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arXiv:2604.11614 · Physical Review A

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

Xinyi Sui, Mankei Tsang

quant-phSubmitted Apr 13, 2026• Announced Apr 14, 2026• 0 citations
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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.

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9 pages, 2 figures

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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.
Measuring the power spectral density of a stochastic process, such as a stochastic force or magnetic field, is a fundamental task in many sensing applications. Quantum noise is becoming a major limiting factor to such a task in future technology, especially in optomechanics for temperature, stochastic gravitational wave, and decoherence measurements. Motivated by this concern, here we prove a measurement-independent quantum limit to the accuracy of estimating the spectrum parameters of a classical stochastic process coupled to a quantum dynamical system. We demonstrate our results by analyzing the data from a continuous-optical-phase-estimation experiment and showing that the experimental performance with homodyne detection is close to the quantum limit. We further propose a spectral photon-counting method that can attain quantum-optimal performance for weak modulation and a coherent-state input, with an error scaling superior to that of homodyne detection at low signal-to-noise ratios.
We study the fundamental limits of noise spectroscopy using estimation theory, Faraday rotation probing of an atomic spin system, and squeezed light. We find a simple and general expression for the Fisher information, which quantifies the sensitivity to spectral parameters such as resonance frequency and linewidth. For optically detected spin noise spectroscopy, we find that shot noise imposes ``local'' standard quantum limits for any given probe power and atom number, and also ``global'' standard quantum limits when the probe power and atom number are taken as free parameters. We confirm these estimation theory results using nondestructive Faraday rotation probing of hot Rb vapor, observing the predicted optima, and finding good quantitative agreement with a first-principles calculation of the spin noise spectra. Finally, we show sensitivity beyond the atom- and photon-number-optimized global standard quantum limit using squeezed light.
I point out the mathematical correspondence between an incoherent imaging model proposed by my group in the study of quantum-inspired superresolution [M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016)] and a noise spectroscopy model also proposed by us [M. Tsang and R. Nair, Phys. Rev. A 86, 042115 (2012); S. Ng et al., Phys. Rev. A 93, 042121 (2016)]. Both can be regarded as random displacement models, where the probability measure for the random displacement depends on unknown parameters. The spatial-mode demultiplexing (SPADE) method proposed for imaging is analogous to the spectral photon counting method proposed by Ng et al. for optical phase noise spectroscopy: Both methods are discrete-variable measurements that are superior to direct displacement measurements (direct imaging or homodyne detection) and can achieve the respective quantum limits. Inspired by SPADE, I propose a modification of spectral photon counting when the input field is squeezed: The output field is simply unsqueezed before spectral photon counting. I show that this method is quantum optimal and far superior to homodyne detection for both parameter estimation and detection, thus solving the open problems in the work of Tsang and Nair and Ng et al.
Precision measurements with quantum systems rely on our ability to trace the differences between experimental signals to variations in unknown physical parameters. In this Letter we derive the Fisher information and the ensuing Cramér-Rao sensitivity limit for parameter estimation by continuous measurements on an open quantum system. We illustrate our theory by its application to resonance fluorescence from a laser-driven two-state atom; we show that photon counting and homodyne detection records yield different sensitivity to the atomic parameters, while none of them exceed our general result.
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