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arXiv:2003.10943 · Physical Review Letters

Heisenberg-Limited Noisy Atomic Clock Using a Hybrid Coherent and Squeezed State Protocol

Luca Pezzè, Augusto Smerzi

Published Nov 19, 2020• 56 citations
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Abstract

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.

DOI
10.1103/physrevlett.125.210503

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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.
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.
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.
Optical atomic clocks are a driving force for precision measurements due to the high accuracy and stability demonstrated in recent years. While further improvements to the stability have been envisioned by using entangled atoms, squeezing the quantum mechanical projection noise, evaluating the overall gain must incorporate essential features of an atomic clock. Here, we investigate the benefits of spin squeezed states for clocks operated with typical Brownian frequency noise-limited laser sources. Based on an analytic model of the closed servo-loop of an optical atomic clock, we report here quantitative predictions on the optimal clock stability for a given dead time and laser noise. Our analytic predictions are in good agreement with numerical simulations of the closed servo-loop. We find that for usual cyclic Ramsey interrogation of single atomic ensembles with dead time, even with the current most stable lasers spin squeezing can only improve the clock stability for ensembles below a critical atom number of about one thousand in an optical Sr lattice clock. Even with a future improvement of the laser performance by one order of magnitude the critical atom number still remains below 100,000. In contrast, clocks based on smaller, non-scalable ensembles, such as ion clocks, can already benefit from squeezed states with current clock lasers.
Optical atomic clocks are a driving force for precision measurements due to the high accuracy and stability demonstrated in recent years. While further improvements to the stability have been envisioned by using entangled atoms, squeezing the quantum mechanical projection noise, evaluating the overall gain must incorporate essential features of an atomic clock. Here, we investigate the benefits of spin squeezed states for clocks operated with typical Brownian frequency noise-limited laser sources. Based on an analytic model of the closed servo-loop of an optical atomic clock, we report here quantitative predictions on the optimal clock stability for a given dead time and laser noise. Our analytic predictions are in good agreement with numerical simulations of the closed servo-loop. We find that for usual cyclic Ramsey interrogation of single atomic ensembles with dead time, even with the current most stable lasers spin squeezing can only improve the clock stability for ensembles below a critical atom number of about one thousand in an optical Sr lattice clock. Even with a future improvement of the laser performance by one order of magnitude the critical atom number still remains below 100,000. In contrast, clocks based on smaller, non-scalable ensembles, such as ion clocks, can already benefit from squeezed states with current clock lasers. © 2020, The Author(s).
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.
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.