arXiv:2409.09922 · Physical review. B./Physical review. B

Bloch oscillations of Fibonacci anyons

Xiaoqi Zhou, Weixuan Zhang, Hao Yuan, Xiangdong Zhang

Published Sep 3, 2024• 6 citations

Abstract

Non-Abelian anyons, which correspond to collective excitations possessing multiple fusion channels and noncommuting braiding statistics, serve as the fundamental constituents for topological quantum computation. Here, we reveal the exotic Bloch oscillations (BOs) induced by non-Abelian fusion of Fibonacci anyons. It is shown that the interplay between fusion-dependent internal energy levels and external forces can induce BOs and Bloch-Zener oscillations (BZOs) of coupled fusion degrees with varying periods. In this case, the golden ratio of the fusion matrix can be determined by the period of BOs or BZOs in conjunction with external forces, giving rise to an effective way to unravel non-Abelian fusion. Furthermore, we experimentally simulate non-Abelian fusion BOs by mapping the Schr\"odinger equation of two Fibonacci anyons onto the dynamical equation of electric circuits. Through the measurement of impedance spectra and voltage evolution, both fusion-dependent BZOs and BOs are simulated. Our findings establish a connection between BOs and non-Abelian fusion, providing a versatile platform for simulating numerous intriguing phenomena associated with non-Abelian physics.

DOI: 10.1103/physrevb.110.094301

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arXiv:2409.09922 · Physical review. B./Physical review. B

Bloch oscillations of Fibonacci anyons

Xiaoqi Zhou, Weixuan Zhang, Hao Yuan, Xiangdong Zhang

Published Sep 3, 2024• 6 citations

arXiv HTML PDF

Abstract

Non-Abelian anyons, which correspond to collective excitations possessing multiple fusion channels and noncommuting braiding statistics, serve as the fundamental constituents for topological quantum computation. Here, we reveal the exotic Bloch oscillations (BOs) induced by non-Abelian fusion of Fibonacci anyons. It is shown that the interplay between fusion-dependent internal energy levels and external forces can induce BOs and Bloch-Zener oscillations (BZOs) of coupled fusion degrees with varying periods. In this case, the golden ratio of the fusion matrix can be determined by the period of BOs or BZOs in conjunction with external forces, giving rise to an effective way to unravel non-Abelian fusion. Furthermore, we experimentally simulate non-Abelian fusion BOs by mapping the Schr"odinger equation of two Fibonacci anyons onto the dynamical equation of electric circuits. Through the measurement of impedance spectra and voltage evolution, both fusion-dependent BZOs and BOs are simulated. Our findings establish a connection between BOs and non-Abelian fusion, providing a versatile platform for simulating numerous intriguing phenomena associated with non-Abelian physics.

DOI: 10.1103/physrevb.110.094301

Related papers

Observation of anyonic Bloch oscillations

Weixuan Zhang, Hao Yuan, Haiteng Wang +5 more

Bloch oscillations are exotic phenomena describing the periodic motion of a wave packet subjected to the external force in a lattice, where the system possessing single- or multipleparticles could exhibit distinct oscillation behaviors. In particular, it has been pointed out that quantum statistics could dramatically affected the Bloch oscillation even in the absence of particle interactions, where the oscillation frequency of two pseudofermions with the anyonic statistical angle being pi becomes half of that for two bosons. However, these statisticdependent Bloch oscillations have never been observed in experiments up to now. Here, we report the first experimental simulation of anyonic Bloch oscillations using electric circuits. By mapping eigenstates of two anyons to modes of designed circuit simulators, the Bloch oscillation of two bosons and two pseudofermions are verified by measuring the voltage dynamics. It is found that the oscillation period in the two-boson simulator is almost twice of that in the two-pseudofermion simulator, which is consistent with the theoretical prediction. Our proposal provides a flexible platform to investigate and visualize many interesting phenomena related to particle statistics, and could have potential applications in the field of the novelty signal control.Bloch oscillations are exotic phenomena describing the periodic motion of a wave packet subjected to the external force in a lattice, where the system possessing single- or multipleparticles could exhibit distinct oscillation behaviors. In particular, it has been pointed out that quantum statistics could dramatically affected the Bloch oscillation even in the absence of particle interactions, where the oscillation frequency of two pseudofermions with the anyonic statistical angle being pi becomes half of that for two bosons. However, these statisticdependent Bloch oscillations have never been observed in experiments up to now. Here, we report the first experimental simulation of anyonic Bloch oscillations using electric circuits. By mapping eigenstates of two anyons to modes of designed circuit simulators, the Bloch oscillation of two bosons and two pseudofermions are verified by measuring the voltage dynamics. It is found that the oscillation period in the two-boson simulator is almost twice of that in the two-pseudofermion simulator, which is consistent with the theoretical prediction. Our proposal provides a flexible platform to investigate and visualize many interesting phenomena related to particle statistics, and could have potential applications in the field of the novelty signal control.

