Researchers from Rochester Institute of Technology have developed Krylov tomography, a preparation- and measurement-aware framework that determines how much of an exceptional point’s (EP) dynamics is actually accessed in finite-precision experiments. By combining time-resolved measurements with theoretical models that include explicit uncertainty bounds, the method distinguishes between dynamics that are physically excited and those that merely exist in the mathematical description of the system.

Exceptional points arise in non-Hermitian systems, where multiple eigenvalues and eigenvectors merge into a single state, producing a nontrivial Jordan structure. These degeneracies are commonly identified through spectral coalescence, response poles, or polynomial-in-time transients. However, such measurements do not automatically reveal how much of the Jordan chain is actually accessed during an experiment because the observed response depends on both the initial preparation of the system and the measurement process.

The researchers explain that every experiment consists of preparation, evolution, and readout. Preparation determines the activated depth—the number of successive nonzero vectors generated by repeatedly applying the system generator, shifted by the exceptional-point eigenvalue, to the prepared state. This sequence forms the preparation-generated Krylov chain. Readout then determines how many of these directions can be independently resolved. In some cases, an existing direction may produce no measurable signal, a phenomenon the authors describe as detector darkness.

As a result, the order of an exceptional point alone may overestimate the amount of dynamics that is experimentally accessible. Without preparation- and measurement-aware analysis, it is impossible to determine whether a missing signal represents the end of the accessible Jordan chain or simply reflects the detector’s inability to observe it. Krylov tomography overcomes this limitation by providing finite-error certification under stated uncertainty bounds, separating the available Jordan order from the activated, visible, and independently resolved depths.

The framework also distinguishes between exact exceptional points, where eigenvalues and eigenvectors perfectly coalesce, and near exceptional points, where calibration uncertainties or small parameter offsets prevent perfect degeneracy. Instead of assuming ideal operating conditions, the method incorporates independently calibrated model parameters together with their uncertainty limits to produce experimentally meaningful conclusions.

To demonstrate the approach, the team studied a linearized red-sideband cavity optomechanical model whose second-moment coherence sector contains a third-order exceptional point. Different preparation protocols activated different portions of the Jordan chain. Exciting only the cavity occupation activated three accessible directions, whereas comparing separate cavity and mechanical excitation runs activated only two. Finite-precision simulations successfully certified both cases despite measurement uncertainty.

The authors describe Krylov tomography as the first finite-error certification framework for preparation- and readout-selected Jordan dynamics. By linking exceptional-point theory with realistic finite-precision measurements, the framework enables researchers to determine which exceptional-point dynamics are genuinely excited, which can be independently resolved, and whether an apparently missing response reflects the physical system or the limitations of the measurement process.

👉 More information
🗞 Krylov Tomography and Finite-Uncertainty Certification of Exceptional-Point Dynamics
✍️ Aritra Ghosh and M. Bhattacharya
🧠 ArXiv: https://arxiv.org/abs/2608.19761

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