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arXiv:2104.14309 · Physical Review A

Improved absolute clock stability by the joint interrogation of two atomic ensembles

Weidong Li, Shuyuan Wu, Augusto Smerzi, Luca Pezzè

Published May 19, 2022• 8 citations
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Abstract

Improving the clock stability is of fundamental importance for the development of quantum-enhanced metrology. One of the main limitations arises from the randomly fluctuating local oscillator (LO) frequency, which introduces ``phase slips'' for long interrogation times and hence the failure of the frequency-feedback loop. Here we propose a strategy to improve the stability of atomic clocks by interrogating two atomic ensembles sharing the same LO. The two ensembles are prepared in coherent spin states pointing along orthogonal directions in the Bloch sphere. While standard Ramsey interrogation can only determine phases unambiguously in the interval $[\ensuremath{-}\ensuremath{\pi}/2,\ensuremath{\pi}/2]$, the joint interrogation allows for an extension to $[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi}]$, resulting in a relaxed restriction of the Ramsey time and improvement of absolute clock stability. Theoretical predictions are supported by ab initio numerical simulation for white and correlated LO noise. While our basic protocol uses uncorrelated atoms, we further extended it to include spin-squeezing and further improving the scaling of clock stability with the number of atoms. Our protocol can be readily tested in current state-of-the-art experiments.

