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arXiv:2506.11810 · Physical Review Letters

Entanglement-Enhanced Optical Ion Clock

Kai Dietze, Lennart Pelzer, Ludwig Krinner, Fabian Dawel, Johannes Krämer, Nicolas Spethmann, Timm Kielinski, Klemens Hammerer, Kurt A. Stahl, J. Klose, Sören Dörscher, Christian Lisdat, Erik Benkler, Piet O. Schmidt

Published Jan 2, 2026• 4 citations
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Abstract

Entangled states hold the promise of improving the precision and accuracy of quantum sensors. We experimentally demonstrate that spectroscopy of an optical clock transition using entangled states can outperform its classical counterpart. Two ^{40}Ca^{+} ions are entangled in a quantum state with vanishing first-order magnetic field sensitivity, extending the coherence time of the atoms and enabling near-lifetime-limited probe times of up to 550 ms. In our protocol, entangled ions reach the same instability as uncorrelated ions, but at half the probe time, enabling faster cycle times of the clock. We run two entangled ^{40}Ca^{+} ions as an optical clock and compare its frequency instability with a ^{87}Sr lattice clock. The instability of the entangled ion clock is below a clock operated with classically correlated states for all probe times. We observe instabilities below the theoretically expected quantum projection noise limit of two uncorrelated ions for interrogation times below 100 ms. The lowest fractional frequency instability of 7×10^{-16}/sqrt[τ/1 s] is reached for 250 ms probe time, limited by residual phase noise of the probe laser. This represents the lowest instability reported to date for a ^{40}Ca^{+} ion clock.

