Closed-loop conditions for spin-squeezing enhancement of optical atomic clocks: excess anti-squeezing, multi-ensemble dead-time-free architectures, and the critical ensemble size

How much can entanglement improve the stability of an optical atomic clock once the essential features of clock operation—a fluctuating local oscillator, a closed servo loop, and dead time between interrogations—are included? Schulte, Lisdat, Schmidt, Sterr and Hammerer (Nature Communications 11, 5955, 2020) showed analytically and by closed-loop simulation that for cyclic Ramsey interrogation of a single ensemble with dead time and Brownian laser frequency noise, spin squeezing improves stability only below a critical atom number (about 1000 for a strontium lattice clock with the best current lasers), and posed four open questions: incorporating imperfect state preparation (excess anti-squeezing) into the colored-noise closed-loop model; whether other interrogation and estimation protocols can circumvent the coherence-time-limit (CTL)/Dick-aliasing restrictions; whether multi-ensemble, cascaded or anti-synchronized dead-time-free architectures remove the critical-ensemble-size limitation; and the experimental demonstration of squeezing-enhanced stability in a dead-time-free dual-ensemble clock. The estimation-protocol question is largely settled by variational/Bayesian closed-loop analyses (Kaubruegger et al. 2021; Kielinski and Hammerer 2025), and the experimental question is substantially settled in the synchronous-comparison variant (Yang et al., PRL 135, 193202, 2025: 2.0(2) dB beyond the standard quantum limit at $1.1\times 10^{-18}$). The surviving open core comprises the closed-loop treatment of realistic (anti-squeezed) state preparation under colored laser noise, a systematic determination of whether the critical ensemble size persists in multi-ensemble and dead-time-free architectures (explicitly placed beyond the scope of the 2025 comprehensive Bayesian treatment), the residual estimation-protocol question of continuous weak measurement, and an anti-synchronized demonstration of improved absolute clock stability.

Background

An optical atomic clock steers a laser—the local oscillator (LO)—onto a narrow atomic reference transition, using repeated interrogations of an ensemble of $N$ atoms to measure and correct the LO frequency. Its performance figure is the fractional frequency instability, quantified by the Allan deviation $\sigma_y(\tau)$ at averaging time $\tau$. With uncorrelated atoms, each measurement outcome carries quantum projection noise (QPN), the random statistics of measuring a collective spin, giving the standard quantum limit (SQL) $\sigma_y \propto 1/\sqrt{N}$ for fixed interrogation time.

Spin-squeezed states reduce the projection noise of one collective-spin quadrature at the price of amplifying the orthogonal (anti-squeezed) one; their metrological value is summarized by the Wineland parameter $\xi_W^2 < 1$, the factor by which the squeezed ensemble outperforms $N$ uncorrelated atoms. Such states are produced, for example, by one-axis twisting (OAT), an effective atom–atom interaction $e^{-i(\mu/2)S_z^2}$ that shears a coherent spin state (a collective state in which all $N$ atoms share the same superposition), or by quantum nondemolition (QND) measurements of the collective population (light passing through a cavity reveals the atoms' spin component without disturbing the clock coherence).

Schulte, Lisdat, Schmidt, Sterr and Hammerer (Nature Communications 11, 5955, 2020) asked what squeezing buys once clock operation is modeled honestly. In each Ramsey interrogation (a $\pi/2$ pulse, free evolution for time $T$, second $\pi/2$ pulse), the LO frequency offset is converted into a measured phase; a servo then corrects the LO, and the cycle of interrogation plus dead time $T_D$ (state preparation and readout, during which the LO runs unobserved) repeats. With realistic Brownian LO frequency noise, three laser-noise mechanisms enter: (i) the laser coherence time limit (CTL), a diffusion-type contribution to $\sigma_y$ from the finite LO coherence time; (ii) the Dick effect, the aliasing of LO noise sampled at harmonics of the cycle rate $1/(T+T_D)$, caused by the dead time; and (iii) fringe hops, rare excursions where the servo locks to a neighboring Ramsey fringe and the clock then runs permanently wrong, which caps the usable Ramsey time. Their analytic model of the closed servo loop, confirmed by Monte Carlo simulation of the loop, showed that squeezing shrinks the QPN contribution but leaves CTL and Dick noise unchanged. Consequently, for cyclic Ramsey interrogation of a single ensemble with dead time, squeezing improves stability only below a critical atom number—about $10^3$ for a strontium optical lattice clock with the most stable present lasers, and still below $10^5$ if lasers improve tenfold—while small non-scalable ensembles (ion clocks) already benefit.

