Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2303.15026 · Physical Review Letters

Probing Complex-Energy Topology via Non-Hermitian Absorption Spectroscopy in a Trapped Ion Simulator

Mengdi Cao, Kai Li, Wen Dong Zhao, Weixuan Guo, Ben Qi, X.-Y. Chang, Z.-C. Zhou, Yong Xu, L.-M. Duan

Published Apr 21, 2023• 57 citations
arXivHTMLPDF

Abstract

Non-Hermitian systems generically have complex energies, which may host topological structures, such as links or knots. While there has been great progress in experimentally engineering non-Hermitian models in quantum simulators, it remains a significant challenge to experimentally probe complex energies in these systems, thereby making it difficult to directly diagnose complex-energy topology. Here, we experimentally realize a two-band non-Hermitian model with a single trapped ion whose complex eigenenergies exhibit the unlink, unknot, or Hopf link topological structures. Based on non-Hermitian absorption spectroscopy, we couple one system level to an auxiliary level through a laser beam and then experimentally measure the population of the ion on the auxiliary level after a long period of time. Complex eigenenergies are then extracted, illustrating the unlink, unknot, or Hopf link topological structure. Our work demonstrates that complex energies can be experimentally measured in quantum simulators via non-Hermitian absorption spectroscopy, thereby opening the door for exploring various complex-energy properties in non-Hermitian quantum systems, such as trapped ions, cold atoms, superconducting circuits, or solid-state spin systems.

DOI
10.1103/physrevlett.130.163001

Related papers

Non-Hermitian Absorption Spectroscopy

Kai Li, Yong Xu

While non-Hermitian Hamiltonians have been experimentally realized in cold atom systems, it remains an outstanding open question of how to experimentally measure their complex energy spectra in momentum space for a realistic system with boundaries. The existence of non-Hermitian skin effects may make the question even more difficult to address given the fact that energy spectra for a system with open boundaries are dramatically different from those in momentum space; the fact may even lead to the notion that momentum-space band structures are not experimentally accessible for a system with open boundaries. Here, we generalize the widely used radio-frequency spectroscopy to measure both real and imaginary parts of complex energy spectra of a non-Hermitian quantum system for either bosonic or fermionic atoms. By weakly coupling the energy levels of a non-Hermitian system to auxiliary energy levels, we theoretically derive a formula showing that the decay of atoms on the auxiliary energy levels reflects the real and imaginary parts of energy spectra in momentum space. We further prove that measurement outcomes are independent of boundary conditions in the thermodynamic limit, providing strong evidence that the energy spectrum in momentum space is experimentally measurable. We finally apply our non-Hermitian absorption spectroscopy protocol to the Hatano-Nelson model and non-Hermitian Weyl semimetals to demonstrate its feasibility.

arXiv:2201.09469•Physical Review Letters•Aug 26, 2022•35 citations

Digital Simulation of Non-Hermitian Knotted Bands on Quantum Hardware

Truman Yu Ng, Yuzhu Wang, Wei Jie Chan +3 more

Knots and links represent a fundamental motif of non-local connectivity that permeates the physical sciences from string theory to protein folds. While spectral braiding has been explored in two-band non-Hermitian models across various platforms, its direct simulation and characterization on programmable quantum hardware, particularly beyond two strands, remains a formidable challenge due to the limitations of variational optimization in these systems. Here, we introduce a family of non-Hermitian multi-band twister models and implement a non-variational protocol to characterize their complex braided band structures on a programmable superconducting quantum processor. By mapping the winding of eigenstates to the spectral topology, we devise an efficient measurement strategy that extracts braid information, including braid words and knot invariants like the Alexander and Jones polynomials, without requiring full spectral tomography or repeated optimization. We experimentally demonstrate the reconstruction of complicated knots and links such as the Hopf chain and Solomon's knot. Our approach provides a general framework for investigating exotic non-Hermitian topology on near-term quantum devices, opening a route to simulate more sophisticated topological structures in knot theory.

quant-pharXiv:2604.26914•Apr 30, 2026•0 citations

Observation of Topological Links Associated with Hopf Insulators in a Solid-State Quantum Simulator

Xinxing Yuan, Li He, Sheng-Tao Wang +8 more

Hopf insulators are intriguing three-dimensional topological insulators characterized by an integer topological invariant. They originate from the mathematical theory of Hopf fibration and epitomize the deep connection between knot theory and topological phases of matter, which distinguishes them from other classes of topological insulators. Here, we implement a model Hamiltonian for Hopf insulators in a solid-state quantum simulator and report the first experimental observation of their topological properties, including nontrivial topological links associated with the Hopf fibration and the integer-valued topological invariant obtained from a direct tomographic measurement. Our observation of topological links and Hopf fibration in a quantum simulator opens the door to probe rich topological properties of Hopf insulators in experiments. The quantum simulation and probing methods are also applicable to the study of other intricate three-dimensional topological model Hamiltonians.