DOI
10.1103/dyqm-k8p6

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Entangled states hold the promise of improving the precision and accuracy of quantum sensors. We experimentally demonstrate that spectroscopy of an optical clock transition using entangled states can outperform its classical counterpart. Two ^{40}Ca^{+} ions are entangled in a quantum state with vanishing first-order magnetic field sensitivity, extending the coherence time of the atoms and enabling near-lifetime-limited probe times of up to 550 ms. In our protocol, entangled ions reach the same instability as uncorrelated ions, but at half the probe time, enabling faster cycle times of the clock. We run two entangled ^{40}Ca^{+} ions as an optical clock and compare its frequency instability with a ^{87}Sr lattice clock. The instability of the entangled ion clock is below a clock operated with classically correlated states for all probe times. We observe instabilities below the theoretically expected quantum projection noise limit of two uncorrelated ions for interrogation times below 100 ms. The lowest fractional frequency instability of 7×10^{-16}/sqrt[τ/1 s] is reached for 250 ms probe time, limited by residual phase noise of the probe laser. This represents the lowest instability reported to date for a ^{40}Ca^{+} ion clock.
A number of optical frequency standards have been evaluated with fractional uncertainties between ${10}^{\ensuremath{-}17}$ and ${10}^{\ensuremath{-}18}$. Reduction of the statistical uncertainty of a clock comparison to this level is challenging, requiring the best possible stability to avoid excessively long averaging times. We report recent improvements of our 88Sr+^{88}\mathrm{Sr}{}^{+}88Sr+ single-ion standard that have reduced its 1-s Allan deviation from $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}$ to $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}$, yielding an order of magnitude decrease in averaging time for a given statistical uncertainty level. Among the improvements made are the implementation of a clear-out laser that transfers the ion from the metastable state to the ground state at each cycle, followed by a state preparation step that transfers the ion to the ground-state magnetic sublevel of the probed transition. With these changes, the ion clock transition interacts with the probe laser at every interrogation cycle. The stability of our optical standard is essentially limited by the quantum projection noise for pulse lengths up to $\ensuremath{\approx}100$ ms.
The use of entangled states was shown to improve the fundamental limits of spectroscopy to beyond the standard-quantum limit. Here, rather than probing the free evolution of the phase of an entangled state with respect to a local oscillator, we probe the evolution of an initially separable two-atom register under an Ising spin Hamiltonian with a transverse field. The resulting correlated spin-rotation spectrum is twice as narrow as that of an uncorrelated rotation. We implement this ideally Heisenberg-limited Rabi spectroscopy scheme on the optical-clock electric-quadrupole transition of ^{88}Sr^{+} using a two-ion crystal. We further show that depending on the initial state, correlated rotation can occur in two orthogonal subspaces of the full Hilbert space, yielding entanglement-enhanced spectroscopy of either the average transition frequency of the two ions or their difference from the mean frequency. The use of correlated spin rotations can potentially lead to new paths for clock stability improvement.
Comparisons of high-accuracy optical atomic clocks \cite{Ludlow2015} are essential for precision tests of fundamental physics \cite{Safronova2018}, relativistic geodesy \cite{McGrew2018, Grotti2018, Delva2019}, and the anticipated redefinition of the SI second \cite{Riehle2018}. The scientific reach of these applications is restricted by the statistical precision of interspecies comparison measurements. The instability of individual clocks is limited by the finite coherence time of the optical local oscillator (OLO), which bounds the maximum atomic interrogation time. In this letter, we experimentally demonstrate differential spectroscopy \cite{Hume2016}, a comparison protocol that enables interrogating beyond the OLO coherence time. By phase-coherently linking a zero-dead-time (ZDT) \cite{Schioppo2017} Yb optical lattice clock with an Al+^++ single-ion clock via an optical frequency comb and performing synchronised Ramsey spectroscopy, we show an improvement in comparison instability relative to our previous result \cite{network2020frequency} of nearly an order of magnitude. To our knowledge, this result represents the most stable interspecies clock comparison to date.
Optical atomic clocks with unrivaled precision and accuracy have advanced the frontier of precision measurement science and opened new avenues for exploring fundamental physics. A fundamental limitation on clock precision is the standard quantum limit (SQL), which stems from the uncorrelated projection noise of each atom. State-of-the-art optical lattice clocks interrogate large ensembles to minimize the SQL, but density-dependent frequency shifts pose challenges to scaling the atom number. The SQL can be surpassed, however, by leveraging entanglement, though it remains an open problem to achieve quantum advantage from spin squeezing at state-of-the-art stability levels. Here, we demonstrate clock performance beyond the SQL, achieving a fractional frequency precision of 1.1×10^{-18} for a single spin-squeezed clock. With cavity-based quantum nondemolition measurements, we prepare two spin-squeezed ensembles of ∼30 000 strontium atoms confined in a two-dimensional optical lattice. A synchronous clock comparison with an interrogation time of 61 ms achieves a metrological improvement of 2.0(2) dB beyond the SQL, after correcting for state preparation and measurement errors. These results establish the most precise entanglement-enhanced clock to date and offer a powerful platform for exploring the interplay of gravity and quantum entanglement.
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 ≈4.5 dB\approx 4.5 \text{ dB}≈4.5 dB compared with the two-level model. Furthermore, we identify an even greater stability enhancement of ≈5.4 dB\approx 5.4 \text{ dB}≈5.4 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 27Al+^{27}\text{Al}^{+}27Al+ optical ion clock.
Abstract We present a novel method for engineering an optical clock transition that is robust against external field fluctuations and is able to overcome limits resulting from field inhomogeneities. The technique is based on the application of continuous driving fields to form a pair of dressed states essentially free of all relevant shifts. Specifically, the clock transition is robust to magnetic field shifts, quadrupole and other tensor shifts, and amplitude fluctuations of the driving fields. The scheme is applicable to either a single ion or an ensemble of ions, and is relevant for several types of ions, such as 40 Ca + , 88 Sr + , 138 Ba + and 176 Lu + . Taking a spherically symmetric Coulomb crystal formed by 400 40 Ca + ions as an example, we show through numerical simulations that the inhomogeneous linewidth of tens of Hertz in such a crystal together with linear Zeeman shifts of order 10 MHz are reduced to form a linewidth of around 1 Hz. We estimate a two-order-of-magnitude reduction in averaging time compared to state-of-the art single ion frequency references, assuming a probe laser fractional instability of 10 − 15 . Furthermore, a statistical uncertainty reaching 2.9 × 10−16 in 1 s is estimated for a cascaded clock scheme in which the dynamically decoupled Coulomb crystal clock stabilizes the interrogation laser for an 27 Al + clock.