to be measured easily and precisely. The contrast is accurately described by a coherence function, which is the Fourier transform of the momentum distribution, produced by the third pulse and by the theory of echo formation. If the third pulse is a traveling wave, loss of grating contrast is observed, an effect also described by a coherence function. The decay of the grating contrast as a function of continuous-wave light intensity is used to infer the cross section for photon absorption.

DOI: 10.1103/physreva.79.021605

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arXiv:cond-mat/0202452Physical Review LettersSep 10, 2002165 citations

{\ensuremath{\omega}}_{r}

as a function of the time at which the third pulse is applied, allowing

{\ensuremath{\omega}}_{r}

{10}^{9}

(ppb) on a time scale of $\ensuremath{\sim}$45 ms in 14 h. Further statistical improvements are anticipated by extending this time scale and narrowing the signal fringe width. However, the total systematic uncertainty is estimated to be $\ensuremath{\sim}$6 parts per

{10}^{6}

(ppm). We describe methods of reducing these systematic errors.

t=0

and

t=T

to diffract and recombine momentum states separated by $2\ensuremath{\hbar}k$ at

t=2T

. Matter wave interference is associated with the formation of a density grating with period $\ensuremath{\lambda}/2$ in the vicinity of this echo time. The grating contrast is measured by recording the intensity of coherently backscattered light. The interferometer is perturbed by an additional pulse applied at $t=2T\ensuremath{-}\ensuremath{\delta}T$ or by continuous-wave background light. If the additional pulse is a standing wave, the momentum states interfering at

t=2T

are displaced and the grating contrast can be completely recovered due to constructive interference. In this case, the contrast shows a periodic modulation at the atomic recoil frequency as a function of $\ensuremath{\delta}T$. In a recent work, it was shown that the atomic recoil frequency can be measured easily and precisely when using coherence functions to model the signal shape. This paper provides an alternative description of the signal shape through an analytical calculation of echo formation in the presence of an additional standing-wave pulse. Using this treatment, it is possible to model the effects of spontaneous emission and spatial profile of the laser beam on the signal shape. Additionally, the theory predicts scaling laws as a function of the pulse area and the number of additional standing-wave pulses. These scaling laws are investigated experimentally and can be exploited to improve precision measurements of the atomic recoil frequency. We also show that coherence functions can be used to make a direct measurement of the populations of momentum states associated with the ground state under conditions where the Doppler-broadened velocity distribution of the sample is much larger than the recoil velocity. These measurements are consistent with Monte Carlo wave-function simulations. If the additional pulse is a traveling wave, we find that the grating contrast measured as a function of $\ensuremath{\delta}T$ can be modeled by a quasiperiodic coherence function as in previous experiments that utilized atomic beams. In this work, we investigate the dependence of the photon scattering rate on the intensity and detuning of the traveling wave. We also study the effects of perturbing the interferometer with continuous-wave light and find that the dependence of the photon scattering rate on the intensity and detuning of the perturbing field is consistent with expectations.