Finite-speed-of-light perturbation in atom gravimeters

The finite-speed-of-light (FSL) effect is a systematic error in atom gravimeters arising from the time delay due to the propagation of the light. It includes the frequency-chirp-independent part and the frequency-chirp-dependent part, which were not considered completely. The FSL effect in atom gravimeters is different from that in corner-cube absolute gravimeters. In the past, this effect has been widely studied in corner-cube absolute gravimeters, whereas little has been discussed about and done with atom gravimeters. In this paper, we mainly propose a complete analytical study based on a coordinate transformation and on a “perturbation” approach to estimate this effect in an atom gravimeter. This also offers the potential to calculate the general relativistic effects in atom gravimeters. In addition, a comparison with a crude “average-path” analysis is given for a particular case of the FSL effect in atom gravimeters.

Figure 1

The schematic for the interaction between the atom and Raman pulses. The green part represents the atom, and the upward (downward) light with wave vector ${\vec{k}}_1$ (${\vec{k}}_2$) is chosen by the particular frequency chirps to form the Raman light.

Figure 2

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 separate hyperfine states) and the photons (black dashed lines). The laser lights used to manipulate the atom are incident from above (${\vec{k}}_2$) and below (${\vec{k}}_1$). (a) and (b) represent the cases where the speed of light is considered to be infinite and finite, respectively; (c) is the space-time diagram, which has been figured after the coordinate transformation for the control light ${\vec{k}}_1$ in (b).

Figure 3

The space-time diagram of a light-pulse atom interferometer after the coordinate transformation for control light with wave vector ${\vec{k}}_1$. $T$ is the pulse separation. ${\vec{k}}_{\alpha j}$ refers to the momentum transfer for the atoms in the $\alpha$ path at the $j\mathrm{th}$ pulse ($i=1,2,3$).

Figure 4

Perturbed and unperturbed paths used to calculate the perturbative phase shift. The upper one denoted by ${\mathrm{\Gamma }}_{\mathrm{cl}}^{\prime }$ represents the perturbative path, and the lower one ${\mathrm{\Gamma }}_{\mathrm{cl}}^{\left(0\right)}$ represents the unperturbative path.

Figure 5

The space-time diagram for the atoms with an unperturbative Lagrangian $L_0^{\prime }$. The upper one denoted by ${\mathrm{\Gamma }}_{\mathrm{cl}}^{\beta }\left(L_0^{\prime }\right)$ represents the unperturbative trajectory for the atoms in the $\beta$ path, while the lower one denoted by ${\mathrm{\Gamma }}_{\mathrm{cl}}^{\alpha }\left(L_0^{\prime }\right)$ is for the atoms in the $\alpha$ path.

Figure 6

The space-time diagram of a light-pulse atom interferometer (a) before and (b) after the crude average-path treatment.