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Nature Communications

Entanglement-enhanced quantum lock-in detection achieving Heisenberg scaling

Jiawei Zhang, M. Zhuang, B. Wang, W.-F. Yuan, Jian Li, G.-Y. Ding, W.-Q. Ding, Liang Chen, Sijing Chen, Fei Zhou, Chaohong Lee, Mang Feng

Published Dec 6, 2025• 1 citation
Publisher

Abstract

Abstract Quantum lock-in detection (QLID), a cornerstone technique in quantum metrology, enables precise extraction of oscillating signals from noise by leveraging dynamical decoupling to suppress unwanted spectral components. Beyond this, many-body quantum entanglement serves as a critical resource for surpassing the standard quantum limit in precision measurements. Here we report the first experimental realization of entanglement-enhanced QLID using two \(^{40}\text{Ca}^+\) ions confined in a linear Paul trap. The system is initialized into a maximally entangled Greenberger-Horne-Zeilinger state via a M$\phi$lmer-S$\phi$rensen gate, followed by the application of periodic multipulse sequences to implement QLID. Compared to non-entangled states (e.g., a product state), our work demonstrates frequency measurement precision scaling as $1/N$ with particle number (N) , achieving the Heisenberg scaling. Remarkably, the precision in the time domain approaches a super-Heisenberg scaling (i.e., $1/T^2$), significantly outperforming conventional bounds. To enhance practical applicability, we further optimize multipulse sequences to improve robustness against rotation angle inaccuracies and detuning errors. This work establishes QLID as a powerful paradigm for quantum sensing and provides a concrete pathway to Heisenberg-limited detection of oscillating signals in realistic settings.

