Evaluating and minimizing distributed cavity phase errors in atomic clocks

Abstract

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 ...

Contributing to TAI with a secondary representation of the SI second

“…We evaluate this shift by applying the approach established in [52] for FO2-Cs, which is based on a theoretical model of the cavity field developed in [53] [54]. In this approach, the leading contributions from the lowest azimuthal terms m = 0, m = 1 and m = 2 of the phase distribution in the cavity are considered.…”

Evaluation of Doppler Shifts to Improve the Accuracy of Primary Atomic Fountain Clocks

“…shifts [1][2][3][4][5][6][7]. These Doppler shifts occur when the cold but moving atoms interact with a field inside the microwave cavity that has a spatial phase variation (Fig.…”

“…1a, and the purity of the standing wave. Uncertainty estimates are as low as ±3×10 −16 , excepting a few cases where it is incorrectly evaluated [1][2][3][4][5][6][7]. Here, we present precise measurements of a primary clock's frequency and large [8].…”

“…Frequency measurements have given indirect information about the phase variations, but often the associated frequency shifts were misattributed to other systematic errors [5]. Recently, a combination of finite element and analytic models have elucidated the behaviors of the fields and their DCP shifts [7,9].…”

“…This requires dense meshing, making direct 3D solutions unfeasible. However, the cylindrical symmetry allows the field to be expressed as an azimuthal Fourier series of 2D finite element solutions [7,9]. We write the microwave magnetic field as the sum of a large standing wave H 0 (r) and a small field, g(r)=Σg m (ρ,z) cos(mφ), which describes the cavity feeds and wall losses [7].…”