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Unexpected results for the non-trivial fusion of Majorana zero modes in interacting quantum-dot arrays

Bradraj Pandey, Satoshi Okamoto, Elbio R. Dagotto

Published Dec 18, 2023• 1 citation
Publisher

Abstract

Abstract Motivated by recent experimental reports of Majorana zero modes (MZMs) in quantum-dot systems at the “sweet spot,” where the electronic hopping th is equal to the superconducting coupling ∆, we study the time-dependent spectroscopy corresponding to the non-trivial fusion of MZMs. For the first time, we employ an experimentally accessible time-dependent real-space local density-of-states (LDOS) method to investigate the non-trivial MZMs fusion outcomes in canonical chains and in a Y-shape array of interacting electrons. In the case of quantum-dot chains where two pairs of MZMs are initially disconnected, after fusion we find equal-height peaks in the electron and hole components of the LDOS, signaling non-trivial fusion into both the vacuum I and fermion Ψ channels with equal weight. For π-junction quantum-dot chains, after the non-trivial fusion, surprisingly we observed the formation of an exotic two-site MZM near the center of the chain, coexisting with another single-site MZM. Furthermore, we also studied the fusion of three MZMs in the Y-shape geometry. In this case, after the fusion we observed the novel formation of another exotic multi-site MZM, with properties depending on the connection and geometry of the central region of the Y-shape quantum-dot array.

