Relativistic effects in atom gravimeters

Atom interferometry is currently developing rapidly, which is now reaching sufficient precision to motivate laboratory tests of general relativity. Thus, it is extremely significant to develop a general relativistic model for atom interferometers. In this paper, we mainly present an analytical derivation process and first give a complete vectorial expression for the relativistic interferometric phase shift in an atom interferometer. The dynamics of the interferometer are studied, where both the atoms and the light are treated relativistically. Then, an appropriate coordinate transformation for the light is performed crucially to simplify the calculation. In addition, the Bordé $ABCD$ matrix combined with quantum mechanics and the “perturbation” approach are applied to make a methodical calculation for the total phase shift. Finally, we derive the relativistic phase shift kept up to a sensitivity of the acceleration $\sim 10^{-14}{\mathrm{m}/\mathrm{s}}^2$ for a $10\text{−}\mathrm{m}$-long atom interferometer.

Figure 1

A space-time diagram of a light-pulse atom interferometer. The curves correspond to the motions of the atoms (red solid lines and blue dot-dashed lines represent the atoms with two hyperfine states, separately) and the photons (dark dashed lines). The laser lights used to manipulate the atom are incident from above (${\overrightarrow{k}}_2$) and below (${\overrightarrow{k}}_1$). (a) represents the case where the speed of light is considered as finite; (b) is the space-time diagram, which has been figured after the coordinate transformation for the control laser ${\overrightarrow{k}}_1$ in (b).

Figure 2

The space-time diagram of a light-pulse atom interferometer after the coordinate transformation for the control laser. $T$ is the pulse separation. ${\overrightarrow{k}}_{\alpha j}$ refers to the momentum transfer for the atoms in the $\alpha$ path at the $j$th pulse ($i=1,2,3$).