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

We demonstrate agreement between measurements and ab initio calculations of the frequency shifts caused by distributed cavity phase variations in the microwave cavity of a primary atomic fountain clock. Experimental verification of the finite element models of the cavities gives the first quantitative evaluation of this leading uncertainty and allows it to be reduced to $\delta \nu /\nu =\pm 8.4\times {10}^{-17}$. Applying these experimental techniques to clocks with improved microwave cavities will yield negligible distributed cavity phase uncertainties, less than $\pm 1\times {10}^{-17}$.

Figure 1

(a) Schematic of distributed cavity phase (DCP) frequency shifts. Feeding a microwave cavity from only one side produces a phase gradient in the cavity and gives a Doppler shift if the atomic fountain is tilted. (b) Frequency differences for feeding at $\varphi =0$ or $\pi$ versus tilt along the cavity feeds for 1, 3, 5, and $7\pi /2$ pulses. All differences are 0 when the fountain has no tilt. (c) The black circles are the frequency differences for $5\pi /2$ pulses and $\pi /2$ pulses versus tilt perpendicular to the feeds, suggesting an inhomogeneous surface resistance. The horizontal dotted line is the predicted $m=0$ DCP shift. The red squares are the differences of the Cs and Rb clock frequencies for $\pi /2$ pulses, which are nonetheless consistent with no $m=1$ DCP shift for perpendicular tilts.

Figure 2

Measured (squares) and calculated (lines) change in transition probability $\delta P$ from DCP shifts versus microwave amplitude $b$, where $1,3,5,\dots$ $\pi /2$ pulses (black dots) are near $b=1,3,5,\dots$ [10]. (a) Azimuthally symmetric $m=0$ DCP shifts produce a negligible shift at $b=1$ (left inset) and have a large amplitude dependence. In the top inset, the DCP frequency shift is large and singular near $b=4$ and 8 ($2\pi$ and $4\pi$ pulses). (b) $m=1$ DCP shifts, given by the clock’s frequency difference between feeding the cavity at $\varphi =0$ and $\pi$ with a fountain tilt of 1.6 mrad. Balancing the feeds and nulling the tilt minimizes this shift. (c) Predicted $m=2$ DCP shift for FO2 with an effective 9.9 mm detection laser beam waist (solid line) and an initial cloud offset of 2 mm at launch with uniform detection (dashed line).

Figure 3

Normalized frequency shift versus phase imbalance of the cavity feeds for a 1.6 mrad tilt along the feeds. As the cavity is tuned to resonance (black curve), the DCP shift goes to 0. The lines are fits to $A\tan (\Delta \psi /2)$, and $A$ agrees with the prediction $\Delta \omega /\Gamma$.