DOI
10.21203/rs.3.rs-3745657/v1

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Abstract Motivated by recent experimental reports of Majorana zero modes (MZMs) in quantum-dot systems at the “sweet spot,” where the electronic hopping th is equal to the superconducting coupling ∆, we study the time-dependent spectroscopy corresponding to the non-trivial fusion of MZMs. For the first time, we employ an experimentally accessible time-dependent real-space local density-of-states (LDOS) method to investigate the non-trivial MZMs fusion outcomes in canonical chains and in a Y-shape array of interacting electrons. In the case of quantum-dot chains where two pairs of MZMs are initially disconnected, after fusion we find equal-height peaks in the electron and hole components of the LDOS, signaling non-trivial fusion into both the vacuum I and fermion Ψ channels with equal weight. For π-junction quantum-dot chains, after the non-trivial fusion, surprisingly we observed the formation of an exotic two-site MZM near the center of the chain, coexisting with another single-site MZM. Furthermore, we also studied the fusion of three MZMs in the Y-shape geometry. In this case, after the fusion we observed the novel formation of another exotic multi-site MZM, with properties depending on the connection and geometry of the central region of the Y-shape quantum-dot array.
Motivated by recent experimental reports of Majorana zero modes (MZMs) in quantum-dot systems at the “sweet spot,” where the electronic hopping t h is equal to the superconducting coupling Δ , we study the time-dependent spectroscopy corresponding to the fusion of MZMs. The term “nontrivial” refers to the fusion of Majoranas from different original pairs of MZMs, each with well-defined parities. We employ an experimentally accessible time-dependent real-space local density-of-states (LDOS) method to investigate the nontrivial MZM fusion outcomes in canonical chains and in a Y-shaped array of interacting electrons. In the case of quantum-dot chains where two pairs of MZMs are initially disconnected, after fusion we find equal-height peaks in the electron and hole components of the LDOS, signaling nontrivial fusion into both the vacuum I and fermion Ψ channels with equal weight. For π -junction quantum-dot chains, where the superconducting phase has opposite signs on the left and right portions of the chain, after the nontrivial fusion we observed the formation of an exotic two-site MZM near the center of the chain, coexisting with another single-site MZM. Furthermore, we also studied the fusion of three MZMs in the Y-shaped geometry. In this case, after the fusion we observed the novel formation of another exotic multisite MZM, with properties depending on the connection and geometry of the central region of the Y-shaped quantum-dot array. Published by the American Physical Society 2024
Motivated by the recent experimental realization of a minimal Kitaev chain in quantum dot systems, we present our theoretical findings on the dynamics and fusion of MZMs at or near the “sweet spot” th = Δ (where the fermionic hopping th and superconducting coupling Δ are equal). We investigated the dynamics and fusion of MZMs using time-dependent real-space local density-of-states methods. The movement of Majoranas and the detection of fusion channels are crucial for topological quantum computations. Additionally, we discuss our recent discovery of exotic “multi-site” MZMs in Y-shaped Kitaev wires, which is important for the potential braiding of Majoranas in Y-junctions formed from arrays of quantum dots. Finally, we present results on ”non-trivial” fusion using canonical Kitaev wires at the sweet spot.
Abstract Recent advances in quantum dot platforms have opened a new platform for realizing Majorana zero modes (MZMs) and simulating topological quantum computation. Here we propose an experimentally feasible setup for implementing topological X\sqrt{X}X​ X - and X -quantum gates in an interacting Y -shaped quantum-dot array. The proposed architecture enables both braiding and charge readout through simple fusion operations controlled by gate-tunable potentials. Using many-body time-dependent exact diagonalization, we analyze MZM braiding and fusion dynamics in the presence of nearest-neighbor Coulomb interactions, hopping-pairing mismatch, onsite charge fluctuations, and pairing disorder. We compute diabatic errors, braiding fidelity, and the time- and space-resolved electron and hole components of the local density of states to monitor the braiding process. Our results show that even weak interactions or pairing disorder induce oscillations in the braiding fidelity, thereby setting an upper bound on the braiding speed. Furthermore, we demonstrate that comparing fusion outcomes before and after double braiding provides a direct and experimentally accessible signature of the non-Abelian nature of MZMs in quantum-dot systems.
In this paper we perform real-time simulations for probing the non-Abelian fusion of a pair of Majorana zero modes (MZMs). The nontrivial fusion outcomes can be either a vacuum, or an unpaired fermion, which reflect the underlying non-Abelian statistics. The two possible outcomes can cause different charge variations in the nearby probing quantum dot (QD), while the charge occupation in the dot is detected by a quantum point contact. In particular, we find that gradual fusion and gradual coupling of the MZMs to the QD (in nearly adiabatic switching-on limit) provide a simpler detection scheme than sudden coupling after fusion to infer the coexistence of two fusion outcomes, by measuring the occupation probability of the QD. For the scheme of sudden coupling (after fusion), we propose and analyze continuous weak measurement for the quantum oscillations of the QD occupancy. From the power spectrum of the measurement currents, one can identify the characteristic frequencies and infer thus the coexistence of the fusion outcomes.
In a recent breakthrough experiment [Nature (London) 614, 445 (2023)], signatures of Majorana zero modes have been observed in tunnel spectroscopy for a minimal Kitaev chain constructed from coupled quantum dots. However, as Ising anyons, Majoranas' most fundamental property of non-Abelian statistics is yet to be detected. Moreover, the minimal Kitaev chain is qualitatively different from topological superconductors in that it supports Majoranas only at a sweet spot. Therefore, it is not obvious whether non-Abelian characteristics such as braiding and fusion can be demonstrated in this platform with a reasonable level of robustness. In this work, we theoretically propose a protocol for detecting the Majorana fusion rules in an artificial Kitaev chain consisting of four quantum dots. In contrast with the previous proposals for semiconductor-superconductor hybrid nanowire platforms, here we do not rely on mesoscopic superconducting islands, which are difficult to implement in quantum dot chains. To show the robustness of the fusion protocol, we discuss the effects of three types of realistic imperfections on the fusion outcomes, e.g., diabatic errors, dephasing errors, and calibration errors. We also propose a fermion parity readout scheme using quantum capacitance. Our work will shed light on future experiments on detecting the non-Abelian properties of Majorana modes in a quantum dot chain.
Abstract We propose and simulate a near-adiabatically coupling probing scheme for nontrivial fusion of a pair of Majorana zero modes (MZMs). The scheme can avoid the complexity of oscillating charge occupation in the probing quantum dot, making thus practical measurements more feasible. We also show how to extract the information of nonadiabatic transition and fermion parity violation caused during moving the MZMs together to fuse, from the initial states prepared with definite fermion parity. All the simulations, including the effective coupling between the fusing MZMs, and their coupling to the probing quantum dot, are based on the lattice model of a Rashba quantum wire in proximity contact with an s -wave superconductor, under the modulation of mini-gate voltage control.