arXiv:1705.00781•Chinese Physics Letters•Jun 1, 2017•31 citations

Detecting Complex-Energy Braiding Topology in a Dissipative Atomic Simulator with Transformer-Based Geometric Tomography

Yang Yue, Nan Li, Xin Zhang +8 more

Machine learning (ML) is shaping our exploration of topological matter, whose existence is inherently tied to the geometry of quantum states or energy spectra. In non-Hermitian systems, distinctive spectral geometry can lead to topological braiding of complex-energy bands, yet directly probing this topology-geometry interplay remains challenging. Here, we introduce a Transformer-based ML framework to capture this interplay and experimentally demonstrate it in a dissipative cold-atom simulator. Using a Bose-Einstein condensate, we engineer tunable dissipative two-level systems whose complex eigenenergies form braids. Owing to the density-dependent dissipation, the instantaneous energy braids exhibit topologically distinct structures at short and long times. The Transformer not only accurately predicts topological invariants for diverse energy braids but also, through its self-attention mechanism, autonomously highlights band crossings as the governing underlying geometric feature. Our work paves the way for ML-guided exploration of non-Hermitian topological phases in cold atoms and beyond.

cond-mat.quant-gasCross-listedarXiv:2603.25775•Mar 27, 2026•0 citations

Quantum simulation for topological Euler insulators

W.-D. Zhao, Yan-Bin Yang, Yue Jiang +9 more

Abstract Although recent studies have established a powerful framework to search for and classify topological phases based on symmetry indicators, there exists a large class of fragile topology beyond the description. The Euler class characterizing the topology of two-dimensional real wave functions is an archetypal fragile topology underlying some important properties. However, as a minimum model of fragile topology, the two-dimensional topological Euler insulator consisting of three bands remains a significant challenge to be implemented in experiments. Here, we experimentally realize a three-band Hamiltonian to simulate a topological Euler insulator with a trapped-ion quantum simulator. Through quantum state tomography, we successfully evaluate the Euler class, Wilson loop flow, entanglement spectra and Berry phases to show the topological properties of the Hamiltonian. The flexibility of the trapped-ion quantum simulator further allows us to probe dynamical topological features including skyrmion-antiskyrmion pairs and Hopf links in momentum-time space from quench dynamics.

arXiv:2201.09234•Communications Physics•Sep 7, 2022•43 citations

Implementing non-Abelian Hatano-Nelson model in electric circuits

Xiangru Chen, Jien Wu, Xingyu Chen +6 more

Non-Hermitian systems generally host complex spectra that bring unique spectral topologies, leading to the spectral braiding and non-Hermitian skin effect. The experimental exploration of non-Hermitian physics is mainly concentrated in artificial systems due to the flexibility in the introduction of the non-Hermiticity, but to date has focused only on the systems without gauge fields or with Abelian gauge fields. Here, we propose a non-Abelian Hatano-Nelson model with a nonreciprocal U(2) gauge field. The gauge field induces two non-Hermitian phenomena: the first is the Hopf-link-shaped complex energy braiding, and the second is the bipolar skin effect arising under the non-Abelian condition. The non-Abelian Hatano-Nelson model is implemented in electric circuits, and the Hopf-link-shaped admittance spectra and bipolar skin admittance modes are observed. Our work enriches the experimental non-Hermitian physics, and provides an approach to designing multifunctional non-Hermitian devices.

