  A technique is demonstrated that allows free space atomic fountain clocks and
interferometers to utilize optical cavity generated spin-squeezed states with
over $390\,000$ ${}^{87}\text{Rb}$ atoms. Fluorescence imaging is used for
population spectroscopy, after a free fall time of 4 milliseconds, to resolve a
single-shot phase sensitivity of $814 (61)$ microradians, which is $5.8(0.6)$
decibels (dB) below the quantum projection limit. The dynamic range is observed
to be 100 milliradians. When operating as a microwave atomic clock with
$240\,000$ atoms at a 3.6 ms Ramsey time, a single-shot fractional frequency
stability of $8.4(0.2)\times10^{-12}$ is reported, $3.8(0.2)$ dB below the
quantum projection limit.
