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. Despite the increasing applications of machine learning in the quantum sciences, applications in the context of topological matter are still limited. Here, the authors demonstrate topological braiding of complex energy bands in a dissipative ultracold atom platform based on a transformer approach, advancing machine-learning-aided search of topological phases.
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. Despite the increasing applications of machine learning in the quantum sciences, applications in the context of topological matter are still limited. Here, the authors demonstrate topological braiding of complex energy bands in a dissipative ultracold atom platform based on a transformer approach, advancing machine-learning-aided search of topological phases.
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.
The pursuit of topological phenomena in non-Hermitian systems has unveiled new physics beyond the conventional Hermitian paradigm, yet their realization in interacting many-body platforms remains a critical challenge. Exploring this interplay is essential to understand how strong interactions and dissipation collectively shape topological phases in open quantum systems. Here, we experimentally demonstrate dynamical spectral topology in a dissipative Rydberg atomic gas and characterize parameter-dependent winding numbers, which quantify a geometric winding of the spectral topology. By increasing the interaction strength, the system evolves from Hermitian to non-Hermitian regime, accompanying emergence of trajectory loop in the complex energy plane. As the scanning time is varied, the spectral topology becomes twisted in the complex energy plane, forming enclosed sub-loops characterized by opposite winding numbers. Furthermore, by changing the scanning direction, we observe the differentiated spectral loops, revealing a signature of scan-direction-dependent nonreciprocity. This work establishes cold Rydberg gases as a versatile platform for exploring the rich interplay between non-Hermitian topology, strong interactions, and dissipative quantum dynamics.
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.
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.
Topological photonic phases are typically identified through band reconstruction, steady-state transmission, or real-space imaging of edge modes. In this work, we present a framework for spectroscopic readout of chiral photonic topology in a single driven optical cavity containing a spin-orbit-coupled Bose-Einstein condensate. We demonstrate that the cavity transmission power spectral density provides a direct and measurable proxy for a momentum- and frequency-resolved photonic Chern marker, enabling topological characteristics to be inferred from spectral data without the need for bulk-band tomography. In the loss-dominated regime, where cavity decay exceeds atomic dissipation, the power spectral density exhibits Dirac-like gapped hybrid modes with a vanishing Chern marker, indicating a trivial phase. When the dissipation imbalance is reversed, a bright, gap-spanning spectral ridge emerges, co-localized with peaks in both the Chern marker and Berry curvature. The complex spectrum reveals parity-time symmetric coalescences and gain-loss bifurcations, marking exceptional points and enabling chiral, gap-traversing transport. By linking noise spectroscopy to geometric and non-Hermitian topology in a minimal cavity-QED architecture, this work provides a framework for spectroscopic detection of topological order in driven quantum systems. This approach offers a pathway to compact, tunable topological photonics across a broad range of light-matter platforms, providing a method for the study and control of topological phases in hybrid quantum systems.
Quantum topology, a fascinating field of physics, offers unique geometric properties that challenge conventional notions of space and symmetry. This paper explores the potential of utilizing quantum topological geometry for rapid computation of complex geometric transformations, driven by the synergistic application of machine learning. We propose a novel computational framework that leverages quantum topological properties to accelerate the analysis and manipulation of geometric shapes. The investigation focuses on demonstrating the efficacy of this approach in solving problems currently intractable with traditional computational methods. Furthermore, we investigate the potential of quantum topological geometry to unveil novel applications within the realm of data analysis and pattern recognition. This work represents a significant step towards integrating advanced mathematical concepts with machine learning, offering a promising avenue for innovation in computational geometry and beyond.