DOI
10.1038/s41467-025-66828-z

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Abstract Quantum lock-in detection (QLID), a cornerstone technique in quantum metrology, enables precise extraction of oscillating signals from noise by leveraging dynamical decoupling to suppress unwanted spectral components. Beyond this, many-body quantum entanglement serves as a critical resource for surpassing the standard quantum limit in precision measurements. Here we report the first experimental realization of entanglement-enhanced QLID using two (^{40}\text{Ca}^+) ions confined in a linear Paul trap. The system is initialized into a maximally entangled Greenberger-Horne-Zeilinger state via a Mϕ\phiϕlmer-Sϕ\phiϕrensen gate, followed by the application of periodic multipulse sequences to implement QLID. Compared to non-entangled states (e.g., a product state), our work demonstrates frequency measurement precision scaling as 1/N1/N1/N with particle number (N) , achieving the Heisenberg scaling. Remarkably, the precision in the time domain approaches a super-Heisenberg scaling (i.e., 1/T21/T^21/T2), significantly outperforming conventional bounds. To enhance practical applicability, we further optimize multipulse sequences to improve robustness against rotation angle inaccuracies and detuning errors. This work establishes QLID as a powerful paradigm for quantum sensing and provides a concrete pathway to Heisenberg-limited detection of oscillating signals in realistic settings.
High-precision measurement of oscillating signal is a ubiquitous issue in fundamental science and a critical task in practical technologies. In quantum metrology, quantum lock-in detection provide an efficient method for measuring such signal. In general, when the initial phase of the oscillating signal is unknown,quantum double lock-in detection can effectively extract complete information about the signal's amplitude, frequency, and initial phase. Conventional quantum double lock-in detection requires two individual quantum interferometry, each of which must undergo state preparation and readout. However, the time for state preparation and readout need not be negligible in practical experiments. In particular, the time for state preparation is longer than the time for sensing. To save experimental resources, it is challenging to achieve quantum double lock-in detection just via a single quantum interferometry while still extracting complete information about the signal's amplitude, frequency, and initial phase. Here, we present a general protocol for achieving a quantum double lock-in detection just via a single quantum interferometry under a sequential orthogonal periodic multipulse sequences. In particular, if the input state is a Greenberger-Horne-Zeilinger state and two interaction-based operations are applied during interferometry, the measurement precisions for frequency, amplitude, and initial phase can both approach the Heisenberg limit. Our study paves a new way for measuring oscillating signals with a single quantum interferometry, and provides a feasible method for achieving Heisenberg-limited detection of alternating signals.
Achieving high-precision detection of time-dependent signals within noise is a ubiquitous issue in physics and a critical task in metrology.Lock-in amplifiers are detectors that can extract alternating signals within extreme noise via a known carrier frequency.Here we present a protocol for achieving an entanglement-enhanced lock-in amplifier via use of many-body multipulse quantum interferometry.The many-body quantum lock-in amplifier is implemented by application of a periodic multi-π -pulse sequence during the interrogation.Our analytic results show that, by our choosing suitable input states and readout operations, the frequency and amplitude of an unknown alternating field can be simultaneously extracted via population measurements.The lock-in point can be determined via the symmetry of the signal during a single interrogation time or the time-averaged signals for multiple interrogation times.We find that the measurement signal at the lock-in point is independent of the interrogation time.In particular, if we input spin cat states and apply interaction-based readout operations, the measurement precisions for frequency and amplitude can both approach the Heisenberg limit.Moreover, our many-body quantum amplifier is also robust with regard to extreme stochastic noise.Our study paves a new way for measuring time-dependent signals with many-body quantum systems, and provides a feasible method for achieving Heisenberg-limited detection of alternating signals.
Entanglement-enhanced quantum metrology explores the utilization of quantum entanglement to enhance measurement precision. When particles in a probe are prepared into a suitable quantum entangled state, they may collectively accumulate information about the physical quantity to be measured, leading to an improvement in measurement precision beyond the standard quantum limit and approaching the Heisenberg limit. The rapid advancement of techniques for quantum manipulation and detection has enabled the generation, manipulation, and detection of multi-particle entangled states in synthetic quantum systems such as cold atoms and trapped ions. This article aims to review and illustrate the fundamental principles and experimental progresses that demonstrate multi-particle entanglement for quantum metrology, as well as discuss the potential applications of entanglement-enhanced quantum sensors.
The Heisenberg scaling is an ultimate precision limit of parameter estimation allowed by the principles of quantum mechanics, with no counterpart in the classical realm, and has been a long-pursued goal in quantum metrology. It has been known that interactions between the probes can help reach the Heisenberg scaling without entanglement. In this paper, we show that interactions between the probes and the additional dimensions of an ancillary system may also increase the precision of parameter estimation to surpass the standard quantum limit and attain the Heisenberg scaling without entanglement, if the measurement scheme is properly designed. The quantum Fisher information exhibits periodic patterns over the evolution time, implying the existence of optimal time points for measurements that can maximize the quantum Fisher information. By implementing optimizations over the Hamiltonian, the initial states of the probes and the ancillary system, the interaction strength, and the time points for measurements, our protocol achieves the Heisenberg scaling for the parameter of the probe Hamiltonian, in terms of both evolution time and probe number. Our protocol features two aspects: (i) the Heisenberg scaling can be achieved by a product state of the probes and (ii) mere local measurement on the ancilla is sufficient, both of which reduce the quantum resources and the implementation complexity to achieve the Heisenberg scaling. The paper is concluded by the investigation of the effects of noise on this protocol.
Abstract The detection of weak time-dependent alternating signals in a strongly noisy background is an important problem in physics and a critical task in metrology. Quantum lock-in amplifier can extract alternating signals within extreme noises by using suitable quantum resources, which has been widely used for magnetic field sensing, vector light shift detection, and force detection. In particular, entanglement-enhanced quantum lock-in amplifier can be realized via many-body quantum interferometry. The many-body lock-in measurement provides a feasible way to achieve high-precision detection of alternating signals, even in noisy environments. In this article, we review general protocol, experiment progresses and potential applications of quantum lock-in measurements.
Entangled quantum probes can achieve Heisenberg-limited measurement precision, but this advantage is typically destroyed by noise. We address this issue by introducing a framework that we call encoded quantum signal processing, which unifies quantum error detection and quantum signal processing into an effective single-qubit framework, and provides a paradigm for constructing logical sensors that are robust to noise while remaining sensitive to the signal of interest. We show that encoding sensor qubits into a repetition code and using syndrome measurements as a signal-processing primitive restores Heisenberg scaling under realistic noise, without applying recovery operations. We prove that product-state sensing with syndrome post-processing is fundamentally limited to standard quantum limit (SQL) scaling, and develop four protocols that overcome this barrier through entanglement or sequential signal amplification, achieving Heisenberg-limited precision with exponential error suppression in code distance. For spatially inhomogeneous fields, Bayesian marginalization preserves Heisenberg scaling provided noise decreases sufficiently with system size. The underlying mechanism, which we formalize as encoded quantum signal processing, reduces multi-qubit metrology to an effective single-qubit problem where syndrome measurement implements nonlinear signal transformations. Numerical simulations validate the theoretical predictions: syndrome-based inference achieves near-Heisenberg scaling at noise levels where bare probes approach the SQL, and a concatenated protocol maintains this scaling under joint transverse noise and longitudinal inhomogeneities.