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arXiv:2503.22792 · Proceedings of the National Academy of Sciences

Detection of anyon braiding through pump–probe spectroscopy

Xu Yang, Ryan Buechele, Nandini Trivedi

Published Nov 12, 2025• 4 citations
arXivHTMLPDF

Abstract

We show that the braiding of anyons in a quantum spin liquid leaves a distinct dynamical signature in the nonlinear pump–probe response. Using a combination of exact diagonalization and matrix product state techniques, we study the nonlinear pump–probe response of the toric code in a magnetic field, a model with mobile electric e and magnetic m anyonic excitations. While the linear response signal oscillates and decays with time like ∼ t − 1.3 , the amplitude of the nonlinear signal for χ XZZ ( 3 ) features a linear-in-time enhancement at early times and a stronger enhancement t α with α > 1 at later times. The comparison between χ XZZ ( 3 ) , which involves nontrivial braiding of e and m anyons, and χ XXX ( 3 ) that involves trivial braiding of the same types of anyons, distinguishes the braiding statistics of anyons. We support our analysis with a hard-core anyon model with statistical gauge fields to develop further insights into the time dependence of the pump–probe response. Pump–probe spectroscopy provides a distinctive new probe of quantum spin liquid states, beyond the inconclusive broad features observed in single spin-flip inelastic neutron scattering.

