The use of correlated states and measurements promises improvements in the accuracy of frequency metrology and the stability of atomic clocks. However, developing strategies robust against dominant noise processes remains challenging. We address the issue of decoherence due to spontaneous decay and show that Greenberger-Horne-Zeilinger (GHZ) states, in conjunction with a correlated measurement and nonlinear estimation strategy, achieve gains of up to 2.25 decibel, comparable to fundamental bounds for up to about 80 atoms in the presence of decoherence. This result is unexpected because GHZ states do not provide any enhancement under dephasing due to white frequency noise compared to the standard quantum limit of uncorrelated states. The gain arises from a veto signal, which allows for the detection and mitigation of errors caused by spontaneous emission events. Through comprehensive Monte Carlo simulations of atomic clocks, we demonstrate the robustness of the GHZ protocol.The use of correlated states and measurements promises improvements in the accuracy of frequency metrology and the stability of atomic clocks. However, developing strategies robust against dominant noise processes remains challenging. We address the issue of decoherence due to spontaneous decay and show that Greenberger-Horne-Zeilinger (GHZ) states, in conjunction with a correlated measurement and nonlinear estimation strategy, achieve gains of up to 2.25 decibel, comparable to fundamental bounds for up to about 80 atoms in the presence of decoherence. This result is unexpected because GHZ states do not provide any enhancement under dephasing due to white frequency noise compared to the standard quantum limit of uncorrelated states. The gain arises from a veto signal, which allows for the detection and mitigation of errors caused by spontaneous emission events. Through comprehensive Monte Carlo simulations of atomic clocks, we demonstrate the robustness of the GHZ protocol.

Jan 1, 20240 citations

GHZ protocols enhance frequency metrology despite spontaneous decay

Timm Kielinski, Piet O. Schmidt, Klemens Hammerer

Science AdvancesOct 23, 202412 citations

Heisenberg-Limited Atom Clocks Based on Entangled Qubits

E. M. Kessler, P. Kómár, Michael Bishof +4 more

We present a quantum-enhanced atomic clock protocol based on groups of sequentially larger Greenberger-Horne-Zeilinger (GHZ) states that achieves the best clock stability allowed by quantum theory up to a logarithmic correction. Importantly the protocol is designed to work under realistic conditions where the drift of the phase of the laser interrogating the atoms is the main source of decoherence. The simultaneous interrogation of the laser phase with a cascade of GHZ states realizes an incoherent version of the phase estimation algorithm that enables Heisenberg-limited operation while extending the coherent interrogation time beyond the laser noise limit. We compare and merge the new protocol with existing state of the art interrogation schemes, and identify the precise conditions under which entanglement provides an advantage for clock stabilization: it allows a significant gain in the stability for short averaging time.We present a quantum-enhanced atomic clock protocol based on groups of sequentially larger Greenberger-Horne-Zeilinger (GHZ) states that achieves the best clock stability allowed by quantum theory up to a logarithmic correction. Importantly the protocol is designed to work under realistic conditions where the drift of the phase of the laser interrogating the atoms is the main source of decoherence. The simultaneous interrogation of the laser phase with a cascade of GHZ states realizes an incoherent version of the phase estimation algorithm that enables Heisenberg-limited operation while extending the coherent interrogation time beyond the laser noise limit. We compare and merge the new protocol with existing state of the art interrogation schemes, and identify the precise conditions under which entanglement provides an advantage for clock stabilization: it allows a significant gain in the stability for short averaging time.

arXiv:1310.6043Physical Review LettersMay 15, 2014126 citations

Extending the fundamental limit of atomic clock stability

Ravid Shaniv, Ayush Agrawal, David B. Hume

Optical atomic clocks have been rapidly developing in recent decades, resulting in major improvements in both precision and accuracy. As a result, they have become instrumental in multiple areas of applied and fundamental research. Despite all atomic frequency references having more than two energy-levels, the commonly used model for evaluating their ultimate limits assumes a two-level atom. This leads to frequency interrogation protocols and theoretical stability bounds that are suboptimal for a true multi-level atom. The most fundamental stability bound assumes two noise sources - quantum projection noise and spontaneous decay from the excited state. In this work, we analyze a model that includes these noise types and is generalized beyond the two-level assumption, where spontaneous decay can branch to more than a single ground state. This model allows for detection and exclusion of atomic frequency interrogations in which the atom decayed, leading to a frequency stability improvement of up to $\approx 4.5 \text{ dB}$ compared with the two-level model. Furthermore, we identify an even greater stability enhancement of $\approx 5.4 \text{ dB}$ for frequency comparisons between atoms in an odd parity Bell state. These enhancements are particularly relevant for the numerous trapped-ion optical clock species that operate close to lifetime-limited stability. We calculate new stability limits for those cases and provide a detailed experimental protocol for frequency interrogation with an $^{27}\text{Al}^{+}$ optical ion clock.Optical atomic clocks have been rapidly developing in recent decades, resulting in major improvements in both precision and accuracy. As a result, they have become instrumental in multiple areas of applied and fundamental research. Despite all atomic frequency references having more than two energy-levels, the commonly used model for evaluating their ultimate limits assumes a two-level atom. This leads to frequency interrogation protocols and theoretical stability bounds that are suboptimal for a true multi-level atom. The most fundamental stability bound assumes two noise sources - quantum projection noise and spontaneous decay from the excited state. In this work, we analyze a model that includes these noise types and is generalized beyond the two-level assumption, where spontaneous decay can branch to more than a single ground state. This model allows for detection and exclusion of atomic frequency interrogations in which the atom decayed, leading to a frequency stability improvement of up to $\approx 4.5 \text{ dB}$ compared with the two-level model. Furthermore, we identify an even greater stability enhancement of $\approx 5.4 \text{ dB}$ for frequency comparisons between atoms in an odd parity Bell state. These enhancements are particularly relevant for the numerous trapped-ion optical clock species that operate close to lifetime-limited stability. We calculate new stability limits for those cases and provide a detailed experimental protocol for frequency interrogation with an $^{27}\text{Al}^{+}$ optical ion clock.