The paper's outlook explicitly lists what remains: excess anti-squeezing from imperfect state preparation (known to degrade stability under white frequency noise) was not included in the colored-noise closed-loop model; whether interrogation and estimation protocols other than cyclic Ramsey (Rabi interrogation, i.e. continuously driving the atomic transition with a single long pulse instead of the two-pulse Ramsey scheme, adaptive or nonlinear estimators, or continuous weak measurement, in which the collective spin is steadily monitored by weak probe light rather than read out in pulses) can circumvent the identified restrictions remained open; multi-ensemble and cascaded architectures—including dead-time-free operation by anti-synchronized interrogation of two ensembles—had not been analyzed with dead time and realistic LO noise included, leaving open whether a similar critical ensemble size appears there; and an experimental stability improvement from squeezed states in a dead-time-free dual-ensemble clock was an open challenge. Subsequent developments define what is still open today: Kaubruegger, Vasilyev, Schulte, Hammerer and Zoller (Phys. Rev. X 11, 041045, 2021) introduced variationally optimized Ramsey protocols scored by clock stability, and Kielinski and Hammerer (Rep. Prog. Phys. 88, 124001, 2025) systematically compared coherent, squeezed, GHZ (maximally entangled multi-atom 'cat') states, phase-optimized inference and variational low-depth quantum circuits inside a full closed servo-loop simulation with colored laser noise and dead time. Experimentally, Rydberg-dressed twisting inside an optical clock (Eckner et al., Nature 621, 734, 2023) and QND-prepared squeezed ensembles enabled Yang et al. (Phys. Rev. Lett. 135, 193202, 2025) to demonstrate 2.0(2) dB of clock precision beyond the SQL at the $10^{-18}$ level in a synchronous comparison of two squeezed ensembles sharing one clock laser—while squeezing on the clock transition itself had already been achieved in 2020 (Pedrozo-Peñafiel et al., Nature 588, 414, 2020).

Problem Statement

In the closed servo-loop framework of Schulte et al. (2020)—cyclic interrogation of atomic ensembles by a laser LO with realistic Brownian (colored) frequency noise, explicit dead time, fringe hops, the laser coherence time limit (CTL) and Dick aliasing—determine under what conditions spin-squeezed states improve the stability of optical atomic clocks. The source outlook poses four linked sub-questions; as of August 2026 the estimation-protocol strand is largely settled and the experimental strand is substantially settled in its synchronous variant. Sub-questions (1)–(3) are analytic and each stands or falls on its own within the closed servo-loop framework; sub-question (4) concerns an experimental milestone, whose treatment here is its design and feasibility conditions:

(1) Imperfect state preparation: incorporate the full covariance of realistic squeezed states—including excess anti-squeezing due to imperfect one-axis-twisting or QND preparation—into the closed-loop model with colored laser noise, and quantify how it modifies the predicted optimal clock stability and the critical atom number (the earlier result that anti-squeezing degrades stability was obtained only for white frequency noise in open-loop-style analyses).

(2) Multi-ensemble architectures: extend the closed-loop analysis (analytic model plus Monte Carlo servo-loop simulation, dead time and colored LO noise explicitly included) to multi-ensemble, cascaded, and anti-synchronized (dead-time-free, alternating-interrogation) clock architectures, and decide whether a critical ensemble size analogous to the ~$10^3$ atoms of the single-ensemble case persists there, or under what conditions squeezing yields a stability advantage for arbitrary ensemble size. Partial results exist for specific two-ensemble joint interrogation and for synchronous differential comparison; the systematic closed-loop determination does not.

(3) Estimation and interrogation protocols: establish whether any protocol not yet analyzed in the closed servo-loop framework—most notably continuous weak measurement, since Rabi interrogation was already excluded and adaptive/Bayesian/variational and GHZ protocols have now been treated—can circumvent the CTL/Dick limitations identified for cyclic Ramsey interrogation.

(4) Anti-synchronized demonstration: specify the operating conditions and feasibility requirements under which a stability improvement from squeezed states in anti-synchronized (dead-time-free) dual-ensemble clock operation would be demonstrated—improving the clock's absolute (LO-locked) stability rather than only a synchronous differential comparison, which has now been achieved at the 2 dB, $10^{-18}$ level. Collectively these decide the source's overarching question: whether, and under what realistic operating conditions, entanglement provides a universal stability advantage over the standard quantum limit in optical atomic clocks.

References

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