DOI
10.1073/pnas.2511917122

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We show that the braiding of anyons in a quantum spin liquid leaves a distinct dynamical signature in the nonlinear pump–probe response. Using a combination of exact diagonalization and matrix product state techniques, we study the nonlinear pump–probe response of the toric code in a magnetic field, a model with mobile electric e and magnetic m anyonic excitations. While the linear response signal oscillates and decays with time like ∼ t − 1.3 , the amplitude of the nonlinear signal for χ XZZ ( 3 ) features a linear-in-time enhancement at early times and a stronger enhancement t α with α > 1 at later times. The comparison between χ XZZ ( 3 ) , which involves nontrivial braiding of e and m anyons, and χ XXX ( 3 ) that involves trivial braiding of the same types of anyons, distinguishes the braiding statistics of anyons. We support our analysis with a hard-core anyon model with statistical gauge fields to develop further insights into the time dependence of the pump–probe response. Pump–probe spectroscopy provides a distinctive new probe of quantum spin liquid states, beyond the inconclusive broad features observed in single spin-flip inelastic neutron scattering.
We show that the presence of anyons in the excitation spectrum of a two-dimensional system can be inferred from nonlinear spectroscopic quantities. In particular, we consider pump-probe spectroscopy, where a sample is irradiated by two light pulses with an adjustable time delay between them. The relevant response coefficient exhibits a universal form that originates from the statistical phase acquired when anyons created by the first pulse braid around those created by the second. This behavior is shown to be qualitatively unchanged by nonuniversal physics including nonstatistical interactions and small nonzero temperatures. In magnetic systems, the signal of interest can be measured using currently available terahertz-domain probes, highlighting the potential usefulness of nonlinear spectroscopic techniques in the search for quantum spin liquids.
Identifying experimental signatures of anyons, which exhibit fractional exchange statistics, remains a central challenge in the study of two-dimensional topologically ordered systems. Previous theoretical work has shown that the threshold behavior in linear response spectroscopy can reveal the fractional exchange statistics between an anyon and its antiparticle. In this work, we extend this framework to nonlinear, two-dimensional coherent spectroscopy. We demonstrate by analyzing time-ordered four-point correlation functions that the threshold behavior of nonlinear response functions encodes the fractional statistics between general pairs of anyons that can combine to any composite topological charge. This feature in particular provides a powerful probe for unambiguously distinguishing non-Abelian anyons, which can form multiple composite charges with distinct nontrivial braid statistics. Our approach is validated using numerical simulations that are consistent with the correct fractional exchange statistics for both the Abelian anyons in the toric code and non-Abelian Ising anyons.
Identifying experimental signatures of anyons, which exhibit fractional exchange statistics, remains a central challenge in the study of two-dimensional topologically ordered systems. Previous theoretical work has shown that the threshold behavior in linear response spectroscopy can reveal the fractional exchange statistics between an anyon and its antiparticle. In this work, we extend this framework to nonlinear, two-dimensional coherent spectroscopy. We demonstrate by analyzing time-ordered four-point correlation functions that the threshold behavior of nonlinear response functions encodes the fractional statistics between general pairs of anyons that can combine to any composite topological charge. This feature in particular provides a powerful probe for unambiguously distinguishing non-Abelian anyons, which can form multiple composite charges with distinct nontrivial braid statistics. Our approach is validated using numerical simulations that are consistent with the correct fractional exchange statistics for both the Abelian anyons in the toric code and non-Abelian Ising anyons.
We study the behavior of linear and nonlinear spectroscopic quantities in two-dimensional topologically ordered systems, which host anyonic excitations exhibiting fractional statistics. We highlight the role that braiding phases between anyons have on the dynamics of such quasiparticles, which as we show dictates the behavior of both linear response coefficients at finite temperatures, as well as nonlinear pump-probe response coefficients. These quantities, which act as probes of temporal correlations in the system, are shown to obey distinctive universal forms at sufficiently long timescales. As well as providing an experimentally measurable fingerprint of anyonic statistics, the universal behavior that we find also demonstrates anomalously fast thermal relaxation: correlation functions decay as a “squished exponential” C ( t ) ∼ exp ( − [ t / τ ] 3 / 2 ) at long times. We attribute this unusual asymptotic form to the nonlocal nature of interactions between anyons, which allows relaxation to occur much faster than in systems with quasiparticles interacting via local, nonstatistical interactions. While our results apply to any Abelian or non-Abelian topological phase in two-dimensions, we discuss in particular the implications for candidate quantum spin liquid materials, wherein the relevant quantities can be measured using pre-existing time-resolved terahertz-domain spectroscopic techniques. Published by the American Physical Society 2024
Optical probes have emerged as versatile tools for detecting exotic fractionalized phases in quantum materials. We calculate the low-energy Raman response arising from mobile, interacting Ising anyons in the chiral Kitaev spin liquid perturbed by symmetry allowed interactions---a phase relevant to $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$ under a magnetic field. At zero temperature, the two-anyon continuum response shows a leading power-law scaling of the intensity near the onset of the signal: $I(\ensuremath{\omega})\ensuremath{\sim}{(\ensuremath{\omega}\ensuremath{-}{E}_{2\ensuremath{\sigma}}^{0})}^{\frac{1}{8}}$ for linear and parallel-circular polarization channels, where ${E}_{2\ensuremath{\sigma}}^{0}$ is the two-particle gap. Strong corrections due to short-range interactions arise at order 1/41/41/4. For cross-circularly polarized channels, the scaling is given by $I(\ensuremath{\omega})\ensuremath{\sim}{(\ensuremath{\omega}\ensuremath{-}{E}_{2\ensuremath{\sigma}}^{0})}^{|l\ifmmode\pm\else\textpm\fi{}1/8|}$, where the value of l=0,1,2l=0,1,2l=0,1,2 is determined by the number of minima in the single anyon dispersion. The exponents are directly related to the topological spin of Ising anyons ${\ensuremath{\theta}}_{\ensuremath{\sigma}}=\ensuremath{\pi}/8$, describing their exchange statistics. Our theory generalizes to spectral probes of anyonic quasiparticles with multiple band minima in other quantum liquids. Interaction between anyons may also induce bound states, resulting in sharp peaks that show strong polarization dependence.
Anyons are quasiparticles occurring in two dimensions, whose topological properties are believed to be robust against local perturbations and may hold promise for fault tolerant quantum computing. Here we present an experiment of demonstrating the path independent nature of anyonic braiding statistics with a superconducting quantum circuit, which represents a 7-qubit version of the toric code model. We dynamically create the ground state of the model, achieving a state fidelity of 0.688±0.015 as verified by quantum state tomography. Anyonic excitations and braiding operations are subsequently implemented with single-qubit rotations. The braiding robustness is witnessed by looping an anyonic excitation around another one along two distinct, but topologically equivalent paths: Both reveal the nontrivial π-phase shift, the hallmark of Abelian 1/2 anyons, with a phase accuracy of ∼99% in the Ramsey-type interference measurement.