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This paper outlines the roadmap towards the redefinition of the second, which was recently updated by the CCTF Task Force created by the CCTF in 2020. The main achievements and the open challenges related to the status of the optical frequency standards, their contribution to time scales and UTC, the possibility of their comparison and the knowledge of the Earth's gravitational potential at the necessary level of uncertainty are discussed. In addition, the mandatory criteria to be achieved before redefinition and their current fulfilment level, together with the redefinition options based on a single or on a set of transitions are described., Comment: 26 pages. This paper is based on the work of a CCTF Task Force on the roadmap to the redefinition of the second

We carried out a 26-day comparison of five simultaneously operated optical clocks and six atomic fountain clocks located at INRIM, LNE-SYRTE, NPL and PTB by using two satellite-based frequency comparison techniques: broadband Two-Way Satellite Time and Frequency Transfer (TWSTFT) and Global Positioning System Precise Point Positioning (GPS PPP). With an enhanced statistical analysis procedure taking into account correlations and gaps in the measurement data, combined overall uncertainties in the range of $1.8 \times 10^{-16}$ to $3.5 \times 10^{-16}$ for the optical clock comparisons were found. The comparison of the fountain clocks yields results with a maximum relative frequency difference of $6.9 \times 10^{-16}$, and combined overall uncertainties in the range of $4.8 \times 10^{-16}$ to $7.7 \times 10^{-16}$.

Mid infrared (MIR) photonics is a key region for molecular physics [1]. High-resolution spectroscopy in the 1--10 {\mu}m region, though, has never been fully tackled for the lack of widely-tunable and practical light sources. Indeed, all solutions proposed thus far suffer from at least one of three issues: they are feasible only in a narrow spectral range; the power available for spectroscopy is limited; the frequency accuracy is poor. Here, we present a setup for high-resolution spectroscopy that can be applied in the whole 1--10 {\mu}m range by combining the power of quantum cascade lasers (QCLs) and the accuracy achievable by difference frequency generation using an OP-GaP crystal. The frequency is measured against a primary frequency standard using the Italian metrological fibre link network. We demonstrate the performance of the setup by measuring a vibrational transition in a highly-excited metastable state of CO around 6 {\mu}m with 11 digits of precision, four orders of magnitude better than the value available in the literature [2]., Comment: 6 pages, 3 figures

We demonstrate a novel way of synthesizing spin-orbit interactions in ultracold quantum gases, based on a single-photon optical clock transition coupling two long-lived electronic states of two-electron $^{173}$Yb atoms. By mapping the electronic states onto effective sites along a synthetic "electronic" dimension, we have engineered synthetic fermionic ladders with tunable magnetic fluxes. We have detected the spin-orbit coupling with fiber-link-enhanced clock spectroscopy and directly measured the emergence of chiral edge currents, probing them as a function of the magnetic field flux. These results open new directions for the investigation of topological states of matter with ultracold atomic gases., Comment: Minor changes with respect to v1 (we have corrected some typos, fixed the use of some mathematical symbols, added one reference)

Four-wave mixing (4WM) is a known source of intense non-classical twin beams. It can be generated when an intense laser beam (the pump) and a weak laser beam (the seed) overlap in a $\chi^{(3)}$ medium (here cesium vapor), with frequencies close to resonance with atomic transitions. The twin beams generated by 4WM have frequencies naturally close to atomic transitions, and can be intense (gain $\gg 1$) even in the CW pump regime, which is not the case for PDC $\chi^{(2)}$ phenomenon in non-linear crystals. So, 4WM is well suited for atom-light interaction and atom-based quantum protocols. Here we present the first realization of a source of 4-wave mixing exploiting $D_2$ line of Cesium atoms., Comment: 10 pages, 10 figures

Global Positioning System (GPS) dissemination of frequency standards is ubiquitous at present, providing the most widespread time and frequency reference for the majority of industrial and research applications worldwide. On the other hand, the ultimate limits of the GPS presently curb further advances in high-precision, scientific and industrial applications relying on this dissemination scheme. Here, we demonstrate that these limits can be reliably overcome even in laboratories without a local atomic clock by replacing the GPS with a 642-km-long optical fiber link to a remote primary caesium frequency standard. Through this configuration we stably address the $^1$S$_0$---$^3$P$_0$ clock transition in an ultracold gas of $^{173}$Yb, with a precision that exceeds the possibilities of a GPS-based measurement, dismissing the need for a local clock infrastructure to perform high-precision tasks beyond GPS limit. We also report an improvement of two orders of magnitude in the accuracy on the transition frequency reported in literature.

It is well known that temperature variations and acoustic noise affect ultrastable frequency dissemination along optical fiber. Active stabilization techniques are in general adopted to compensate for the fiber-induced phase noise. However, despite this compensation, the ultimate link performances remain limited by the so called delay-unsuppressed fiber noise that is related to the propagation delay of the light in the fiber. In this paper, we demonstrate a data post-processing approach which enables us to overcome this limit. We implement a subtraction algorithm between the optical signal delivered at the remote link end and the round-trip signal. In this way, a 6 dB improvement beyond the fundamental limit imposed by delay-unsuppressed noise is obtained. This result enhances the resolution of possible comparisons between remote optical clocks by a factor of 2. We confirm the theoretical prediction with experimental data obtained on a 47 km metropolitan fiber link, and propose how to extend this method for frequency dissemination purposes as well.

To significantly improve the frequency references used in radio-astronomy and precision measurements in atomic physics, we provide frequency dissemination through a 642 km coherent optical fiber link, that will be also part of a forthcoming European network of optical links. We obtained a resolution of 3e-19 at 1000 s on the frequency transfer, and an accuracy of 5e-19. The ultimate link performance has been evaluated by doubling the link to 1284 km, demonstrating a new characterization technique based on the double round-trip on a single fiber. The arming of a second fiber is avoided: this is beneficial to long hauls realizations in view of a continental fiber network for frequency and time metrology. The data analysis is based on the Allan deviation; its expression is theoretically derived for the observed noise power spectrum, which is seldom found in the literature.