quant-pharXiv:2603.24642•Mar 26, 2026
Non-Hermitian systems generically have complex energies, which may host topological structures, such as links or knots. While there has been great progress in experimentally engineering non-Hermitian models in quantum simulators, it remains a significant challenge to experimentally probe complex energies in these systems, thereby making it difficult to directly diagnose complex-energy topology. Here, we experimentally realize a two-band non-Hermitian model with a single trapped ion whose complex eigenenergies exhibit the unlink, unknot, or Hopf link topological structures. Based on non-Hermitian absorption spectroscopy, we couple one system level to an auxiliary level through a laser beam and then experimentally measure the population of the ion on the auxiliary level after a long period of time. Complex eigenenergies are then extracted, illustrating the unlink, unknot, or Hopf link topological structure. Our work demonstrates that complex energies can be experimentally measured in quantum simulators via non-Hermitian absorption spectroscopy, thereby opening the door for exploring various complex-energy properties in non-Hermitian quantum systems, such as trapped ions, cold atoms, superconducting circuits, or solid-state spin systems.
While non-Hermitian Hamiltonians have been experimentally realized in cold atom systems, it remains an outstanding open question of how to experimentally measure their complex energy spectra in momentum space for a realistic system with boundaries. The existence of non-Hermitian skin effects may make the question even more difficult to address given the fact that energy spectra for a system with open boundaries are dramatically different from those in momentum space; the fact may even lead to the notion that momentum-space band structures are not experimentally accessible for a system with open boundaries. Here, we generalize the widely used radio-frequency spectroscopy to measure both real and imaginary parts of complex energy spectra of a non-Hermitian quantum system for either bosonic or fermionic atoms. By weakly coupling the energy levels of a non-Hermitian system to auxiliary energy levels, we theoretically derive a formula showing that the decay of atoms on the auxiliary energy levels reflects the real and imaginary parts of energy spectra in momentum space. We further prove that measurement outcomes are independent of boundary conditions in the thermodynamic limit, providing strong evidence that the energy spectrum in momentum space is experimentally measurable. We finally apply our non-Hermitian absorption spectroscopy protocol to the Hatano-Nelson model and non-Hermitian Weyl semimetals to demonstrate its feasibility.
Knots and links represent a fundamental motif of non-local connectivity that permeates the physical sciences from string theory to protein folds. While spectral braiding has been explored in two-band non-Hermitian models across various platforms, its direct simulation and characterization on programmable quantum hardware, particularly beyond two strands, remains a formidable challenge due to the limitations of variational optimization in these systems. Here, we introduce a family of non-Hermitian multi-band twister models and implement a non-variational protocol to characterize their complex braided band structures on a programmable superconducting quantum processor. By mapping the winding of eigenstates to the spectral topology, we devise an efficient measurement strategy that extracts braid information, including braid words and knot invariants like the Alexander and Jones polynomials, without requiring full spectral tomography or repeated optimization. We experimentally demonstrate the reconstruction of complicated knots and links such as the Hopf chain and Solomon's knot. Our approach provides a general framework for investigating exotic non-Hermitian topology on near-term quantum devices, opening a route to simulate more sophisticated topological structures in knot theory.
Hopf insulators are intriguing three-dimensional topological insulators characterized by an integer topological invariant. They originate from the mathematical theory of Hopf fibration and epitomize the deep connection between knot theory and topological phases of matter, which distinguishes them from other classes of topological insulators. Here, we implement a model Hamiltonian for Hopf insulators in a solid-state quantum simulator and report the first experimental observation of their topological properties, including nontrivial topological links associated with the Hopf fibration and the integer-valued topological invariant obtained from a direct tomographic measurement. Our observation of topological links and Hopf fibration in a quantum simulator opens the door to probe rich topological properties of Hopf insulators in experiments. The quantum simulation and probing methods are also applicable to the study of other intricate three-dimensional topological model Hamiltonians.
Machine learning (ML) is shaping our exploration of topological matter, whose existence is inherently tied to the geometry of quantum states or energy spectra. In non-Hermitian systems, distinctive spectral geometry can lead to topological braiding of complex-energy bands, yet directly probing this topology-geometry interplay remains challenging. Here, we introduce a Transformer-based ML framework to capture this interplay and experimentally demonstrate it in a dissipative cold-atom simulator. Using a Bose-Einstein condensate, we engineer tunable dissipative two-level systems whose complex eigenenergies form braids. Owing to the density-dependent dissipation, the instantaneous energy braids exhibit topologically distinct structures at short and long times. The Transformer not only accurately predicts topological invariants for diverse energy braids but also, through its self-attention mechanism, autonomously highlights band crossings as the governing underlying geometric feature. Our work paves the way for ML-guided exploration of non-Hermitian topological phases in cold atoms and beyond.
Abstract Although recent studies have established a powerful framework to search for and classify topological phases based on symmetry indicators, there exists a large class of fragile topology beyond the description. The Euler class characterizing the topology of two-dimensional real wave functions is an archetypal fragile topology underlying some important properties. However, as a minimum model of fragile topology, the two-dimensional topological Euler insulator consisting of three bands remains a significant challenge to be implemented in experiments. Here, we experimentally realize a three-band Hamiltonian to simulate a topological Euler insulator with a trapped-ion quantum simulator. Through quantum state tomography, we successfully evaluate the Euler class, Wilson loop flow, entanglement spectra and Berry phases to show the topological properties of the Hamiltonian. The flexibility of the trapped-ion quantum simulator further allows us to probe dynamical topological features including skyrmion-antiskyrmion pairs and Hopf links in momentum-time space from quench dynamics.
Non-Hermitian systems generally host complex spectra that bring unique spectral topologies, leading to the spectral braiding and non-Hermitian skin effect. The experimental exploration of non-Hermitian physics is mainly concentrated in artificial systems due to the flexibility in the introduction of the non-Hermiticity, but to date has focused only on the systems without gauge fields or with Abelian gauge fields. Here, we propose a non-Abelian Hatano-Nelson model with a nonreciprocal U(2) gauge field. The gauge field induces two non-Hermitian phenomena: the first is the Hopf-link-shaped complex energy braiding, and the second is the bipolar skin effect arising under the non-Abelian condition. The non-Abelian Hatano-Nelson model is implemented in electric circuits, and the Hopf-link-shaped admittance spectra and bipolar skin admittance modes are observed. Our work enriches the experimental non-Hermitian physics, and provides an approach to designing multifunctional non-Hermitian devices.