Theory of the cold collision frequency shift in 1S--2S spectroscopy of Bose-Einstein-condensed and non-condensed hydrogen

We show that a correct formulation of the cold collision frequency shift for two photon spectroscopy of Bose-condensed and cold non-Bose-condensed hydrogen is consistent with experimental data. Our treatment includes transport and inhomogeneity into the theory of a non-condensed gas, which causes substantial changes in the cold collision frequency shift for the ordinary thermal gas, as a result of the very high frequency (3.9kHz) of transverse trap mode. For the condensed gas, we find substantial corrections arise from the inclusion of quasiparticles, whose number is very large because of the very low frequency (10.2Hz) of the longitudinal trap mode. These two effects together account for the apparent absence of a "factor of two" between the two possibilities. Our treatment considers only the Doppler-free measurements, but could be extended to Doppler-sensitive measurements. For Bose-condensed hydrogen, we predict a characteristic "foot" extending into higher detunings than can arise from the condensate alone, as a result of a correct treatment of the statistics of thermal quasiparticles.