The use of ultra-precise optical clocks in space ("master clocks") will allow for a range of new applications in the fields of fundamental physics (tests of Einstein's theory of General Relativity, time and frequency metrology by means of the comparison of distant terrestrial clocks), geophysics (mapping of the gravitational potential of Earth), and astronomy (providing local oscillators for radio ranging and interferometry in space). Within the ELIPS-3 program of ESA, the "Space Optical Clocks" (SOC) project aims to install and to operate an optical lattice clock on the ISS towards the end of this decade, as a natural follow-on to the ACES mission, improving its performance by at least one order of magnitude. The payload is planned to include an optical lattice clock, as well as a frequency comb, a microwave link, and an optical link for comparisons of the ISS clock with ground clocks located in several countries and continents. Undertaking a necessary step towards optical clocks in space, the EU-FP7-SPACE-2010-1 project no. 263500 (SOC2) (2011-2015) aims at two "engineering confidence", accurate transportable lattice optical clock demonstrators having relative frequency instability below 1\times10^-15 at 1 s integration time and relative inaccuracy below 5\times10^-17. This goal performance is about 2 and 1 orders better in instability and inaccuracy, respectively, than today's best transportable clocks. The devices will be based on trapped neutral ytterbium and strontium atoms. One device will be a breadboard. The two systems will be validated in laboratory environments and their performance will be established by comparison with laboratory optical clocks and primary frequency standards. In this paper we present the project and the results achieved during the first year., Comment: Contribution to European Frequency and Time Forum 2012, Gothenburg, Sweden

Caesium fountain frequency-standards realize the second in the International System of Units with a relative uncertainty approaching 10^-16. Among the main contributions to the accuracy budget, cold collisions play an important role because of the atomic density shift of the reference atomic transition. This paper describes an application of the Bayesian analysis of the clock frequency to estimate the density shift and describes how the Bayes theorem allows the a priori knowledge of the sign of the collisional coefficient to be rigourously embedded into the analysis. As an application, data from the INRIM caesium fountain are used and the Bayesian and orthodox analyses are compared. The Bayes theorem allows the orthodox uncertainty to be reduced by 28% and demonstrates to be an important tool in primary frequency-metrology., Comment: 13 pages, 5 figures, submitted to Metrologia, application of the Bayes theorem to metrology

In this paper the Bayesian analysis is applied to assign a probability density to the value of a quantity having a definite sign. This analysis is logically consistent with the results, positive or negative, of repeated measurements. Results are used to estimate the atom density shift in a caesium fountain clock. The comparison with the classical statistical analysis is also reported and the advantages of the Bayesian approach for the realization of the time unit are discussed., Comment: 10 pages, 4 figures, submitted to Metrologia

We report the theoretical evaluations of the static scalar polarizability of the 133Cs ground state and of the black body radiation shift induced on the transition frequency between the two hyperfine levels with m_F = 0. This shift is of fundamental importance in the evaluation of the accuracy of the primary frequency standards based on atomic fountains and employed in the realization of the SI second in the International Atomic Time (TAI) scale at the level of 1e-15. Our computed value for the polarizability is alpha_0=6.600(16)e-39 Cm^2/V in agreement at the level of 1e-3 with recent theoretical and experimental values. As regards the black body radiation shift we .nd for the relative hyper.ne transition frequency beta=-1.49 (7)e-14 at T = 300 K in agreement with frequency measurements reported by our group and by Bauch and Schroder [Phys. Rev. Lett. 78, 622, (1997)]. This value is lower by 2e-15 than that obtained with measurements based on the dc Stark shift and than the value commonly accepted up to now., Comment: 18 pages, 2 figures, 6 tables

We used a Cs atomic fountain frequency standard to measure the Stark shift on the ground state hyperfine transiton frequency in cesium (9.2 GHz) due to the electric field generated by the blackbody radiation. The measures relative shift at 300 K is -1.43(11)e-14 and agrees with our theoretical evaluation -1.49(07)e-14. This value differs from the currently accepted one -1.69(04)e-14. The difference has a significant implication on the accuracy of frequency standards, in clocks comparison, and in a variety of high precision physics tests such as the time stability of fundamental constants., Comment: 4 pages, 2 figures, 2 tables

Over five years we have compared the hyperfine frequencies of 133Cs and 87Rb atoms in their electronic ground state using several laser cooled 133Cs and 87Rb atomic fountains with an accuracy of ~10^{-15}. These measurements set a stringent upper bound to a possible fractional time variation of the ratio between the two frequencies : (d/dt)ln(nu_Rb/nu_Cs)=(0.2 +/- 7.0)*10^{-16} yr^{-1} (1 sigma uncertainty). The same limit applies to a possible variation of the quantity (mu_Rb/mu_Cs)*alpha^{-0.44}, which involves the ratio of nuclear magnetic moments and the fine structure constant., Comment: 4 pages, 3 figures, 1 table submitted to Phys. Rev. Lett

At BNM-SYRTE laboratories in Paris two 133Cs fountains and a 87Rb one have been already demonstrated. This paper illustrates the construction of a 87Rb and Cs dual fountain realized within the same structure to work simultaneously. Up to now only the Cs part is under operation but Rb part, already completed, will be soon operational. The experiment has been projected to improve present fountain frequency standards and to perform a more precise laboratory test (at 10E-16 level) on a possible time variation of the fine structure constant alpha, as recent cosmological evidences strongly motivates., Comment: 3 pages, 7 figures, 1 table