Atom interferometers in weakly curved spacetimes using Bragg diffraction and Bloch oscillations

We present a systematic approach to determine all relativistic phases up to $\mathcal{O}(c^{-2})$ in light-pulse atom interferometers in weakly curved spacetime that are based on elastic scattering—namely, Bragg diffraction and Bloch oscillations. Our analysis is derived from first principles using the parametrized post-Newtonian formalism. In the treatment developed here, we derive algebraic expressions for relativistic phases for arbitrary interferometer geometries in an automated manner. As case studies, we consider symmetric and antisymmetric Ramsey-Bordé interferometers, as well as a symmetric double diffraction interferometer with baseline lengths of 10 m and 100 m. We compare our results to previous calculations conducted for a Mach-Zehnder interferometer.

Figure 1

Schematic pictures of atomic trajectories (green lines) for three different IF geometries in the freely falling frame. Bragg pulses are depicted in red dashed lines with a momentum transfer of $\pm \hslash k_R$ and Bloch oscillations in solid blue with a momentum transfer of $\hslash k_B$. Finite-speed-of-light effects are neglected in this picture. (a) Symmetric Ramsey-Bordé Interferometer (SRBI), (b) Symmetric Double Diffraction Interferometer (SDDI), (c) Asymmetric Ramsey-Bordé Interferometer (ARBI). Note that (a) and (c) can be realized using single Bragg diffraction, whereas (b) relies on double Bragg diffraction.

Figure 2

Phase shift contributions in the three IF geometries SRBI, SDDI, and ARBI for 10 m and 100 m baselines. Solid curves correspond to phase shifts of order $\mathcal{O}(2)$ and dashed curves to $\mathcal{O}(3)$ shifts. (a)–(f) The Bloch time $T_B$ is set to zero, and all nonzero phase shift contributions are plotted with respect to time $T=2T_R$. (f)–(l) Colored phase shift contributions depend nontrivially on $T_B$ and are plotted against $T_B$ for a fixed time $T=2T_R+T_B$ of 3 seconds (9 seconds) for the 10 m (100 m) baseline; the gray curves correspond to the Bloch-time-independent phase contributions from (a)–(f). In addition to the numerical values in Table 1, the plots assume numerical values of $g=9.81{\mathrm{ms}}^{-2}$ and $\mathrm{\Gamma }=1.54\times {10}^{-6}{\mathrm{s}}^{-2}$.

Figure 3

Graphical description of interaction points (black dots) of photon paths (red dashed lines) and atoms (green solid lines) in the laboratory frame. The photon flight times are then defined by the emission height of $z_{\mathrm{up}}$ or $z_{\mathrm{low}}$ and the interaction points.