Krylov Tomography and Finite-Uncertainty Certification of Exceptional-Point Dynamics

AI summary

Overview

Research area: Quantum physics, specifically non-Hermitian open systems and exceptional-point (EP) dynamics, with an illustrative implementation in cavity optomechanics.

Technical level: Advanced. The paper uses Jordan-block / nilpotent-operator theory, Riesz spectral projectors, restricted induced operator norms, singular-value rank tests, and generalized least-squares fitting.

Scope (one sentence): The paper introduces "Krylov tomography," a preparation- and measurement-aware certification framework that uses time-resolved data with stated uncertainty bounds to determine how much of an exceptional point's Jordan structure an experiment actually excites, how much a detector independently resolves, and whether a vanishing signal means true termination or detector blindness — illustrated with finite-precision simulations of a red-sideband optomechanical model.

What This Paper Is About

Finding an exceptional point in a system's generator does not tell you what an actual experiment does with it: a given preparation may excite only part of the Jordan chain, a given detector may fail to distinguish the directions that are excited, and a small measured coefficient may be zero, nonzero-but-uncertain, or invisible to the detector. The authors build a framework that separates the structure available in the generator from the structure activated by preparation and the structure resolved by readout, and that certifies these quantities under explicit finite-error bounds. They then demonstrate the framework on a red-sideband optomechanical model whose second-moment coherence sector contains a third-order EP.

Key Contributions

1. A three-stage decomposition of exceptional dynamics. The framework distinguishes four quantities and orders them in a depth hierarchy: the available Jordan order ν_λ, the preparation-activated depth d_act, the visible depth d_vis^(z), and the independently resolved depth d_rank^(z), with d_rank^(z) ≤ d_vis^(z) ≤ d_act ≤ ν_λ.

2. Finite-error certification tests. The paper gives a coefficient test (a measured coefficient exceeding its uncertainty radius certifies activated depth, and a nonzero terminal coefficient certifies nontermination without requiring readout injectivity), a singular-value based projected-rank test for independent resolution, and an upper bound on the normalized termination residual that requires a positive detector-sensitivity lower bound and a calibrated rate lower bound.

3. An analytic bound on offset-induced "reopening" of a terminated endpoint. For a bounded departure from the exact EP (drift perturbation norm δ_A and preparation error norm δ_x, made dimensionless as ε and ζ), the reopened response is bounded by χ_r / [(1−ε)(ρ_{r−1} − χ_{r−1})], which for small errors behaves as (rε + ζ)/ρ_{r−1} plus higher-order terms. This addresses the practical difficulty that exact-EP operation is hard to achieve and stabilize.

4. A concrete optomechanical instantiation. A linearized resonant red-sideband model with a field-amplitude EP2 at G_EP = (κ − γ_m)/4 produces a size-three Jordan block in the three-coordinate second-moment coherence sector, with a 3 ⊕ 1 partition, and the paper certifies depth-three and depth-two preparations from simulated data.

Main Findings

• Available structure versus activated structure differ for the same generator. In the optomechanical model, N_coh³ = 0 and N_coh² ≠ 0, so the three-coordinate coherence sector carries a size-three Jordan block. A cavity-noise preparation gives x₀^(3) = n_p(1,0,0)^T and activates depth three, while a differential preparation x₀^(2) = n_p(1,−1,0)^T activates only depth two.

• Closed-form preparation-dependent responses. With d = (κ − γ_m)/2 = 2G_EP and Γ̄ = (κ + γ_m)/2, the cavity responses are δn_c^(3)(t) = n_p e^{−Γ̄t}(1 − dt + d²t²/4) and δn_c^(2)(t) = n_p e^{−Γ̄t}(1 − dt).

• Certification succeeds for nonzero predecessors and ranks, but termination only partly. Using separate joint 95% regions, the predecessor and terminal ratios are ‖p̂₂‖/ε₂ > 4.51 and ‖p̂₃‖/ε₃ < 0.59 for depth three, and ‖p̂₁‖/ε₁ > 31.9 and ‖p̂₂‖/ε₂ < 0.25 for depth two. The projected-rank tests certify rank P₃ = 3 and rank P₂ = 2. No 95% coverage is claimed for the intersection of these preparation-specific events.

• Termination bound meets the 10% tolerance only for depth two. With predeclared 2% calibration margins (‖ΔC‖ ≤ 0.02, α_C^L = 0.98, β_C^U = 1.02, Ω_K^L = 0.98 Ω_*), the Eq. (6) right-hand sides are B₃ ≈ 0.313495 and B₂ ≈ 0.031269. The exact identities N_coh³ = 0 and N_coh²x₀^(2) = 0 make both terminal vectors vanish analytically at the exact EP, so B₃ > 0.10 reflects limited finite-data precision rather than a physical nonzero endpoint.

• Scalar readout is insufficient. A scalar cavity-output map is noninjective on the three-dimensional terminal sector and cannot bound its state-space termination residual; a three-output model (cavity and mechanical populations plus a coherence obtained by differencing two simulated channels with opposite coherence signs) is used instead, with nominal C = C̃ = I₃.

• Departure from the EP reopens the depth-two endpoint in a preparation-specific way. For the differential preparation, q₂^(2) = (d² − 4G²) q₀^(2) and R₂^(2) = |d² − 4G²|/(d² + 4G²), which equals the squared mode splitting normalized by d² + 4G². The condition R₂^(2) ≤ b (for 0 ≤ b < 1) is equivalent to √((1−b)/(1+b)) ≤ 4G/(κ − γ_m) ≤ √((1+b)/(1−b)). By contrast, the cavity-noise preparation spans three independent response directions for every G ≠ 0.