arXiv:2110.05967Oct 12, 20210 citations

Observation of Bloch oscillations dominated by effective anyonic particle statistics

Weixuan Zhang, Hao Yuan, Haiteng Wang +5 more

Bloch oscillations are exotic phenomena describing the periodic motion of a wave packet subjected to an external force in a lattice, where a system possessing single or multiple particles could exhibit distinct oscillation behaviors. In particular, it has been pointed out that quantum statistics could dramatically affect the Bloch oscillation even in the absence of particle interactions, where the oscillation frequency of two pseudofermions with an anyonic statistical angle of [Formula: see text] becomes half of that for two bosons. However, these statistically dependent Bloch oscillations have never been observed in experiments until now. Here, we report the experimental simulation of anyonic Bloch oscillations using electric circuits. By mapping the eigenstates of two anyons to the modes of the designed circuit simulators, the Bloch oscillations of two bosons and two pseudofermions are verified by measuring the voltage dynamics. The oscillation period in the two-boson simulator is almost twice of that in the two-pseudofermion simulator, that is consistent with the theoretical prediction. Our proposal provides a flexible platform to investigate and visualize many interesting phenomena related to particle statistics and could have potential applications in the field of the signal control.Bloch oscillations are exotic phenomena describing the periodic motion of a wave packet subjected to an external force in a lattice, where a system possessing single or multiple particles could exhibit distinct oscillation behaviors. In particular, it has been pointed out that quantum statistics could dramatically affect the Bloch oscillation even in the absence of particle interactions, where the oscillation frequency of two pseudofermions with an anyonic statistical angle of [Formula: see text] becomes half of that for two bosons. However, these statistically dependent Bloch oscillations have never been observed in experiments until now. Here, we report the experimental simulation of anyonic Bloch oscillations using electric circuits. By mapping the eigenstates of two anyons to the modes of the designed circuit simulators, the Bloch oscillations of two bosons and two pseudofermions are verified by measuring the voltage dynamics. The oscillation period in the two-boson simulator is almost twice of that in the two-pseudofermion simulator, that is consistent with the theoretical prediction. Our proposal provides a flexible platform to investigate and visualize many interesting phenomena related to particle statistics and could have potential applications in the field of the signal control.

Nature CommunicationsMay 2, 202240 citations

Topological Qubits: Braiding Non-Abelian Anyons with Fibonacci Fusion and Flux Readout — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon topological qubits rely on braiding non-Abelian anyons in 2D systems, with flux qubits enabling readout via Josephson vortices, and Fibonacci anyons providing universal gate sets through braiding operations. MATH: Braiding group generators $\sigma_i$ satisfy $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ (Yang–Baxter equation). Fibonacci anyon fusion rules: $\tau \times \tau = 1 + \tau$ , with quantum dimension $d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618$ . Single-qubit braiding gates compiled from Fibonacci anyons use matrices with entries involving $\phi^{-1} \approx 0.618$ . Chern–Simons term in Ginzburg–Landau model: $S_{CS} = \frac{k}{4\pi} \int A \wedge dA$ , with level $k$ integer. CONNECTION: Fibonacci anyon quantum dimension $d_\tau = \phi$ directly ties to the golden ratio 1.618 and its inverse 0.618. Braiding matrices for Fibonacci anyons involve $\phi^{-1}$ and $\phi^{-2} \approx 0.382$ . The Yan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon topological qubits rely on braiding non-Abelian anyons in 2D systems, with flux qubits enabling readout via Josephson vortices, and Fibonacci anyons providing universal gate sets through braiding operations. MATH: Braiding group generators $ \sigma_i $ satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $ (Yang–Baxter equation). Fibonacci anyon fusion rules: $ \tau \times \tau = 1 + \tau $, with quantum dimension $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. Single-qubit braiding gates compiled from Fibonacci anyons use matrices with entries involving $ \phi^{-1} \approx 0.618 $. Chern–Simons term in Ginzburg–Landau model: $ S_{CS} = \frac{k}{4\pi} \int A \wedge dA $, with level $ k $ integer. CONNECTION: Fibonacci anyon quantum dimension $ d_\tau = \phi $ directly ties to the golden ratio 1.618 and its inverse 0.618. Braiding matrices for Fibonacci anyons involve $ \phi^{-1} $ and $ \phi^{-2} \approx 0.382 $. The Yan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Andrew Stewart Caldin

