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An experimental proposal to observe non-abelian statistics of Majorana-Shockley fermions in an optical lattice 1 DONG-LING DENG,

Sheng-Tao Wang, Kai Sun, L.-M. Duan

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

in the current investigation on quantum physics. Despite the recent great progress, it remains technically elusive to braid the quasi-particles in materials to verify their conjectured non-abelian statistics. 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. Observation of the non-abelian statistics is of both fundamental interest and practical importance, in particular for topological quantum computation.

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in the current investigation on quantum physics. Despite the recent great progress, it remains technically elusive to braid the quasi-particles in materials to verify their conjectured non-abelian statistics. 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. Observation of the non-abelian statistics is of both fundamental interest and practical importance, in particular for topological quantum computation.
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.
Topological quantum states of matter, both Abelian and non-Abelian, are characterized by excitations whose wavefunctions undergo nontrivial statistical transformations as one excitation is moved (braided) around another. Topological quantum computation proposes to use the topological protection and the braiding statistics of a non-Abelian topological state to perform quantum computation. The enormous technological prospect of topological quantum computation provides new motivation for experimentally observing a topological state. Here, we explicitly work out a realistic experimental scheme to create and braid the Abelian topological excitations in the Kitaev model built on a tunable robust system, a cold atom optical lattice. We also demonstrate how to detect the key feature of these excitations: their braiding statistics. Observation of this statistics would directly establish the existence of anyons, quantum particles that are neither fermions nor bosons. In addition to establishing topological matter, the experimental scheme we develop here can also be adapted to a non-Abelian topological state, supported by the same Kitaev model but in a different parameter regime, to eventually build topologically protected quantum gates.
Majorana zero modes (MZMs) exhibit a fundamental property of non-Abelian statistics, which makes them potentially useful as ideal quantum bits to manipulate certain topological quantum tasks. Rapid theoretical developments have greatly reduced the technological requirements and made it possible to experimentally observe MZMs. However, until now, only a few positive signatures of the formation of MZMs have been reported in solid-state systems, and there has been no investigation of their crucial non-Abelian aspects. Here, we experimentally investigate the non-Abelian character of MZMs in the Kitaev chain model using a photonic quantum simulator. Through the Jordan-Wigner transformation, the geometry phases of the braiding of two MZMs can be directly obtained in an optical system. The local phase-error immunity of MZMs is further studied. This work will be useful for the study of quantum non-Abelian statistics and may provide a novel way to investigate topological quantities of quantum systems.
We propose an experimental scheme to simulate the transverse field Ising model with cold atoms trapped in one-dimensional optical lattice. Majorana fermions are created at the ends of the optical lattice segment in topological phase. By controlling the addressing lasers, one can move, fuse and braid them. We also show that the non-Abelian braiding statistics of Majorana fermions can be demonstrated unambiguously through the construction of two braiding operations and distinguishing the resulting two output orthogonal collective spin states. A nice feature of the scheme is that the strong fluorescence provided by the collective spin state can be readily detected in experiment.
We propose an alternative experimental scheme to detect the non-Abelian statistics of the Majorana fermions (MF) with a trapped atomic bosonic gas subject to a spin-orbit coupling in 1D optical lattices. The spin-orbit coupling leads to a faithful simulation of an arbitrary tunable XY spin model, where the anisotropy of the effective spin coupling can be controlled by adjusting the optical potentials, Feshbach resonance, and spin-orbit coupling. MF are created at the ends of the optical lattice segment in topological phase. By controlling the different addressing laser beams, one can move, braid, and exchange them. We also show that the non-Abelian statistics of MF can be demonstrated through distinguishing the states before and after braiding by detecting the parity operator of collective spin states.
It has recently been realized that a general class of non-Abelian defects can be created in conventional topological states by introducing extrinsic defects, such as lattice dislocations or superconductor-ferromagnet domain walls in conventional quantum Hall states or topological insulators. In this paper, we begin by placing these defects within the broader conceptual scheme of extrinsic twist defects associated with symmetries of the topological state. We explicitly study several classes of examples, including Z2{Z}_{2}Z2​ and Z3{Z}_{3}Z3​ twist defects, where the topological state with NNN twist defects can be mapped to a topological state without twist defects on a genus $g\ensuremath{\propto}N$ surface. To emphasize this connection we refer to the twist defects as genons. We develop methods to compute the projective non-Abelian braiding statistics of the genons, and we find the braiding is given by adiabatic modular transformations, or Dehn twists, of the topological state on the effective genus ggg surface. We study the relation between this projective braiding statistics and the ordinary non-Abelian braiding statistics obtained when the genons become deconfined, finite-energy excitations. We find that the braiding is generally different, in contrast to the Majorana case, which opens the possibility for fundamentally novel behavior. We find situations where the genons have quantum dimension 2 and can be used for universal topological quantum computing (TQC), while the host topological state is by itself nonuniversal for TQC.