physics.atom-phCross-listedarXiv:2604.01099Apr 2, 20260 citations

Mitigating detuning-induced systematic errors in entanglement-enhanced metrology

Shingo Kukita, Yuichiro Matsuzaki

Quantum sensing leverages non-classical resources to enhance precision. In particular, Greenberger-Horne-Zeilinger (GHZ) states can, in principle, attain the Heisenberg limit that surpasses the standard quantum limit. While many studies have examined how open-system noise-typically modeled with Lindblad master equations-degrades GHZ-based metrology, coherent control imperfections during state preparation and readout have received less attention. Here, we analyze the effect of detuning between actual and nominal spin frequencies in a GHZ-state preparation scheme employing a frequency selective pulse. We show that detuning induces coherent, systematic error that prevents GHZ sensing from reaching the Heisenberg limit. To mitigate this effect, we design a composite-pulse protocol that compensates for detuning-induced errors and improves the sensitivity under the effect of coherent error.Quantum sensing leverages non-classical resources to enhance precision. In particular, Greenberger-Horne-Zeilinger (GHZ) states can, in principle, attain the Heisenberg limit that surpasses the standard quantum limit. While many studies have examined how open-system noise-typically modeled with Lindblad master equations-degrades GHZ-based metrology, coherent control imperfections during state preparation and readout have received less attention. Here, we analyze the effect of detuning between actual and nominal spin frequencies in a GHZ-state preparation scheme employing a frequency selective pulse. We show that detuning induces coherent, systematic error that prevents GHZ sensing from reaching the Heisenberg limit. To mitigate this effect, we design a composite-pulse protocol that compensates for detuning-induced errors and improves the sensitivity under the effect of coherent error.

arXiv:2510.16739Physical Review AMar 4, 20260 citations

Spin Grid States for Quantum Metrology in Atomic Clocks Limited by Spontaneous Emission

Marius Burgath, Klemens Hammerer

Entanglement can enhance precision in phase estimation and in frequency estimation with atomic clocks, but it remains a central challenge to identify useful states and measurements under realistic noise processes. Here, we study quantum metrology with an ensemble of atoms subject to spontaneous emission, which limits frequency estimation by constraining the useful interrogation time. We identify spin grid states (SGSs) as the relevant near-optimal probes beyond a crossover at an ensemble size of 51, where GHZ-like states cease to be optimal. SGSs display a periodic grid on the Bloch sphere and can be generated with two one-axis-twisting (OAT) operations separated by a collective rotation. By optimizing the quantum Fisher information over permutationally symmetric states, we find that SGSs perform close to the globally optimal probes for intermediate and large ensembles, which turn out to be spin GKP states. We further present a sequential readout strategy that nearly saturates the corresponding quantum Fisher information. This strategy uses an OAT echo to map spontaneous-emission events onto collective rotations, making the corresponding jump sectors distinguishable and allowing the measurement basis to be conditioned on the number of decay events. Together, these results show that SGSs provide a practical route to quantum-enhanced phase and frequency metrology in atomic ensembles limited by spontaneous emission.Entanglement can enhance precision in phase estimation and in frequency estimation with atomic clocks, but it remains a central challenge to identify useful states and measurements under realistic noise processes. Here, we study quantum metrology with an ensemble of atoms subject to spontaneous emission, which limits frequency estimation by constraining the useful interrogation time. We identify spin grid states (SGSs) as the relevant near-optimal probes beyond a crossover at an ensemble size of 51, where GHZ-like states cease to be optimal. SGSs display a periodic grid on the Bloch sphere and can be generated with two one-axis-twisting (OAT) operations separated by a collective rotation. By optimizing the quantum Fisher information over permutationally symmetric states, we find that SGSs perform close to the globally optimal probes for intermediate and large ensembles, which turn out to be spin GKP states. We further present a sequential readout strategy that nearly saturates the corresponding quantum Fisher information. This strategy uses an OAT echo to map spontaneous-emission events onto collective rotations, making the corresponding jump sectors distinguishable and allowing the measurement basis to be conditioned on the number of decay events. Together, these results show that SGSs provide a practical route to quantum-enhanced phase and frequency metrology in atomic ensembles limited by spontaneous emission.

quant-pharXiv:2609.16975Sep 16, 20260 citations

\approx 4.5 \text{ dB}

compared with the two-level model. Furthermore, we identify an even greater stability enhancement of

\approx 5.4 \text{ dB}

for frequency comparisons between atoms in an odd parity Bell state. These enhancements are particularly relevant for the numerous trapped-ion optical clock species that operate close to lifetime-limited stability. We calculate new stability limits for those cases and provide a detailed experimental protocol for frequency interrogation with an

^{27}\text{Al}^{+}

optical ion clock.