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arXiv:2212.00308 · Physical Review A

Wave-front curvature in optical atomic beam clocks

Aidan Strathearn, Rachel F. Offer, Ashby Paul Hilton, Elizaveta Klantsataya, Andre Nicholas Luiten, R. P. Anderson, Ben M. Sparkes, Thomas M. Stace

Published Jul 11, 2023• 5 citations
arXivHTMLPDF

Abstract

Atomic clocks provide a reproducible basis for our understanding of time and frequency. Recent demonstrations of compact optical clocks, employing thermal atomic beams, have achieved short-term fractional frequency instabilities of order ${10}^{\ensuremath{-}16}$, competitive with the best international frequency standards available. However, a serious challenge inherent in compact clocks is the necessarily smaller optical beams, which results in rapid variation in interrogating wave fronts. This can cause inhomogeneous excitation of the thermal beam leading to long-term drifts in the output frequency. Here we develop a model for Ramsey-Bord\'e interferometry using optical fields with curved wave fronts and simulate the $^{40}\mathrm{Ca}$ beam clock experiment described by Olson et al. [Phys. Rev. Lett. 123, 073202 (2019)]. The results of Olson et al. showed surprising and unexplained behavior in the response of the atoms in the interrogation. Our model predicts signals consistent with experimental data and can account for the significant sensitivity to laser geometry that was reported. We find the signal-to-noise ratio is maximized when the laser is uncollimated at the interrogation zones to minimize inhomogeneity and also identify an optimal waist size determined by both laser inhomogeneity and the velocity distribution of the atomic beam. We investigate the shifts and stability of the clock frequency, showing that the Gouy phase is the primary source of frequency variations arising from laser geometry.

DOI
10.1103/physreva.108.013105

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Atomic clocks provide a reproducible basis for our understanding of time and frequency. Recent demonstrations of compact optical clocks, employing thermal atomic beams, have achieved short-term fractional frequency instabilities of order 10\ensuremath−16{10}^{\ensuremath{-}16}10\ensuremath−16, competitive with the best international frequency standards available. However, a serious challenge inherent in compact clocks is the necessarily smaller optical beams, which results in rapid variation in interrogating wave fronts. This can cause inhomogeneous excitation of the thermal beam leading to long-term drifts in the output frequency. Here we develop a model for Ramsey-Bord'e interferometry using optical fields with curved wave fronts and simulate the 40Ca^{40}\mathrm{Ca}40Ca beam clock experiment described by Olson et al. [Phys. Rev. Lett. 123, 073202 (2019)]. The results of Olson et al. showed surprising and unexplained behavior in the response of the atoms in the interrogation. Our model predicts signals consistent with experimental data and can account for the significant sensitivity to laser geometry that was reported. We find the signal-to-noise ratio is maximized when the laser is uncollimated at the interrogation zones to minimize inhomogeneity and also identify an optimal waist size determined by both laser inhomogeneity and the velocity distribution of the atomic beam. We investigate the shifts and stability of the clock frequency, showing that the Gouy phase is the primary source of frequency variations arising from laser geometry.
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.
We demonstrate an optical clock based on Ramsey-Bordé interferometry in a laser-cooled ^{40}Ca beam. The mean velocity is reduced by an order of magnitude relative to a thermal beam and the transverse temperature approaches the Doppler limit, enabling the measurement of sub-kHz linewidth fringes in a compact interferometer. Using tailored phase and intensity modulation of the spectroscopy laser to add uniform frequency sidebands, we interrogate atoms throughout the transverse velocity distribution, increasing the Ramsey-Bordé fringe amplitude by a factor of 14 and improving the Allan deviation to 3.4×10^{-15} at one second averaging time.
Field-grade atomic clocks capable of primary standard performance in compact physics packages would be of significant value in applications ranging from network synchronization to inertial navigation. A coherent-population-trapping clock featuring laser-cooled 87Rb^{87}\mathrm{Rb}87Rb atoms and pulsed Ramsey interrogation is a strong candidate for this technology if the frequency biases can be minimized and controlled. Here we characterize the light shift in a cold-atom coherent-population-trapping clock, explaining observed shifts in terms of phase shifts that arise during the formation of dark-state coherences combined with optical-pumping effects caused by unwanted incoherent light in the interrogation spectrum. Measurements are compared with existing and new theoretical treatments, and a laser configuration is identified that would reduce clock frequency uncertainty from light shifts to a fractional frequency level of $\ensuremath{\Delta}\ensuremath{\nu}/\ensuremath{\nu}=4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}$ per 100 kHz of laser frequency uncertainty.
Chip-scale microwave atomic systems based on thermal atomic beams offer a promising approach to realize low-power and low-drift clocks for timing holdover applications. Miniature beam clocks are expected to suppress many of the shifts that commonly limit existing chip-scale atomic clocks based on coherent population trapping, including collisional shifts and some light shifts. However, the beam geometry can amplify some challenges such as Doppler shifts, which generate a strong sensitivity to laser frequency variation. Using a centimeter-scale 87 Rb atom beam clock, we identify a surprisingly strong competition between Doppler shifts and resonant light shifts arising from asymmetric decay in the clock spectroscopy Λ -system. Leveraging this competition between Doppler and resonant light shifts, we demonstrate clock operation at specific, convenient experimental parameters consistent with zero sensitivity to laser frequency variation and white-noise-limited clock frequency averaging for 1000 s of integration.
Active optical frequency standards provide interesting alternatives to their passive counterparts. Particularly, such a clock alone continuously generates highly stable narrow-line laser radiation. Thus, a local oscillator is not required to keep the optical phase during a dead time between interrogations as in passive clocks, but only to boost the active clock's low output power to practically usable levels with the current state of technology. Here we investigate the spectral properties and the stability of active clocks, including homogeneous and inhomogeneous broadening effects. We find that for short averaging times the stability is limited by photon shot noise from the limited emitted laser power and at long averaging times by phase diffusion of the laser output. Operational parameters for best long-term stability were identified. Using realistic numbers for an active clock with 87Sr^{87}\mathrm{Sr}87Sr, we find that optimized stability of ${\ensuremath{\sigma}}_{y}(\ensuremath{\tau})\ensuremath{\approx}4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}/\sqrt{\ensuremath{\tau}[\mathrm{s}]}$ is achievable.
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.