused on the upper layer. A uniform condensate is loaded on the torus initially, then the stirring potential along cycle 1 ( $V_{1}$ ) or cycle 2 ( $V_{2}$ ) is ramped up and down. (b) Quantization of vorticity in two cycles. Dotted curves in the upper plots indicate the ramping sequences of $V_{1}$ and $V_{2}$ . Solid lines in the upper plots indicate the number of completed cycles ( $m$ ) in the stirring process. The lower plots show vorticities ( $v_{i}$ ) changing over time. Steady-state wave functions of the different sequences are shown below.
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Figure 4
(a),(b) A scheme for FQH Hamiltonian. Different Raman beam triplets $T_{i = 1 - 4}$ give the different tunneling phases in Eq. (6). Schematic beam configuration of $T_{1}$ is shown for an example. Zeeman energy difference $Δ_{x} (Δ_{y})$ in the $x (y)$ direction is matched with detuning of Raman beams in triplets $T_{i = 1, 3} (T_{i = 2, 4})$ to give tunneling terms in the same direction. To address each layer independently, beam $i +$ and $i -$ in triplet $T_{i = 1, 2} (T_{i = 3, 4})$ destructively interfere at the lower (upper) layer. (c) Exact diagonalization of the FQH Hamiltonian for 3 hardcore bosonic atoms on a $6 \times 6$ square lattice ( $N_{x} = N_{y} = 6$ ) with periodic boundary conditions and $ϕ = π / 3$ , magnetic length $l_{B} \equiv \sqrt{2 π / ϕ}$ . $E_{s} (| ψ_{s} ⟩)$ indicates the $s$ th lowest eigenenergy (eigenstate). (d) Energy spectrums with distinct intralayer ( $J$ ) and interlayer $(J^{'})$ tunnelings. (e) Spectrum with a random disorder of scale $0.05 J$ . Energy splitting between the ground states is $5 \times 10^{- 3} J$ . (f) Inserting flux $Φ_{x}$ through the handle of torus is equivalent to the boundary condition with twist angle $α_{x}$ . (g) With additional potential $V (y) = (0.01 J / N_{y}) y$ , the spectral flows in $α_{x}$ can be detected by measuring the $y$ coordinates of the states.
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