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This paper addresses the idea of the applicability of mathematics, using, as a case study, a construction and software package that partition the unit sphere into regions of equal area. The paper assesses the applicability of this construction and software by examining citing works, including papers, dissertations and software., Comment: 29 pages, 24 figures, accepted by Journal of Approximation Software

The next generation of space-based networks will contain optical clocks embedded within satellites. To fully realize the capabilities of such clocks, high-precision clock synchronization across the networks will be necessary. Current experiments have shown the potential for classical frequency combs to synchronize remote optical clocks over free-space. However, these classical combs are restricted in precision to the standard quantum limit. Quantum frequency combs, however, which exhibit quantum properties such as squeezing and entanglement, provide pathways for going beyond the standard quantum limit. Here, we present our perspective on the prospects for practical clock synchronization in space using both classical and quantum frequency combs. We detail the current outcomes achievable with a classical frequency comb approach to synchronization, before quantifying the potential outcomes offered by quantum frequency combs. Challenges to be overcome in deploying frequency combs in space are presented, and the implications of almost-perfect synchronization for future space-based applications and experiments discussed., Comment: 23 pages, 11 figures, perspective, comments welcome

In this effort, we demonstrate the performance of a highly stable time reference for the proposed Black Hole Explorer (BHEX) mission, a space-based extension to the Event Horizon Telescope (EHT) Very Long Baseline Interferometry (VLBI) project. This precision timing system is based on the use of a space-qualified, ultra-low noise laser developed as part of the Laser Interferometer Space Antenna (LISA) mission as the timing reference, and an optical frequency comb to transfer the stability of this laser to the microwave regime for instrumentation use. We describe the implementation of this system and experimental setup to characterize the stability performance. We present the results of this experiment that demonstrate the performance of this system meets requirements for the BHEX mission., Comment: To be published in the proceedings of SPIE Astronomical Telescopes + Instrumentation 2024

Optical frequency combs are an essential tool for precision metrology experiments ranging in application from remote spectroscopic sensing of trace gases to the characterization and comparison of optical atomic clocks for precision time-keeping and searches for physics beyond the standard model. Here we describe the architecture and fully characterize a telecom-band, self-modelocking frequency comb based on a free-space laser with an Er/Yb co-doped glass gain medium. The laser provides a robust and cost-effective alternative to Er:fiber laser based frequency combs, while offering stability and noise performance similar to Ti:sapphire laser systems. Finally, we demonstrate the Er/Yb:glass frequency comb's utility in high-stability frequency synthesis using two ultra-stable optical references at 1157 nm and 1070 nm and in low-noise photonic microwave generation by dividing these references to the microwave domain.

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.

We describe a high-performance, compact optical frequency standard based on a microfabricated Rb vapor cell and a low-noise, external cavity diode laser operating on the Rb two-photon transition at 778 nm. The optical standard achieves an instability of 1.8x10$^{-13}$/$\sqrt{\tau}$ for times less than 100 s and a flicker noise floor of 1x10$^{-14}$ out to 6000 s. At long integration times, the instability is limited by variations in optical probe power and the AC Stark shift. The retrace was measured to 5.7x10$^{-13}$ after 30 hours of dormancy. Such a simple, yet high-performance optical standard could be suitable as an accurate realization of the SI meter or, if coupled with an optical frequency comb, as a compact atomic clock comparable to a hydrogen maser., Comment: 5 pages, 5 figures

We use frequency comb-based optical two-way time-frequency transfer (O-TWTFT) to measure the optical frequency ratio of state-of-the-art ytterbium and strontium optical atomic clocks separated by a 1.5 km open-air link. Our free-space measurement is compared to a simultaneous measurement acquired via a noise-cancelled fiber link. Despite non-stationary, ps-level time-of-flight variations in the free-space link, ratio measurements obtained from the two links, averaged over 30.5 hours across six days, agree to $6\times10^{-19}$, showing that O-TWTFT can support free-space atomic clock comparisons below the $10^{-18}$ level.

