Microwave lensing frequency shift of the PHARAO laser-cooled microgravity atomic clock

We evaluate the microwave lensing frequency shift of the microgravity laser-cooled caesium clock PHARAO. We find microwave lensing frequency shifts of δ_ν/ν = 11 * 10^-17 to 13 * 10^-17, larger than the shift of typical fountain clocks. The shift has a weak dependence on PHARAO parameters, including the atomic temperature, size of the atomic cloud, detection laser intensities, and the launch velocity. We also find the lensing frequency shift to be insensitive to selection and detection spatial inhomogeneities and the expected low-frequency vibrations. We conservatively assign a nominal microwave lensing frequency uncertainty of ±4 * 10^-17. (authors)