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arXiv:1307.0839 · Physical Review B

Correlated parity measurements as a probe of non-Abelian statistics in one-dimensional superconducting wires

F. J. Burnell

Published Jun 17, 2014• 12 citations
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Abstract

We propose a method to detect a signature of non-Abelian statistics in a one-dimensional (1D) superconducting wire by tuning the effective coupling between a pair of Majorana particles. Our experiment requires a single wire with two segments in the topological superconducting phase and a total of four Majorana particles. We show that an appropriately timed ``pulse'' in the coupling between the two middle Majoranas leads to a coherent rotation of the two qubits associated with the pairs of Majoranas in the two topological superconducting segments, in much the same way that an appropriate-length pulse of a transverse magnetic field can be used to rotate spins in nuclear magnetic resonance. This can be exploited both to probe the correlations of the Majoranas and to manipulate the Majorana qubits in these 1D wires. We discuss the experimental requirements for such a coherent rotation to be observable when the wire system is coupled to a fermionic bath.

DOI
10.1103/physrevb.89.224510

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We propose a method to detect a signature of non-Abelian statistics in a one-dimensional (1D) superconducting wire by tuning the effective coupling between a pair of Majorana particles. Our experiment requires a single wire with two segments in the topological superconducting phase and a total of four Majorana particles. We show that an appropriately timed ``pulse'' in the coupling between the two middle Majoranas leads to a coherent rotation of the two qubits associated with the pairs of Majoranas in the two topological superconducting segments, in much the same way that an appropriate-length pulse of a transverse magnetic field can be used to rotate spins in nuclear magnetic resonance. This can be exploited both to probe the correlations of the Majoranas and to manipulate the Majorana qubits in these 1D wires. We discuss the experimental requirements for such a coherent rotation to be observable when the wire system is coupled to a fermionic bath.
Zero-energy Majorana fermion states (Majoranas) can arise at the ends of a semiconducting wire in proximity with a superconductor. A first generation of experiments has detected a zero bias conductance peak in these systems that strongly suggests these Majoranas do exist; however, a definitive demonstration of the long-ranged entanglement that is crucial for potential applications in quantum computing has yet to be carried out. This work discusses two possible measurement schemes to detect this long-ranged entanglement in a wire system with two coupled pairs of Majoranas, by varying the coupling between one pair while measuring the fermion parity of the second pair. First, in a system with two coupled pairs of Majoranas, we discuss how varying the coupling of one pair in time, while measuring temporal fermion parity correlations of the second pair, allows for an experimental probe of long-ranged Majorana entanglement. Second, we show that the power spectrum of the charge noise (fermion parity noise) of one pair contains signatures of these correlations, as well as allowing one to infer the parity relaxation rate.
The one-dimensional (1D) J1−J2J_1-J_2J1​−J2​ quantum spin model can be viewed as a strong-coupling analogue of the Schrieffer-Su-Heeger model with two inequivalent alternating Ising couplings along the wire, associated to the physics of resonating valence bonds. Similar to the quantum Ising model, which differently presents a long-range N\' eel ordered phase, this model also maps onto a p-wave superconducting wire which shows a topological phase transition with the emergence of low-energy Majorana fermions. We show how signatures of the topological phase transition for the p-wave superconducting wire, i.e. a half Skyrmion, are revealed through local (short-range) spin observables and their derivatives related to the capacitance of the pairing fermion model. Then, we present an edge correspondence through the edge spin susceptibility in the J1−J2J_1-J_2J1​−J2​ model revealing that the topological phase transition is a metal of Majorana fermions. We justify that the spin magnetization at an edge at very small transverse magnetic field is a good marker of the topological invariant and of Majorana zero modes. We identify a correspondence between the quantum information of resonating valence bonds and the charge fluctuations in a p-wave superconductor through our method "the bipartite fluctuations". Physical properties of this 1D model are in fact robust when including additional interactions, which is optimistic for practical applications e.g. in quantum circuits.
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.
Majorana fermions hold promise for quantum computation, because their non-Abelian braiding statistics allows for topologically protected operations on quantum information. Topological qubits can be constructed from pairs of well-separated Majoranas in networks of nanowires. The coupling to a superconducting charge qubit in a transmission line resonator (transmon) permits braiding of Majoranas by external variation of magnetic fluxes. We show that readout operations can also be fully flux controlled, without requiring microscopic control over tunnel couplings. We identify the minimal circuit that can perform the initialization-braiding-measurement steps required to demonstrate non-Abelian statistics. We introduce the Random Access Majorana Memory (RAMM), a scalable circuit that can perform a joint parity measurement on Majoranas belonging to a selection of topological qubits. Such multiqubit measurements allow for the efficient creation of highly entangled states and simplify quantum error correction protocols by avoiding the need for ancilla qubits.
Topological materials are of great interest for applications in quantum computing, providing intrinsic robustness against environmental noises. A popular direction is to look for Majorana modes in integrated systems interfaced with superconducting materials. However, is superconductivity necessary for materials to exhibits non-abelian statistics? Here we predict with strong theoretical and numerical evidences that there exist topologically phases in a class of one-dimensional single crystals, which contain large bandgaps and are within experimental reach. Specifically, the nontrivial Zak phases are associated with gapless boundary states, which provides the non-Abelian statistics required for constructing topologically-protected quantum gates, even without superconductivity and magnetic field. Another anomalous feature of these materials is that the electric polarization is quantized and a transverse field induces a longitudinal polarization. This novel material property provides an experimentally-accessible mechanism for braiding the non-Abelian anyons.
Among the broad spectrum of systems predicted to exhibit topological superconductivity and Majorana fermions, one-dimensional wires with strong spin-orbit coupling provide one of the most promising experimental candidates. Here we investigate the fate of the topological superconducting phase in such wires when repulsive interactions are present. Using a combination of density matrix renormalization group, bosonization, and Hartree--Fock techniques, we demonstrate that while interactions degrade the bulk gap---consistent with recent results of Gangadharaiah et al.---they also greatly expand the parameter range over which the topological phase arises. In particular, we show that with interactions this phase can be accessed over a broader chemical potential window, thereby leading to greater immunity against disorder-induced chemical potential fluctuations in the wire. We also suggest that in certain wires strong interactions may allow Majorana fermions to be generated without requiring a magnetic field.