Atomic clocks occupy a unique position in measurement science, exhibiting higher accuracy than any other measurement standard and underpinning six out of seven base units in the SI system. By exploiting higher resonance frequencies, optical atomic clocks now achieve greater stability and lower frequency uncertainty than existing primary standards. Here, we report frequency ratios of the $^{27}$Al$^+$, $^{171}$Yb and $^{87}$Sr optical clocks in Boulder, Colorado, measured across an optical network spanned by both fiber and free-space links. These ratios have been evaluated with measurement uncertainties between $6\times10^{-18}$ and $8\times10^{-18}$, making them the most accurate reported measurements of frequency ratios to date. This represents a critical step towards redefinition of the SI second and future applications such as relativistic geodesy and tests of fundamental physics., Comment: 51 pages, 12 figures, 6 tables

Optical atomic clocks are poised to redefine the SI second, thanks to stability and accuracy more than one hundred times better than the current microwave atomic clock standard. However, the best optical clocks have not seen their performance transferred to the electronic domain, where radar, navigation, communications, and fundamental research rely on less stable microwave sources. By comparing two independent optical-to-electronic signal generators, we demonstrate a 10 GHz microwave signal with phase that exactly tracks that of the optical clock phase from which it is derived, yielding an absolute fractional frequency instability of 1*10-18 in the electronic domain. Such faithful reproduction of the optical clock phase expands the opportunities for optical clocks both technologically and scientifically for time-dissemination, navigation, and long-baseline interferometric imaging., Comment: 19 page, 10 figures (including Supplementary Text)

We demonstrate Ramsey-Bord\'e (RB) atom interferometry for high performance laser stabilization with fractional frequency instability $<2 \times 10^{-16}$ for timescales between 10 and 1000s. The RB spectroscopy laser interrogates two counterpropagating $^{40}$Ca beams on the $^1$S$_0$ -- $^3$P$_1$ transition at 657 nm, yielding 1.6 kHz linewidth interference fringes. Fluorescence detection of the excited state population is performed on the (4s4p) $^3$P$_1$ -- (4p$^2$) $^3$P$_0$ transition at 431 nm. Minimal thermal shielding and no vibration isolation are used. These stability results surpass performance from other thermal atomic or molecular systems by one to two orders of magnitude, and further improvements look feasible.

We demonstrate a time scale based on a phase stable optical carrier that accumulates an estimated time error of $48\pm94$ ps over 34 days of operation. This all-optical time scale is formed with a cryogenic silicon cavity exhibiting improved long-term stability and an accurate $^{87}$Sr lattice clock. We show that this new time scale architecture outperforms existing microwave time scales, even when they are steered to optical frequency standards. Our analysis indicates that this time scale is capable of reaching a stability below $1\times10^{-17}$ after a few months of averaging, making timekeeping at the $10^{-18}$ level a realistic prospect., Comment: 13 pages, 12 Figures

We have measured the quadratic Zeeman coefficient for the ${^{1}S_{0} \leftrightarrow {^{3}P_{0}}}$ optical clock transition in $^{27}$Al$^{+}$, $C_{2}=-71.944(24)$~MHz/T$^{2}$, and the unperturbed hyperfine splitting of the $^{25}$Mg$^{+}$ $^{2}S_{1/2}$ ground electronic state, $\Delta W / h = 1~788~762~752.85(13)$~Hz, with improved uncertainties. Both constants are relevant to the evaluation of the $^{27}$Al$^{+}$ quantum-logic clock systematic uncertainty. The measurement of $C_{2}$ is in agreement with a previous measurement and a new calculation at the $1~\sigma$ level. The measurement of $\Delta W$ is in good agreement with a recent measurement and differs from a previously published result by approximately $2\sigma$. With the improved value for $\Delta W$, we deduce an improved value for the nuclear-to-electronic g-factor ratio $g_{I}/g_{J} = 9.299 ~308 ~313(60) \times 10^{-5}$ and the nuclear g-factor for the $^{25}$Mg nucleus $g_{I} = 1.861 ~957 ~82(28) \times 10^{-4}$. Using the values of $C_{2}$ and $\Delta W$ presented here, we derive a quadratic Zeeman shift of the $^{27}$Al$^{+}$ quantum-logic clock of $\Delta \nu / \nu = -(9241.8 \pm 3.7) \times 10^{-19}$, for a bias magnetic field of $B \approx 0.12$~mT., Comment: 7 pages, 5 figures, updated

