Spectrum analysis with quantum dynamical systems. II. Finite-time analysis

Abstract: The prequel to this work [Ng et al., Phys. Rev. A 93, 042121 (2016)] proposes the method of spectral photon counting to enhance noise spectroscopy with an optical interferometer. While the predicted enhancement over homodyne detection is promising, the results there are derived by taking an asymptotic limit of infinite observation time; their validity for a finite time remains unclear. To validate the theory, here we perform a numerical study of a finite-time model. Assuming that the signal is an Ornstein--Uhlenbeck process with an unknown variance parameter, we evaluate the Fisher information for homodyne detection, a lower bound on the Fisher information for spectral photon counting, and a quantum upper bound, all without taking the infinite-time limit. To confirm that the Fisher-information quantities are satisfactory precision measures, we also compute the errors of the maximum-likelihood estimator by Monte-Carlo simulations. The results demonstrate that the Fisher-information quantities and the estimation errors all smoothly approach their asymptotic limits, and the advantage of spectral photon counting over homodyne detection can remain substantial for finite times.

• The paper validates that spectral photon counting (SPC) dramatically surpasses homodyne detection in finite-time Fisher information comparisons, especially in weak-signal regimes.
• It employs a rigorous finite-time analysis using discretized measurement models and Fisher information derivations via exact matrix expressions and Poisson process mapping.
• Numerical simulations reveal that maximum-likelihood estimators closely approach the Cramér–Rao bound at moderate observation times, confirming practical measurement advantages.

Finite-Time Analysis of Spectrum Estimation in Quantum Dynamical Systems

Introduction

This work presents a systematic finite-time analysis of spectrum estimation involving quantum dynamical systems, focusing predominantly on the comparative performance of spectral photon counting (SPC) versus homodyne detection. The approach directly interrogates the validity of Fisher information-based precision benchmarks outside the asymptotic regime, addressing two persistent open questions from prior literature: (1) the reliability of stationary process and long observation time (SPLOT) approximated Fisher-information expressions for finite measurement durations, and (2) the tightness of the Cramér–Rao bound for finite-time error estimation, given that its achievability is strictly guaranteed only in the asymptotic limit.

Theoretical Framework

The analysis adopts a simplified model representing an optical interferometer where the signal undergoing phase modulation is an Ornstein–Uhlenbeck (OU) process with an unknown variance parameter. Both homodyne detection and SPC scenarios are considered for coherent optical input states. The theoretical treatment details:

• Discretization of measurement time, explicit construction of the observation process, and computation of the covariance matrix $\Sigma(\theta)$.
• Derivation of the Fisher information for homodyne detection via exact matrix expressions and its SPLOT approximation.
• Introduction of the SPC process where detection is mapped to a doubly stochastic Poisson process, with Stein et al.'s lower bound employed for Fisher information calculations.
• Quantum upper bounds on Fisher information, using extended-convexity properties and explicit model construction for coherent states, further specialized to circulant matrix cases in the SPLOT limit.

The resulting Fisher information and likelihood functions for estimation are numerically tractable only in the SPLOT limit; exact finite-time calculations are non-trivial and generally computationally intensive.

Numerical Evaluation

Fisher Information in Finite Time

Numerical simulations systematically compare:

• Exact finite-time Fisher information for both homodyne detection and SPC (using Stein et al.'s bound) to their SPLOT counterparts.
• Quantum Fisher information upper bounds computed per the extended-convexity formalism.

Strong numerical evidence is provided that, while SPLOT approximations diverge appreciably from the exact expressions at short observation times, convergence is smooth as measurement duration increases, requiring only moderate values of normalized observation time ($T$) to achieve near-asymptotic fidelity.

Numerical results corroborate that, in the weak-signal regime (characterized by low $C = 8\theta N$), SPC remains substantially superior to homodyne detection, with the Fisher information for SPC exceeding its homodyne counterpart by orders of magnitude as $C \to 0$, independent of finite-time effects.

Maximum-Likelihood Estimation and Cramér–Rao Bounds

Monte Carlo simulations of both homodyne and SPC measurement processes are performed with SPLOT-based likelihood functions to evaluate maximum-likelihood estimator performance and associated mean-square error (MSE). Bootstrap confidence intervals quantify estimation precision.

Key findings include:

• The MSE for both detection methods approaches the Cramér–Rao bound as $T$ increases, confirming efficiency and the adequacy of SPLOT-based Fisher information as a performance benchmark for finite measurement duration.
• For intermediate $T$ and low $C$, estimator bias may cause mild local violations of the bound, but convergence is always achieved for larger $T$.
• The ratio of homodyne-to-SPC MSE confirms SPC's advantage persists for finite times, aligning well with Fisher information ratios.

A general rescaling argument demonstrates that both MSE and Fisher information scale universally in $\theta$ and $N$, facilitating extension of results to arbitrary values.

Implications

The results demonstrate that SPC's fundamental advantage over homodyne detection survives realistic finite-time constraints, provided weak-signal regimes are accessible. This assertion is substantiated—not only at the level of Fisher information and quantum bounds, but also in actual estimator errors—lending robustness to predictions derived from SPLOT approximations.

Practically, these findings reinforce the motivation for adopting SPC in noise spectroscopy applications where weak signals predominate (e.g., stochastic gravitational wave detection, tests of novel quantum gravity models, and thermometry). Theoretically, the work validates the SPLOT regime as an adequate benchmark for finite-time estimation and delineates the sub-optimality of homodyne-based measurements relative to quantum limits.

Future directions include generalization to more complex signal models (non-Gaussian/nonstationary), multiparameter estimation, and integration with quantum-enhanced states, all of which may necessitate new formalisms for optimal measurement and quantum limit calculation.

Conclusion

This study provides a rigorous finite-time validation of the superiority of spectral photon counting over homodyne detection for quantum noise spectroscopy, confirming the adequacy of SPLOT-based Fisher information and Cramér–Rao bounds as finite-time performance benchmarks. The numerical evidence establishes that the key theoretical claims persist beyond asymptotic assumptions, underpinning the practical utility of SPC in weak-signal measurement regimes and inviting further extensions into broader quantum estimation landscapes ["Spectrum analysis with quantum dynamical systems. II. Finite-time analysis" (2604.11614)].