DOI
10.1103/physreva.105.053116

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Improving the clock stability is of fundamental importance for the development of quantum-enhanced metrology. One of the main limitations arises from the randomly fluctuating local oscillator (LO) frequency, which introduces ``phase slips'' for long interrogation times and hence the failure of the frequency-feedback loop. Here we propose a strategy to improve the stability of atomic clocks by interrogating two atomic ensembles sharing the same LO. The two ensembles are prepared in coherent spin states pointing along orthogonal directions in the Bloch sphere. While standard Ramsey interrogation can only determine phases unambiguously in the interval [\ensuremath−\ensuremathπ/2,\ensuremathπ/2][\ensuremath{-}\ensuremath{\pi}/2,\ensuremath{\pi}/2][\ensuremath−\ensuremathπ/2,\ensuremathπ/2], the joint interrogation allows for an extension to [\ensuremath−\ensuremathπ,\ensuremathπ][\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi}][\ensuremath−\ensuremathπ,\ensuremathπ], resulting in a relaxed restriction of the Ramsey time and improvement of absolute clock stability. Theoretical predictions are supported by ab initio numerical simulation for white and correlated LO noise. While our basic protocol uses uncorrelated atoms, we further extended it to include spin-squeezing and further improving the scaling of clock stability with the number of atoms. Our protocol can be readily tested in current state-of-the-art experiments.
We propose a hybrid quantum-classical atomic clock where the interrogation of atoms prepared in a spin-coherent (or weakly squeezed) state is used to feed back one or more highly spin-squeezed atomic states toward their optimal phase-sensitivity point. The hybrid clock overcomes the stability of a single Ramsey clock using coherent or optimal spin-squeezed states and reaches a Heisenberg-limited stability while avoiding nondestructive measurements. When optimized with respect to the total number of particles, the protocol surpasses the state-of-the-art proposals that use Greenberger-Horne-Zeilinger or NOON states. We compare analytical predictions with numerical simulations of clock operations, including correlated 1/f local oscillator noise.
Frequency metrology is a cornerstone of modern precision measurements and optical atomic clocks have emerged as one of the most precise measurement devices. In this progress report, we explore various Ramsey interrogation schemes tailored to optical atomic clocks primarily limited by laser noise. To incorporate frequency fluctuations directly into the theoretical model, we consider a Bayesian framework. In this context, we review fundamental bounds arising in Bayesian estimation theory, which serve as a benchmark throughout this work. We investigate the trade-off between entanglement-enhanced sensitivity and robustness against laser noise in order to identify optimal initial states, measurement schemes and estimation strategies. Beside standard protocols based on coherent spin states, spin-squeezed states and Greenberger-Horne-Zeilinger states, we consider variational Ramsey protocols implemented via low-depth quantum circuits based on one-axis twisting operations to approach optimal stability. In particular, we review known and identify new optimal Ramsey interrogation schemes for a variety of scenarios, including different experimental platforms, ensemble sizes and regimes characterized by a wide range of interrogation durations and dead times. Hence, this work establishes a comprehensive theoretical framework for optimizing Ramsey interrogation schemes, providing guidance for the development of next-generation optical atomic clocks.
The stability of an optical atomic clock is a critical figure of merit for almost all clock applications. To this end, much optical atomic clock research has focused on reducing clock instability by increasing the atom number, lengthening the coherent interrogation times, and introducing entanglement to push beyond the standard quantum limit. In this work, we experimentally demonstrate an alternative approach to reducing clock instability using a phase estimation approach based on individually controlled atomic ensembles in a strontium (Sr) optical lattice clock. We first demonstrate joint Ramsey interrogation of two spatially resolved atom ensembles that are out of phase with respect to each other, which we call “quadrature Ramsey spectroscopy,” resulting in a factor of 1.36(5) reduction in absolute clock instability as measured with interleaved self-comparisons. We then leverage the rich hyperfine structure of Sr87 to realize independent coherent control over multiple ensembles with only global laser addressing. Finally, we utilize this independent control over four atom ensembles to implement a form of phase estimation, achieving a factor of greater than 3 enhancement in coherent interrogation time and a factor of 2.08(6) reduction in instability over an otherwise identical single-ensemble clock with the same local oscillator and the same number of atoms. We expect that multiensemble protocols similar to those demonstrated here will result in reduction in the instability of any optical lattice clock with an interrogation time limited by the local oscillator. Published by the American Physical Society 2024
Abstract The frequency stability of many optical atomic clocks is limited by the coherence of their local oscillator. Here, we present a measurement protocol that overcomes the laser coherence limit. It relies on engineered dynamical decoupling of laser phase noise and near-synchronous interrogation of two clocks. One clock coarsely tracks the laser phase using dynamical decoupling; the other refines this estimate using a high-resolution phase measurement. While the former needs to have a high signal-to-noise ratio, the latter clock may operate with any number of particles. The protocol effectively enables minute-long Ramsey interrogation for coherence times of few seconds as provided by the current best ultrastable laser systems. We demonstrate implementation of the protocol in a realistic proof-of-principle experiment, where we interrogate for 0.5 s at a laser coherence time of 77 ms. Here, a single lattice clock is used to emulate synchronous interrogation of two separate clocks in the presence of artificial laser frequency noise. We discuss the frequency instability of a single-ion clock that would result from using the protocol for stabilisation, under these conditions and for minute-long interrogation, and find expected instabilities of σy(τ) = 8 × 10−16(τ/s)−1/2 and σy(τ) = 5 × 10−17(τ/s)−1/2, respectively.
Atomic clocks have reached the Quantum Projection Noise (QPN) limit of stability and it has been proposed to use the entangled atomic ensemble in order to overcome the QPN limit. We have proposed a new method to lock the phase of the Local Oscillator (LO) to the atomic phase and call, "atomic phase lock (APL)." This APL could possibly overcome the QPN limit without need of preparing the entangled atomic ensemble. Traditional Ramsey method destroys the coherence of the atomic spin due to the projection measurement at each cycle. This destruction and initialization of the phase at each cycle introduce additional noise and the performance of the atomic clock is limited at the white frequency noise level. APL employs dispersion measurements, i.e. Faraday rotation, in order to measure the phase difference without destroying the atomic phase. By repeating the measurement cycle with sufficiently small dead time, one can suppress the LO noise down to white phase noise level, achieving the τ-1 dependence of the Allan variance that can eventually overcome the QPN limit in long term. We are preparing a proof-of-principle experiment using the ensemble of trapped 171Yb+ ions with hyperfine splitting (12.6 GHz) as a clock transition and report the current status of the experiment as well.
We show that the stability of an optical atomic clock can only be improved by spin squeezed states for ensembles below a critical particle number. This limitation results from a trade-off between measurement noise and instabilities due to limited laser coherence. Our results apply to the case of cyclic Ramsey interrogations on a single atomic ensemble with dead time between each measurement. The combination of analytical models for projection noise, dead time noise (Dick effect) and laser phase noise allows quantitative predictions of the critical particle number and the optimal clock stability for a given dead time and laser noise. Our analytical predictions are confirmed by numerical simulations of the closed servo loop of an optical atomic clock.