Abstract
Exceptional points (EPs) are singularities in non-Hermitian systems, where k (k ≥ 2) eigenvalues and eigenstates coalesce. High-order EPs exhibit richer topological characteristics and better sensing performance than second-order EPs. Theory predicts even richer non-Hermitian topological phases for high-order EP geometries, such as lines or rings formed entirely by high-order EPs. However, experimental exploration of high-order EP geometries has hitherto proved difficult due to the demand for more degrees of freedom in the Hamiltonian’s parameter space or a higher level of symmetries. Here we observe a third-order exceptional line in an atomic-scale system. To this end, we use a nitrogen-vacancy spin in diamond and introduce multiple symmetries in the non-Hermitian Hamiltonian realized with the system. Furthermore, we show that the symmetries play an essential role in the occurrence of high-order EP geometries. Our approach can in future be further applied to explore high-order EP-related topological physics at the atomic scale and, potentially, for applications of high-order EPs in quantum technologies.
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Data availability
The data supporting the findings of this study are available within this article and its Supplementary Information. Source data are provided with this paper.
Code availability
The codes that were used in this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the National Key R&D Program of China (grant number 2018YFA0306600 (J.D.)), the National Natural Science Foundation of China (grant numbers 12174373 (Yang Wu), 92265204 (Ya Wang) and 12261160569 (X.R.)). J.D. acknowledges funding from the Chinese Academy of Sciences (grant numbers XDC07000000 and GJJSTD20200001), the Innovation Program for Quantum Science and Technology (grant number 2021ZD0302200), the Anhui Initiative in Quantum Information Technologies (grant number AHY050000) and the Hefei Comprehensive National Science Center. Ya Wang thanks the Fundamental Research Funds for the Central Universities for their support. W.L. is funded by Beijing University of Posts and Telecommunications Innovation Group. This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.
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J.D. and X.R. proposed the idea and supervised the experiments. X.R., Yang Wu and Yunhan Wang designed the experiments. Yunhan Wang and Yang Wu performed the experiments. X.Y. and Ya Wang prepared the sample. Yunhan Wang, Yang Wu, W.L., Z.N. and C.-K.D. carried out the calculations. All authors analysed the data, discussed the results and wrote the manuscript.
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Extended data
Extended Data Fig. 1 Measurement of the parameters \({\mathbf{\gamma }}_{\mathbf{exp }}\), \({\mathbf{h}}_{\mathbf{exp }}\), \({\mathbf{\mu }}_{\mathbf{exp }}\) and \({\mathbf{\nu }}_{\mathbf{exp }}\) of the non-Hermitian Hamiltonian.
a, The measured quantities CPT and the retrieved parameter \({\nu }_{\exp }\). b, The measured quantities CpsCh and the retrieved parameter \({\mu }_{\exp }\). c-d, Population evolution under Hμ,ν(γ,h) with two different sets of initial states and measurement bases. \({\mu }_{\exp }\) and \({\nu }_{\exp }\) are obtained via linear fitting, and \({\gamma }_{\exp }\) and \({{{{\rm{h}}}}}_{\exp }\) are obtained from the evolution as described in the text. Red dots with error bars are experimental results, and blue lines are the theoretical predictions. All errors shown are one standard deviation with one million averages.
Extended Data Fig. 2 Eigenstates of the non-Hermitian Hamiltonian at the EP3 where μ=ν=0, h=0, γ=1.
a-f, Real (a,c,e) and imaginary (b,d,f) parts of the measured density matrices \({{\rho }_{1}}^{\exp }\) (a,b), \({{\rho }_{2}}^{\exp }\) (c,d) and \({{\rho }_{3}}^{\exp }\) (e,f) of three eigenstates (labelled by 1,2 and 3) obtained by quantum state tomography. g,h,\({{\rho }_{1}}^{{{{\rm{theo}}}}}\)=\({{\rho }_{2}}^{{{{\rm{theo}}}}}\)=\({{\rho }_{3}}^{{{{\rm{theo}}}}}\)=ρtheo is the real (g) and imaginary (h) parts the density matrix of theoretically predicted eigenstate. All errors shown are one standard deviation with one million averages.
Supplementary information
Supplementary Information
Supplementary discussion including Supplementary Figs. 1–9 and Table 1.
Supplementary Data 3
The result of the Ramsey experiment and the Rabi oscillation driven by the AC electric fields.
Supplementary Data 5
The measured quantities CpsCh and CPT and the retrieved parameters \({\mu }_{\exp }\) and \({\nu }_{\exp }\).
Supplementary Data 6
Population evolution under Hμ,ν(γ,h) with two different sets of initial states and measurement bases.
Supplementary Data 7
Eigenstates of the NH Hamiltonian at the EP3 where μ=ν=0, h=0.35 and γ=1.06.
Supplementary Data 8
Eigenstates of the NH Hamiltonian at the EP2 where μ=0.2, ν=0, h=-0.35 and γ=0.73.
Supplementary Data 9
Eigenstates of the NH Hamiltonian with μ=0.2, ν=0.05, h=0 and γ=0.96.
Source data
Source Data Fig. 3
Source Data for Fig. 3
Source Data Fig. 4
Source Data for Fig. 4
Source Data Extended Data Fig./Table 1
Source Data for Extended Data Fig. 1.
Source Data Extended Data Fig./Table 2
Source Data for Extended Data Fig. 1.
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Wu, Y., Wang, Y., Ye, X. et al. Third-order exceptional line in a nitrogen-vacancy spin system. Nat. Nanotechnol. 19, 160–165 (2024). https://doi.org/10.1038/s41565-023-01583-0
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DOI: https://doi.org/10.1038/s41565-023-01583-0
