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arXiv:2605.06263

Beating noise in frequency estimation with squeezing and memory in continuous-variable systems

Ayan Patra, Manju, Aditi Sen De, Matteo G. A. Paris

quant-phSubmitted May 7, 2026• Announced May 8, 2026• 0 citations
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Abstract

Quantum metrology promises precision beyond classical limits, yet environmental noise typically degrades the quantum resources required for such enhancement. In this work, we investigate frequency estimation in noisy continuous-variable systems, focusing on two complementary strategies to mitigate decoherence: Hamiltonian engineering and the exploitation of non-Markovian dynamics. By embedding squeezing directly into the system Hamiltonian, we show that the quantum Fisher information (QFI) may acquire a tunable higher-order time dependence, leading to enhanced sensitivity in the short-time regime. Moving beyond the Markovian approximation, we employ the quantum Brownian motion model to demonstrate that structured environments with finite memory can induce information backflow, temporarily restoring and even improving estimation precision relative to the unitary limit. We further assess the achievability of these bounds via Gaussian measurements, identifying regimes where homodyne, heterodyne, and optimized general-dyne measurements saturate the QFI, and noting that stronger squeezing widens the gap, potentially requiring non-Gaussian measurement strategies. Our results establish that jointly tailoring system Hamiltonian and environmental memory offers a viable route toward robust quantum-enhanced frequency estimation in open systems.

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16 Pages, 7 Figures

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Quantum metrology promises precision beyond classical limits, yet environmental noise typically degrades the quantum resources required for such enhancement. In this work, we investigate frequency estimation in noisy continuous-variable systems, focusing on two complementary strategies to mitigate decoherence: Hamiltonian engineering and the exploitation of non-Markovian dynamics. By embedding squeezing directly into the system Hamiltonian, we show that the quantum Fisher information (QFI) may acquire a tunable higher-order time dependence, leading to enhanced sensitivity in the short-time regime. Moving beyond the Markovian approximation, we employ the quantum Brownian motion model to demonstrate that structured environments with finite memory can induce information backflow, temporarily restoring and even improving estimation precision relative to the unitary limit. We further assess the achievability of these bounds via Gaussian measurements, identifying regimes where homodyne, heterodyne, and optimized general-dyne measurements saturate the QFI, and noting that stronger squeezing widens the gap, potentially requiring non-Gaussian measurement strategies. Our results establish that jointly tailoring system Hamiltonian and environmental memory offers a viable route toward robust quantum-enhanced frequency estimation in open systems.
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.
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We derive ultimate precision bounds for estimating parameters encoded in \emph{time-dependent} Hamiltonians in the presence of general Markovian noise, allowing for arbitrary adaptive protocols with fast controls and noiseless ancillas. Extending the minimization-over-purifications framework to time-varying continuous channels, we obtain a differential upper bound on the achievable quantum Fisher information (QFI) that can be evaluated at all times via semidefinite programming. For parameter-independent noise, we prove a universal long-time scaling law: if the coherent (noiseless) dynamics yields Qcoh(T)∼T2kQ_{\mathrm{coh}}(T)\sim T^{2k}Qcoh​(T)∼T2k, then under Markovian noise the QFI scales at most as Q(T)∼T2kQ(T)\sim T^{2k}Q(T)∼T2k in the DHNLS regime, whereas in the DHLS regime it is fundamentally limited to Q(T)∼T2k−1Q(T)\sim T^{2k-1}Q(T)∼T2k−1. We illustrate these behaviors on paradigmatic driven-qubit sensors, exhibiting T4T^{4}T4 and T3T^{3}T3 scalings under dephasing and spontaneous emission, respectively. Finally, we provide explicit continuous exact and approximate quantum error correction constructions -- supplemented by spin-squeezed probes -- that asymptotically saturate the bounds, establishing their tightness.
Quantum metrology faces major challenges in noisy environments, where decoherence rapidly degrades useful quantum resources. We investigate the dynamics of the precision limits given by the quantum Fisher information (QFI) for phase estimation under spatially correlated dephasing. We characterize the dynamics of the QFI by the sensitivity and degradation indicators that can be obtained as analytical expressions derived using perturbative theory treatment. These short-time and weak-noise formulas yield analytic insight into how collective-spin moments govern both (i) the noiseless sensitivity and (ii) the leading noise-induced degradation of metrological usefulness. We identify a trade-off that is intrinsic to our commuting encoding-noise structure. We analyze the QFI dynamics for the Gaussian spin state (GSS) superpositions, encompassing spin coherent state (SCS), Dicke state superpositions, spin-squeezed states, and GHZ-like states. Predictions from indicators of the QFI dynamics are compared to both the quantum Cramér--Rao bound and the measurement-specific sensitivity bounds for an optimal parameter and interrogation time under a finite total time resource. When possible, we analytically derive the measurement-specific sensitivity bounds for spin-projection and parity-based measurements.
This manuscript investigates the enhancement of parameter-estimation precision (P-EP) in photonic systems subjected to pure dephasing within Markovian and non-Markovian environments. We employ the quantum Fisher information (QFI) to quantify precision and analyze its dynamics in a system where a single photon's polarization interacts with a structured frequency environment. This environment is engineered using a Fabry-Pérot cavity and characterized by parameters such as the tilt angle, spectral width, and refractive index difference. Our results demonstrate that for specific configurations, particularly at certain angles, the system exhibits pronounced non-Markovian behavior, leading to oscillatory revivals in QFI, which signify periods of enhanced P-EP. Furthermore, we find that narrower spectral widths and smaller refractive index differences amplify these effects, prolonging periods of high QFI and improving estimation precision. These findings are corroborated by von Neumann entropy analysis, which reveals corresponding oscillations in quantum purity. Our study highlights the potential of engineered environments to improve the precision of quantum measurements despite environmental interactions, offering a promising strategy for advancing quantum technologies.
Non-classical resources enable measurements to achieve a precision that exceeds the limits predicted by the central limit theorem. However, environmental noise arising from system-environment interactions severely limits the performance of such resources through decoherence. While significant progress has been made in mitigating Markovian noise, the extent to which non-Markovian noise can be mitigated remains poorly understood. We demonstrate that Heisenberg Scaling, the ultimate quantum limit on measurement precision, can be recovered in quantum metrology under non-Markovian noise by leveraging carefully designed Dynamical Decoupling Techniques. Importantly, our approach does not rely on assumptions of Markovian dynamics. By imposing appropriate conditions on the control Hamiltonian, we show that HS can be achieved irrespective of whether the noise is Markovian or non-Markovian. We also prove necessary and sufficient conditions for the existence of such control Hamiltonians. As an illustrative example, we apply our framework to the damped Jaynes-Cummings model, successfully mitigating memory effects and maintaining measurement precision in complex, non-Markovian environments. These findings highlight the power of quantum control to overcome decoherence challenges and enhance metrological performance in realistic, noisy quantum systems.