We give an overview of the work done with the Laboratoire National de Métrologie et d'Essais-Systèmes de Référence Temps-Espace (LNE-SYRTE) fountain ensemble during the last five years. After a description of the clock ensemble, comprising three fountains, FO1, FO2, and FOM, and the newest developments, we review recent studies of several systematic frequency shifts. This includes the distributed cavity phase shift, which we evaluate for the FO1 and FOM fountains, applying the techniques of our recent work on FO2. We also report calculations of the microwave lensing frequency shift for the three fountains, review the status of the blackbody radiation shift, and summarize recent experimental work to control microwave leakage and spurious phase perturbations. We give current accuracy budgets. We also describe several applications in time and frequency metrology: fountain comparisons, calibrations of the international atomic time, secondary representation of the SI second based on the (87)Rb hyperfine frequency, absolute measurements of optical frequencies, tests of the T2L2 satellite laser link, and review fundamental physics applications of the LNE-SYRTE fountain ensemble. Finally, we give a summary of the tests of the PHARAO cold atom space clock performed using the FOM transportable fountain.

We evaluate the distributed cavity phase and microwave lensing frequency shifts, which were the two largest sources of uncertainty for the NPL-CsF2 cesium fountain clock. We report measurements that confirm a detailed theoretical model of the microwave cavity fields and the frequency shifts of the clock that they produce. The model and measurements significantly reduce the distributed cavity phase uncertainty to 1.1 × 10 −16 . We derive the microwave lensing frequency shift for a cylindrical cavity with circular apertures. An analytic result with reasonable approximations is given, in addition to a full calculation that indicates a shift of 6.2 × 10 −17 . The measurements and theoretical models we report, along with improved evaluations of collisional and microwave leakage induced frequency shifts, reduce the frequency uncertainty of the NPL-CsF2 standard to 2.3 × 10 −16 , nearly a factor of two lower than its most recent complete evaluation.

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We perform 3D finite element calculations of the fields in microwave cavities and analyze the distributed cavity phase errors of atomic clocks that they produce. The fields of cylindrical cavities are treated as an azimuthal Fourier series. Each of the lowest components produces clock errors with unique characteristics that must be assessed to establish a clock's accuracy. We describe the errors and how to evaluate them. We prove that sharp structures in the cavity do not produce large frequency errors, even at moderately high powers, provided the atomic density varies slowly. We model the amplitude and phase imbalances of the feeds. For larger couplings, these can lead to increased phase errors. We show that phase imbalances produce a novel distributed cavity phase error that depends on the cavity detuning. We also design improved cavities by optimizing the geometry and tuning the mode spectrum so that there are negligible phase variations, allowing this source of systematic error to be dramatically reduced. ( ), , r z ρ φ = [10]. This led to an analytic solution for cavities 2 with no endcap holes, giving insight as well as an often accurate description of the phase in the center of real cavities. This solution showed that endcap losses produce the large longitudinal phase variation and a radial phase variation of order 5 μrad, which cannot be reduced by adding more azimuthally distributed feeds. The Fourier series in φ also enabled efficient finite element solutions for cavities with holes in the endcaps using dense meshes. We presented the azimuthally symmetric (m=0) fields in [10] and a key result was that the fields and the losses are very large near the apertures of the endcap holes -the phase of the field approaches π/4 [10]. The existence of large phase shifts in the cavity is inconsistent with the premise of the early accuracy evaluations of fountain clocks [4], as well as some more recent evaluations [11,12]. In 2005, two groups reported calculations of the power dependence of DCP errors due to azimuthally symmetric (m=0) longitudinal phase variations [13,14]. We used our finite element solutions from [10]. Ref [14] instead used a phenomenological field that is not a solution of Maxwell's equations. The m=0 phase variation that they analyzed is of order 100 times smaller than the dominant m=0 DCP error, has a dramatically different shape as a function of power, and arises from different physical effects [13]. It is important to note that this power dependent template is used by a number of groups to evaluate their clock's DCP uncertainty [3,12], including the NIST and INRIM clocks which report DCP uncertainties smaller than 3×10 −17 [15,16], an order of magnitude smaller than those for other clocks. Another recent treatment used an incomplete mode expansion that does not satisfy Maxwell's equations at 9.2 GHz or the boundary conditions [2]. A number of DCP uncertainty estimates no doubt include some element of intuition, with a goal of being conservative. Except for the NIST and INRIM clocks, the DCP ...

We analyse the phase variations of the microwave field in a TE 011 microwave cavity and how these variations affect the frequency of an atomic clock. We analytically solve for the microwave fields in a TE 011 cavity. These analytic solutions show significant new terms that are not present in previous two-dimensional treatments. The new terms show that cavities with small radii, near 2.1 cm for a 9.2 GHz cavity, have smaller phase shifts than cavities with larger radii. We also show that the three-dimensional phase variations near the axis of the cavity can be efficiently calculated with a rapidly converging series of two-dimensional finite element calculations. The cavities used in atomic clocks have holes in the endcaps, and we use finite element methods to study the large fields and phase shifts near these holes. The effects of the phase variations on atoms traversing a cavity are analysed using the sensitivity function, and we present a cavity design that has small phase shifts for all atomic trajectories. For two π/2 pulses, the proposed cavity has transverse variations of the effective phase that are within ±0.1 µrad and produce no systematic frequency error for a nearly homogeneous and expanding cloud of atoms.