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arXiv:2409.04515 · Physica Scripta

Cavity control of topological qubits: fusion rule, anyon braiding, and Majorana-Schrödinger cat states

Luis Quiroga, Fernando J. Gómez-Ruiz, Ivan A. Bocanegra-Garay, Ferney J. Rodríguez, C. Tejedor

Published Aug 1, 2025• 0 citations
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Abstract

Abstract We investigate the effects of coupling a local electromagnetic cavity to a segment of a topological Kitaev chain (KC), with particular emphasis on the interplay between photons and Majorana zero modes (MZMs). In addition to the well-known scissor effect-which effectively partitions the chain and isolates free MZMs in the bulk-we provide evidence of non-trivial fusion rules and braiding operations, hallmark signatures of non-Abelian anyons, enabled by spatially selective ultrastrong KC-cavity coupling. We propose that these distinctive MZM properties can be experimentally probed via fermionic parity measurements and photon-induced Berry phases. Furthermore, we demonstrate that, in the so-called sweet-spot regime, the coupled system can be mapped onto a Rabi-like model with a homodyne-rotated quadrature, offering a simplified yet powerful theoretical description. Exploiting the symmetry of fermionic modes within a two-site cavity configuration, we also show the feasibility of generating hybrid MZM-polariton Schrödinger cat states. Our findings offer a novel approach to manipulating topological quantum matter through local light–matter interactions and provide theoretical tools for future experimental realizations in platforms such as quantum materials or superconducting circuits.

DOI
10.1088/1402-4896/adf782

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Abstract We investigate the effects of coupling a local electromagnetic cavity to a segment of a topological Kitaev chain (KC), with particular emphasis on the interplay between photons and Majorana zero modes (MZMs). In addition to the well-known scissor effect-which effectively partitions the chain and isolates free MZMs in the bulk-we provide evidence of non-trivial fusion rules and braiding operations, hallmark signatures of non-Abelian anyons, enabled by spatially selective ultrastrong KC-cavity coupling. We propose that these distinctive MZM properties can be experimentally probed via fermionic parity measurements and photon-induced Berry phases. Furthermore, we demonstrate that, in the so-called sweet-spot regime, the coupled system can be mapped onto a Rabi-like model with a homodyne-rotated quadrature, offering a simplified yet powerful theoretical description. Exploiting the symmetry of fermionic modes within a two-site cavity configuration, we also show the feasibility of generating hybrid MZM-polariton Schrödinger cat states. Our findings offer a novel approach to manipulating topological quantum matter through local light–matter interactions and provide theoretical tools for future experimental realizations in platforms such as quantum materials or superconducting circuits.
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.
FINDING: Majorana zero modes (MZMs) in topological superconductors yield non-Abelian anyonic statistics, enabling topologically protected quantum error correction via ground-state degeneracy in the Kitaev chain. | MATH: Kitaev chain Hamiltonian H=∑j[−t(cj†cj+1+h.c.)+Δ(cjcj+1+h.c.)−μ(cj†cj−1/2)] H = \sum_j [-t(c^\dagger_j c_{j+1} + h.c.) + \Delta(c_j c_{j+1} + h.c.) - \mu(c^\dagger_j c_j - 1/2)] H=∑j​[−t(cj†​cj+1​+h.c.)+Δ(cj​cj+1​+h.c.)−μ(cj†​cj​−1/2)]. In the topological phase (∣μ∣<2t|\mu| < 2t∣μ∣<2t), unpaired Majorana operators γ1,γ2\gamma_1, \gamma_2γ1​,γ2​ satisfy γi†=γi\gamma_i^\dagger = \gamma_iγi†​=γi​, {γi,γj}=2δij\{\gamma_i,\gamma_j\} = 2\delta_{ij}{γi​,γj​}=2δij​, and the nonlocal fermion f=(γ1+iγ2)/2f = (\gamma_1 + i\gamma_2)/2f=(γ1​+iγ2​)/2 gives a two-fold ground-state degeneracy (qubit). Braiding MZMs implements the Clifford group; the fusion rules follow the Ising anyon model: σ×σ=1+ψ\sigma \times \sigma = 1 + \psiσ×σ=1+ψ, with quantum dimension dσ=2d_\sigma = \sqrt{2}dσ​=2​. | CONNECTION: The Ising anyon fusion rule σ×σ=1+ψ\sigma \times \sigma = 1 + \psiσ×σ=1+ψ maps directly to the golden-ratio-adjacent structure of the Fibonacci anyon model (though MZMs are Ising, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.
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 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.
We investigate the ability of selective cavity coupling to a topological chain for tailoring the connectivity of Majorana fermions. We show how topological qubits (TQs), associated with nonlocal Majorana fermion pairing, can be displaced from the edges to the bulk of a topological chain through selective access to light-matter interaction with specific physical sites. In particular, we present a comprehensive density matrix renormalization group study of ground-state features in different chain-cavity coupling geometries, and we validate analytical insights in the strong-coupling regime. This type of Majorana fermion correlation generation process comes with emergent cavity photon features. Moreover, by considering the time evolution after a sudden quench of the cavity-matter coupling strength, we show that the development of high nontrivial matter (Majorana) correlations leaves behind measurable nonclassical photon imprints in the cavity. Innovative ways to dynamically generate TQ nonlocal correlations in topological chains of arbitrary length are thus provided, opening alternative routes to controllable long-range entanglement in hybrid photonic solid-state systems.