We have observed the effects of magnetic field gradients and gravitational acceleration on grating echoes in a time-domain single state atom interferometer that uses laser cooled Rb atoms. These observations are compared to theoretical predictions based on a simplified model. The oscillatory dependence of the echo amplitude due to the magnetic field gradient is in agreement with the predicted quadratic scaling as a function of the time between excitation pulses. We also observe a linear dependence of this oscillation frequency as a function of the magnetic field gradient which is predicted by theory. In the presence of gravity, the calculations predict a quadratic dependence for the echo phase on the time between excitation pulses as well as a change in the shape of the echo envelope. We have observed both of these effects in the experiment, and we find that the change in shape is qualitatively consistent with our prediction. It is necessary to understand these effects in order to carry out high precision studies of the atomic fine-structure constant and gravitational acceleration using this interferometric technique. We also present an improved measurement of gravitational acceleration using this technique that is precise to $\ensuremath{\sim}15\phantom{\rule{0.3em}{0ex}}\mathrm{ppm}$ by exploiting the quadratic phase dependence.
We have developed two configurations of an echo interferometer that rely on standing-wave excitation of a laser-cooled sample of rubidium atoms. Both configurations can be used to measure acceleration
a along the axis of excitation. For a two-pulse configuration, the signal from the interferometer is modulated at the recoil frequency and exhibits a sinusoidal frequency chirp as a function of pulse spacing. In comparison, for a three-pulse stimulated-echo configuration, the signal is observed without recoil modulation and exhibits a modulation at a single frequency as a function of pulse spacing. The three-pulse configuration is less sensitive to effects of vibrations and magnetic field curvature, leading to a longer experimental time scale. For both configurations of the atom interferometer (AI), we show that a measurement of acceleration with a statistical precision of 0.5% can be realized by analyzing the shape of the echo envelope that has a temporal duration of a few microseconds. Using the two-pulse AI, we obtain measurements of acceleration that are statistically precise to 6 parts per million (ppm) on a 25 ms time scale. In comparison, using the three-pulse AI, we obtain measurements of acceleration that are statistically precise to 0.4 ppm on a time scale of 50 ms. A further statistical enhancement is achieved by analyzing the data across the echo envelope so that the statistical error is reduced to 75 parts per billion (ppb). The inhomogeneous field of a magnetized vacuum chamber limited the experimental time scale and resulted in prominent systematic effects. Extended time scales and improved signal-to-noise ratio observed in recent echo experiments using a nonmagnetic vacuum chamber suggest that echo techniques are suitable for a high-precision measurement of gravitational acceleration
g. We discuss methods for reducing systematic effects and improving the signal-to-noise ratio. Simulations of both AI configurations with a time scale of 300 ms suggest that an optimized experiment with improved vibration isolation and atoms selected in the
mF=0 state can result in measurements of
g statistically precise to 0.3 ppb for the two-pulse AI and 0.6 ppb for the three-pulse AI.