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Optimal quantum frequency estimation

Katarzyna Macieszczak, Rafał Demkowicz-Dobrzański, Martin Fraas

Published Nov 21, 2013• 4 citations
OpenAlex

Abstract

We advocate a Bayesian approach to optimal quantum frequency estimation - an important problem for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of the prior knowledge in determining the optimal interrogation times and optimal estimation strategies. We propose a general framework capable of describing the local oscillator noise as well as additional atomic dephasing effects. We also demonstrate an important fact: the average Bayesian cost can be expressed explicitly using the quantum Fisher information and thus establish a direct link between the two, often competing, approaches to quantum estimation theory.

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We advocate a Bayesian approach to optimal quantum frequency estimation - an important problem for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of the prior knowledge in determining the optimal interrogation times and optimal estimation strategies. We propose a general framework capable of describing the local oscillator noise as well as additional atomic dephasing effects. We also demonstrate an important fact: the average Bayesian cost can be expressed explicitly using the quantum Fisher information and thus establish a direct link between the two, often competing, approaches to quantum estimation theory.
We advocate a Bayesian approach to optimal quantum frequency estimation—an important issue for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of prior knowledge in determining the optimal interrogation times and optimal estimation strategies. We propose a general framework capable of describing local oscillator noise as well as additional collective atomic dephasing effects. For a Gaussian noise, the average Bayesian cost can be expressed using the quantum Fisher information. Thus we establish a direct link between the two, often competing, approaches to quantum estimation theory.
We advocate a Bayesian approach to optimal quantum frequency estimation—an important issue for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of prior knowledge in determining the optimal interrogation times and optimal estimation strategies. We propose a general framework capable of describing local oscillator noise as well as additional collective atomic dephasing effects. For a Gaussian noise, the average Bayesian cost can be expressed using the quantum Fisher information. Thus we establish a direct link between the two, often competing, approaches to quantum estimation theory.
Frequency metrology is a cornerstone of modern precision measurements and optical atomic clocks have emerged as one of the most precise measurement devices. In this progress report, we explore various Ramsey interrogation schemes tailored to optical atomic clocks primarily limited by laser noise. To incorporate frequency fluctuations directly into the theoretical model, we consider a Bayesian framework. In this context, we review fundamental bounds arising in Bayesian estimation theory, which serve as a benchmark throughout this work. We investigate the trade-off between entanglement-enhanced sensitivity and robustness against laser noise in order to identify optimal initial states, measurement schemes and estimation strategies. Beside standard protocols based on coherent spin states, spin-squeezed states and Greenberger-Horne-Zeilinger states, we consider variational Ramsey protocols implemented via low-depth quantum circuits based on one-axis twisting operations to approach optimal stability. In particular, we review known and identify new optimal Ramsey interrogation schemes for a variety of scenarios, including different experimental platforms, ensemble sizes and regimes characterized by a wide range of interrogation durations and dead times. Hence, this work establishes a comprehensive theoretical framework for optimizing Ramsey interrogation schemes, providing guidance for the development of next-generation optical atomic clocks.
Atomic clocks are crucial for fundamental science and practical technology, but their sensitivity is often constrained by the standard quantum limit. With a cold-atom coherent-population-trapping clock, the authors design and experimentally demonstrate an adaptive Bayesian quantum frequency estimation protocol that approaches the Heisenberg scaling with respect to total interrogation time. This atomic clock also yields improved fractional frequency stability and enhanced robustness against technical noises. This work provides a high-precision approach to atomic clock locking, and holds promising applications in various interferometry-based quantum sensors.
Quantum frequency estimation under general noise is investigated in the Bayesian parameter estimation approach. To evaluate the accuracy of estimation, the Bayes cost is obtained analytically which can be applied to the common noisy channels, such as the phase-damping channel, the amplitude-damping channel, and the depolarizing channel. The Bayes cost formula clearly shows that the prior information imposes a restriction upon the effect of noise in the estimation process. Three examples of frequency estimations are provided to illustrate the roles of prior probability in the estimation process. It is found that, due to the restriction of prior information on the noise, the estimation accuracy is less sensitive to the noise in the Bayesian approach than that in the Cram\'er-Rao bound approach. More prior information can help us to use the non-Gaussianity of the noise channel to improve the estimation accuracy in the phase-damping channel.
Frequency estimation, a cornerstone of quantum metrology, has been significantly enhanced by advanced quantum sensing strategies. However, most protocols rely either on static or time-independent encoding mechanisms, inherently limiting their achievable precision scaling, or on control strategies requiring changing the Hamiltonian and/or implementing feedback mechanisms. To overcome this, we investigate a simpler dynamical encoding protocol where the quantum oscillator is driven by a general continuous temporal frequency modulation Ω(t)=ω0f(t)Ω(t) = ω_0 f(t)Ω(t)=ω0​f(t). We analytically demonstrate that for a given modulation profile f(t)f(t)f(t) and its corresponding time-integral F(t)F(t)F(t), the quantum Fisher information (QFI) scales as O(F(t)2)\mathcal{O}(F(t)^2)O(F(t)2). This enhancement stems from the fact that temporal encoding fundamentally alters the mechanism of dynamical phase accumulation. Crucially, when evaluated under the energy and evolution-time constraints, this framework reveals a genuine precision enhancement over the conventional time-independent baseline. By analyzing explicit polynomial and exponential modulations, we establish that arbitrary precision scaling can be deterministically engineered, with ultimate bounds that are asymptotically saturable via optimal homodyne detection. Our framework provides a universal paradigm for exploiting time-dependent quantum control in next-generation sensors.