• Simulation parameters. ω_m/2π = 1 MHz, κ/2π = 0.2 MHz, γ_m/2π = 50 Hz, T_m = 4 K, EP condition G = G_EP = (κ − γ_m)/4, each physical run adds n_p = 10. Fits are cubic for depth three and quadratic for depth two, using the full noise covariance.

• Model assumptions are explicit. Closure of the moment sector, isolation of the λ contribution, and calibration of the readout map C are independent modeling inputs, and a centering error Δλ produces a leading spurious terminal term q_r^(λ̂) = −r Δλ q_{r−1} plus higher-order corrections that must enter every coefficient radius.

Methodology in Plain English

The authors model a small, closed set of "moments" (for example, photon and phonon occupations and a coherence) that evolve linearly among themselves under a constant matrix A. They isolate the part of that evolution belonging to one eigenvalue λ using a standard projector, and then repeatedly apply the shifted generator (A − λI) to the prepared state. Each application produces a new "Krylov" direction; the count of successive nonzero directions is the activated depth, and the count of directions that produce linearly independent detector outputs is the resolved depth.

Because real data is noisy, they never treat a small fitted coefficient as exactly zero. Instead, they reconstruct the time-polynomial coefficients from the measured record, attach a simultaneous uncertainty radius to each, and declare a coefficient genuinely nonzero only when its measured magnitude exceeds that radius. Independent resolution is tested by a smallest-singular-value condition on a rescaled matrix of reconstructed coefficient vectors, with a free scaling rate Ω_sc that can be optimized after fitting without changing the fit or the uncertainty radii. To distinguish a true endpoint from a detector that simply cannot see a direction, they compute the weakest and strongest output magnitude per unit state norm over a predeclared set of allowed terminal directions; a strictly positive weakest gain rules out blindness. Finally, they bound how large the "should-be-zero" next response can become when the system sits slightly away from the exact EP, in terms of relative drift and preparation errors.

For the illustration, they simulate the linearized resonant red-sideband optomechanical interaction with added Gaussian noise, run the depth-three and depth-two preparations separately, fit cubic and quadratic polynomials to the demodulated records, and apply the certification tests and termination bound.

Why This Matters

The work reframes exceptional-point experiments as a question of what is prepared and read out, not just what the generator supports. It provides finite-error guarantees rather than asymptotic statements, and it makes explicit the failure mode in which a missing signal is mistaken for chain termination when it is actually detector blindness. It also quantifies how much a realistic imperfect EP degrades the very polynomial responses that EP sensing schemes rely on.

Real-world applications:

• Optomechanical and non-Hermitian sensing: determining whether a measured enhanced response truly comes from an accessed high-order Jordan chain, and how large a spurious response a small offset from the EP can generate.
• Detector and readout design for quantum devices: using the depth hierarchy and the blindness test to decide how many independent output channels are needed before claiming a given number of resolved directions.
• Characterization of Gaussian bosonic and fermionic platforms: the closed-moment-sector requirement is noted to be common in Gaussian bosonic and fermionic systems and in symmetry-resolved open spin dynamics.
• Experimental calibration under stated uncertainty: the framework takes calibration information obtained independently of the tested record and keeps it separate from quantities inferred from that record, which suits laboratory practice with declared tolerances.

Industry relevance: quantum sensing hardware, optomechanical transducers and cavity-optomechanics experiments, and any metrology program that operates near an exceptional point or relies on higher-order coalescence for signal enhancement.

Future Directions

• Experimental realization of the proposed preparations. The paper notes that related band-limited microwave-noise injection has already been demonstrated experimentally; carrying out the cavity-noise and differential preparations on a real optomechanical device is the natural next step.
• Improving finite-data precision so that depth-three termination can be certified. The reported B₃ ≈ 0.313495 exceeds the stated 10% tolerance, which the authors attribute to limited finite-data precision (longer records, better time sampling, or tighter calibration margins are the implied levers).
• Extending beyond the single tested block. The framework is developed for an isolated EP contribution in a closed moment sector; multi-block structures, higher-order EPs, and bounded spectral leakage from neighboring eigenvalues are open extensions.
• Joint coverage across preparation-specific events. The authors explicitly do not claim 95% coverage for the intersection of the depth-two and depth-three certification events, leaving a multi-event simultaneous confidence construction as open work.

Target Audience

Researchers working on non-Hermitian physics and exceptional points, quantum metrology and sensing, and cavity optomechanics; experimentalists who need to state what their EP experiment actually certifies; and mathematically inclined readers interested in finite-error, preparation- and readout-aware certification of Jordan dynamics. The paper assumes comfort with operator theory, spectral projectors, and matrix perturbation bounds, so it is best suited to graduate-level and professional readers rather than beginners.

Authors’ abstract

Exceptional points (EPs) can produce striking responses, but locating one does not reveal how much of its dynamics an excitation accesses, which responses a detector distinguishes, or whether a missing signal is absent or undetected. We introduce Krylov tomography, a preparation- and measurement-aware framework that uses time-resolved data and models with stated uncertainty limits to answer these questions, while also bounding offset-induced departures from exact-EP behavior. As an explicit illustration of the general framework, we present finite-precision simulations of a red-sideband optomechanical model, whose second-moment coherence sector contains a third-order EP, certifying preparation-sensitive access to two and three directions. Krylov tomography thus bridges EP structure and finite-precision measurements, providing a general framework for non-Hermitian systems.