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Jul 21, 20260 citations

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Andrew Stewart Caldin

FINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter equation); non-Abelian anyons carry unitary representations $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)$ for $|i-j|\geq 2$ . Fibonacci anyons: fusion rule $\tau \times \tau = 1 + \tau$ , Hilbert space dimension grows as Fibonacci sequence $F_n$ , golden ratio $\phi = (1+\sqrt{5})/2 \approx 1.618$ appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus- $g$ surfaces: dimension $= k^g$ for level- $k$ Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio $F_{n+1}/F_n \to 1.618$ — direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter equation); non-Abelian anyons carry unitary representations $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)$ for $|i-j|\geq 2$. Fibonacci anyons: fusion rule $\tau \times \tau = 1 + \tau$, Hilbert space dimension grows as Fibonacci sequence $F_n$, golden ratio $\phi = (1+\sqrt{5})/2 \approx 1.618$ appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-$g$ surfaces: dimension $= k^g$ for level-$k$ Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio $F_{n+1}/F_n \to 1.618$ — direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Non-Abelian Anyons and Braid Group Representations for Topological Quantum Computation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})$ ; Fibonacci anyons have fusion rule $\tau \times \tau = 1 + \tau$ , yielding quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ (golden ratio); topological protection arises from ground-state degeneracy on genus- $g$ surfaces: $\dim \mathcal{H} = 4^g$ for toric code, or $2^g$ for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio $\phi = 1.618$ , and its inverse $\phi^{-1} = 0.618$ appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})$; Fibonacci anyons have fusion rule $\tau \times \tau = 1 + \tau$, yielding quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ (golden ratio); topological protection arises from ground-state degeneracy on genus-$g$ surfaces: $\dim \mathcal{H} = 4^g$ for toric code, or $2^g$ for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio $\phi = 1.618$, and its inverse $\phi^{-1} = 0.618$ appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 21, 20260 citations

\sigma_i

satisfy

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}

(Yang–Baxter equation). Fibonacci anyon fusion rules:

\tau \times \tau = 1 + \tau

, with quantum dimension

d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618

. Single-qubit braiding gates compiled from Fibonacci anyons use matrices with entries involving

\phi^{-1} \approx 0.618

. Chern–Simons term in Ginzburg–Landau model:

S_{CS} = \frac{k}{4\pi} \int A \wedge dA

, with level

k

integer. CONNECTION: Fibonacci anyon quantum dimension

d_\tau = \phi

directly ties to the golden ratio 1.618 and its inverse 0.618. Braiding matrices for Fibonacci anyons involve

\phi^{-1}

and

\phi^{-2} \approx 0.382

. The Yan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling topological quantum computation via braiding operations. MATH: - Braid group

B_n

generators

\sigma_i

satisfy

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}

and

\sigma_i \sigma_j = \sigma_j \sigma_i

for

|i-j|>1

. - Hecke algebra relations:

(\sigma_i - q^{1/2})(\sigma_i + q^{-1/2}) = 0

, with

q = e^{2\pi i/k}

for SU(2)_k anyons. - Fusion rules:

\phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c

, where

N_{ab}^c

are non-negative integers (fusion coefficients). - Fibonacci anyon fusion:

\tau \times \tau = 1 + \tau

, with quantum dimension

d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618

. CONNECTION: - Fibonacci anyon quantum dimension

d_\tau = \phi = 1.618

(golden ratio). - Braid matrix eigenvalues for Fibonacci anyons:

e^{\pm 3\pi i/5}

, related to golden ratio via \( 2\cos(3\pi/5) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

B_n

generators

\sigma_i

satisfy

\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}

(Yang–Baxter equation); non-Abelian anyons carry unitary representations

U(\sigma_i)

with

U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)

for

|i-j|\geq 2

. Fibonacci anyons: fusion rule

\tau \times \tau = 1 + \tau

, Hilbert space dimension grows as Fibonacci sequence

F_n

, golden ratio

\phi = (1+\sqrt{5})/2 \approx 1.618

appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-

g

surfaces: dimension

= k^g

for level-

k

Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio

F_{n+1}/F_n \to 1.618

— direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group

B_n

generators

\sigma_i

satisfy

\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}

(Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices

U(\sigma_i)

with

U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})

; Fibonacci anyons have fusion rule

\tau \times \tau = 1 + \tau

, yielding quantum dimension

\phi = (1+\sqrt{5})/2 \approx 1.618

(golden ratio); topological protection arises from ground-state degeneracy on genus-

g

surfaces:

\dim \mathcal{H} = 4^g

for toric code, or

2^g

for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio

\phi = 1.618

, and its inverse

\phi^{-1} = 0.618

appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com