Proposal for observing non-Abelian statistics of Majorana-Shockley fermions in an optical lattice

Dong-Ling Deng, Sheng-Tao Wang, Kai Sun, and Lu-Ming Duan

Phys. Rev. B 91, 094513 – Published 23 March 2015

Abstract

Besides the conventional bosons and fermions, in synthetic two-dimensional (2D) materials there could exist more exotic quasiparticles with non-Abelian statistics, meaning that the quantum states in the system will be transformed by noncommuting unitary operators when we adiabatically braid the particles one around another. Here, we propose an experimental scheme to observe non-Abelian statistics with cold atoms in a 2D optical lattice. We show that the Majorana-Schockley modes associated with line defects can be braided with non-Abelian statistics through adiabatic shift of the local potentials. We also demonstrate that the braiding operations are robust against typical experimental imperfections and the readout of topological qubits can be accomplished by local measurement of the atom number.

Received 22 November 2014
Revised 3 March 2015

DOI:https://doi.org/10.1103/PhysRevB.91.094513

Authors & Affiliations

Dong-Ling Deng^1,2, Sheng-Tao Wang^1,2, Kai Sun¹, and Lu-Ming Duan^1,2

¹Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
²Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China

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Vol. 91, Iss. 9 — 1 March 2015

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Images

Figure 1
Creation and manipulation of the MSFs in an optical lattice. (a) Cold fermionic atoms are loaded into a 2D optical lattice. $J$ and $Δ$ denote the nearest neighbor hopping rate and pairing strength. A line defect with different local chemical potential binds two zero-energy MSFs $γ_{1}$ and $γ_{2}$ (red circles) at its edges. (b) Energy spectrum of the Hamiltonian $H$ on a square lattice of size $18 a \times 10 a$ with open boundaries. The length of the line defect is $14 a$ . The zero-energy MSFs have tiny energy splitting due to the small size of the line defect, which is numerically found to be $< 10^{- 10} J$ for our parameters. (c) The amplitude of the mode function for $γ_{1}$ and $γ_{2}$ . The black line indicates the line defect with chemical potential $μ_{d}$ . The parameters are chosen as $Δ = J$ , $μ_{0} = 10 J$ , and $μ_{d} = 0.1 J .$
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Figure 2
Braiding of two MSFs bound to the same line defect. (a) The black line indicates the line defect with chemical potential $μ_{d}$ . Sequentially tuning the local chemical potentials at one end from $μ_{d}$ to $μ_{0}$ shortens the line defect and transports $γ_{1}$ along the $x$ direction. The red arrow shows the moving direction of the MSF. Similar operations along a T-junction path realize adiabatic exchange of $γ_{1}$ and $γ_{2}$ , with steps illustrated in (a)–(f). (g) The evolution of the energy gap $E_{g}$ throughout the braiding process. The system is always gapped with the minimum gap $E_{g} > 0.5 J$ . (h) Time evolution of the MSF modes $γ_{1}, γ_{2}$ and their correlations (see Appendixes A and B). All the parameters are the same as in Fig. 1. The total evolution time $T \sim 1.44 \times 10^{4} J$ .
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Figure 3
Braiding of two MSFs bound to different line defects. (a) Illustration of braiding two MSFs from different line defects along the T-junction path. (b) Time evolution of the MSF modes $γ_{1}, γ_{2}, γ_{3}, γ_{4}$ and their correlations. The MSFs $γ_{2}$ and $γ_{3}$ are braided. The parameters are taken as follows: the lattice size $12 a \times 36 a$ , two horizontal line defects each of length $9 a$ and distance $16 a$ , $Δ = 0.91 J$ , $μ_{0} = 10 Δ$ , and $μ_{d} = 0.1 Δ$ . The energy gap is also maintained during the braiding process $E_{g} > 0.5 J$ . The total evolution time $T \sim 1.92 \times 10^{4} J$ .
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Figure 4
Robustness to experimental noise and imperfections. The lattice size is $20 a \times 12 a$ and other parameters are the same as in Fig. 1. $α, V_{T}, λ_{R}$ denote the parameters characterizing respectively the laser beam crosstalk, the strength of the global harmonic trap, and the magnitude of random fluctuation of the chemical potential (see the main text). The total evolution time $T \sim 1.76 \times 10^{4} J$ .
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Figure 5
Detection of the topological qubit. (a) Two MSFs $γ_{1}$ and $γ_{2}$ are fused through adiabatic shortening of the line defect to a single lattice site $r_{0}$ . (b) Transformation of the MSF modes $γ_{1}$ and $γ_{2}$ under adiabatic merging. For simplicity, we plot evolution of the magnitude of the mode overlap between $c_{r_{0}}$ and $[γ_{1} (t) + i γ_{2} (t)] / 2$ . At the end of merging, $γ_{1}$ and $γ_{2}$ are mapped dominantly to the local modes $c_{r_{0}}^{†} + c_{r_{0}}$ and $i (c_{r_{0}}^{†} - c_{r_{0}})$ , respectively, which enables detection of the initial nonlocal topological qubit by a simple measurement of the atom number $c_{r_{0}}^{†} c_{r_{0}}$ on a single lattice site after the adiabatic merging. All the parameters are the same as in Fig. 1. The total evolution time $T \sim 1.04 \times 10^{4} J$ .
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