A time scale is a procedure for accurately and continuously marking the passage of time. It is exemplified by Coordinated Universal Time (UTC), and provides the backbone for critical navigation tools such as the Global Positioning System (GPS). Present time scales employ microwave atomic clocks, whose attributes can be combined and averaged in a manner such that the composite is more stable, accurate, and reliable than the output of any individual clock. Over the past decade, clocks operating at optical frequencies have been introduced which are orders of magnitude more stable than any microwave clock. However, in spite of their great potential, these optical clocks cannot be operated continuously, which makes their use in a time scale problematic. In this paper, we report the development of a hybrid microwave-optical time scale, which only requires the optical clock to run intermittently while relying upon the ensemble of microwave clocks to serve as the flywheel oscillator. The benefit of using clock ensemble as the flywheel oscillator, instead of a single clock, can be understood by the Dick-effect limit. This time scale demonstrates for the first time sub-nanosecond accuracy for a few months, attaining a fractional frequency uncertainty of 1.45*10-16 at 30 days and reaching the 10-17 decade at 50 days, with respect to UTC. This time scale significantly improves the accuracy in timekeeping and could change the existing time-scale architectures.

The pursuit of ever more precise measures of time and frequency is likely to lead to the eventual redefinition of the second in terms of an optical atomic transition. To ensure continuity with the current definition, based on a microwave transition between hyperfine levels in ground-state $^{133}$Cs, it is necessary to measure the absolute frequency of candidate standards, which is done by comparing against a primary cesium reference. A key verification of this process can be achieved by performing a loop closure$-$comparing frequency ratios derived from absolute frequency measurements against ratios determined from direct optical comparisons. We measure the $^1$S$_0\!\rightarrow^3$P$_0$ transition of $^{171}$Yb by comparing the clock frequency to an international frequency standard with the aid of a maser ensemble serving as a flywheel oscillator. Our measurements consist of 79 separate runs spanning eight months, and we determine the absolute frequency to be 518 295 836 590 863.71(11) Hz, the uncertainty of which is equivalent to a fractional frequency of $2.1\times10^{-16}$. This absolute frequency measurement, the most accurate reported for any transition, allows us to close the Cs-Yb-Sr-Cs frequency measurement loop at an uncertainty of $<$3$\times10^{-16}$, limited by the current realization of the SI second. We use these measurements to tighten the constraints on variation of the electron-to-proton mass ratio, $\mu=m_e/m_p$. Incorporating our measurements with the entire record of Yb and Sr absolute frequency measurements, we infer a coupling coefficient to gravitational potential of $k_\mathrm{\mu}=(-1.9\pm 9.4)\times10^{-7}$ and a drift with respect to time of $\frac{\dot\mu}{\mu}=(5.3 \pm 6.5)\times10^{-17}/$yr.

Analog photonic links require high-fidelity, high-speed optical-to-electrical conversion for applications such as radio-over-fiber, synchronization at kilometer-scale facilities, and low-noise electronic signal generation. Photodetector nonlinearity is a particularly vexing problem, causing signal distortion and excess noise, especially in systems utilizing ultrashort optical pulses. Here we show that photodetectors designed for high power handling and high linearity can perform optical-to-electrical conversion of ultrashort optical pulses with unprecedented linearity over a large photocurrent range. We also show that the broadband, complex impedance of the circuit following the photodiode modifies the linearity significantly. By externally manipulating the circuit impedance, we extend the detector's linear range to higher photocurrents, with over 50 dB rejection of amplitude-to-phase conversion for photocurrents up to 40 mA. This represents a 1000-fold improvement over state-of-the-art photodiodes and significantly extends the attainable microwave power by a factor of four. As such, we eliminate the long-standing requirement in ultrashort pulse detection of precise tuning of the photodiode's operating parameters (average photocurrent, bias voltage or temperature) to coincide with a nonlinearity minimum. These results should also apply more generally to reduce nonlinear distortion in a range of other microwave photonics applications.

The quasi-Clifford algebras, and their Wedderburn structure and representation theory, as described by Gastineau-Hills in 1980 and 1982, should be better known, and have only recently been rediscovered. These algebras and their representation theory provide effective tools to address certain questions relating to plug-in constructions for Hadamard matrices. The key question addressed is: Given $\lambda$, a pattern of amicability / anti-amicability, with $\lambda_{j,k}=\lambda_{k,j}=\pm 1$, find a set of $n$ monomial $\{-1,0,1\}$ matrices $\{D_j\}$ of minimal order such that $$ D_j D_k^T - \lambda_{j,k} D_k D_j^T = 0 \quad (j \neq k). $$, Comment: 11 pages, 4 tables. Revised for publication. Accepted by 'Advances in Applied Clifford Algebras' as part of AGACSE 2018 IMECC -- UNICAMP (7th Conference on Applied Geometric Algebras in Computer Science and Engineering, Campinas Brazil)