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arXiv:2507.12772 · Europhysics Letters (EPL)

Probing nontrivial fusion of Majorana zero modes via near-adiabatic coupling

Jing Bai, Luting Xu, Wei Feng, Xin-Qi Li

Published Jul 1, 2025• 0 citations
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Abstract

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.

DOI
10.1209/0295-5075/ade6a6

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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.
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.
We perform simulations for transport and nontrivial fusion of Majorana zero modes in topological superconducting quantum wires. We uncover interesting behaviors of nonadiabatic transition associated with the transport through mini-gate-controlled multiple-segment modulations. Owing to breaking of the initial fermion parity induced by nonadiabatic transitions, a deviation from the statistics of outcomes of nontrivial fusion arises and is analyzed. Moreover, we develop a measurement scheme to infer the amount of fermion parity breaking and nonadiabatic transition probability to excited states, based on the characteristic spectrum of measurement current by a quantum-point-contact detector, by measurement of the charge occupation dynamics in a fusion-outcome-probing quantum dot.
Abstract Fusing Majorana zero modes (MZMs) leads to multiple outcomes, a property unique to non-Abelian anyons. The successful demonstration of this nontrivial fusion rule would be a hallmark for the development of topological quantum computation. Here, we propose that such fusion can be achieved by simply attaching a fermionic mode to a single Majorana zero mode. By modulation of the energy level of the fermionic mode (FM) as well as its coupling with MZM in different sequences, we show that a zero or integer charge pumping can be realized when different fusion loops are chosen. Such fusion loops are intimately related with the nontrivial fusion rule of Majorana modes and are solely determined by the crossings at zero energy in the parameter space. Finally, we demonstrate our proposal in a nanowire-based topological superconductor coupled to a quantum dot. We show that charge pumping is robust for MZMs in the realistic system irrespective of the initial condition of FM state.
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
In this work we perform real-time simulations for the dynamics of braiding a pair of Majorana zero modes (MZMs) through a quantum dot in a minimal setup of pure one-dimensional realization. We reveal the strong nonadiabatic effect when the dot energy level approaches zero in order to achieve a geometric phase $\ensuremath{\pi}/4$ which is required for a full exchange between the MZMs. Rather than the strategies of nonuniformly manipulating the system according to adiabatic conditions and shortcuts to adiabaticity, we propose and illustrate a more feasible scheme to suppress the nonadiabatic transition, which allows for a full exchange between the Majorana modes.
Braiding and fusion of Majorana zero modes are key elements of any future topological Majorana-based quantum computer. Here, we investigate the fusion dynamics of Majorana zero modes in the spinless Kitaev model, as well as in a spinful model describing magnet-superconductor hybrid structures. We consider various scenarios allowing us to reproduce the fusion rules of the Ising anyon model. Particular emphasis is given to the charge of the fermion obtained after fusing two Majorana zero modes: as long as it remains on the superconductor, charge quantization is absent. When moving the fermion to a non-superconducting region, such as a quantum dot, nearly-quantized charge can be measured. Our findings confirm for both platforms that fusion dynamics of Majorana zero modes can indeed be used for the readout of Majorana qubits.