Atom-interferometric test of the universality of gravitational redshift and free fall

Introducing internal-state transitions simultaneously on each branch of a light-pulse atom interferometer, we propose a scheme that is concurrently sensitive to both violations of the universality of free fall and gravitational redshift, two premises of general relativity. In contrast to redshift tests with quantum clocks, a superposition of internal states is not necessary but merely transitions between them, leading to a generalized concept of clocks in this context. The experimental realization seems feasible with already demonstrated techniques.

Figure 1

Redshift-sensitive geometry. The interferometer is based on a diffraction scheme where each laser pulse, apart from transferring momentum, changes the internal state. Initially, the input wave packet is split into two components by a $\pi /2$ pulse, which move in opposite directions with momentum $\pm \hslash k$, where $k$ is the effective wave number of the laser fields. At the second and third laser pulse the wave packets are redirected and finally recombined at the fourth laser pulse by further $\pi /2$ pulses. If initially in the ground state $m_{-}$ (green), the atoms reside in the excited state $m_{+}$ (blue) during the first and third segment but occupy the ground state during the central segment of the interferometer. The atoms are always in the same internal state at equal times (in the laboratory frame).

Figure 2

Origin of the clock phase. We compare clock phases accumulated between the upper branch (violet arrow) and lower branch (red arrow) along the three segments of the interferometer. The accumulation of phases on each branch is depicted by a rotation on the equatorial plane, where the Compton frequency $m{c}^2/\hslash$ is chosen as a reference. Hence, atoms in the excited state rotate counterclockwise with $\mathrm{\Omega }/2$ around the blue axial vector, whereas the axial vector flips when the atoms are in the ground state (green vector) so that they rotate clockwise with $-\mathrm{\Omega }/2$. (a) If the internal state is not changed during the central segment, the sum of all proper-time differences vanishes so that the clock phase $\mathrm{\Omega }\mathrm{\Delta }\tau =0$. (b) If the internal state flips during the central segment, the phase difference does not vanish although $\mathrm{\Delta }\tau =0$. Instead, the phases accumulated during each segment add, and one finds the clock phase $\mathrm{\Omega }\mathrm{\Delta }{\tau }_2$, represented by the shaded area. Because the atoms fall during the central segment in parallel at different heights, the phase is proportional to the gravitational redshift.

Figure 3

Energy-momentum diagrams of state- and momentum-changing diffraction schemes. (a) Double-Raman diffraction is based on two two-photon processes mediated by a virtual state (dashed line) and generated by two counterpropagating optical lattices (symbolized by the red and blue arrows). Initially in the ground state, denoted by its mass $m_{-}$ (green solid line), the atom is elevated to the excited state with $m_{+}$ (blue solid line) and simultaneously diffracted into a superposition of two opposite momenta $\pm \hslash k$. The parabolas highlight the dispersion relation in each state and the condition for a resonant transition. (b) A sequence of double-Bragg diffraction (red and blue arrows) that first diffracts within the ground state from two counterpropagating lattices into two opposing momentum states in combination with Doppler-free E1-M1 two-photon transitions (purple arrows) reproduces the double-Raman scheme. However, the recoil-less transitions can be performed with optical frequencies so that a different range of clock frequencies can be addressed compared to conventional double-Raman diffraction.

Figure 4

Origin of the phase in the freely falling frame. Because of the different masses involved, the classical trajectories (solid lines) deviate from those corresponding to $m_{\pm }=m$ (dashed lines). In particular, at time $T_1$ where the laser pulses change the mass from $m_{+}$ (blue) to $m_{-}$ (green), momentum conservation in the laboratory frame implies $p_{\mathrm{f}}=p_{\mathrm{i}}-\hslash k$ where $p_{\mathrm{i}}$ ($p_{\mathrm{f}}$) is the momentum before (after) the laser pulse. Performing the Galilei transformation, we find $p_{\mathrm{f}}^{\prime }-m_{-}g{T}_1=p_{\mathrm{i}}^{\prime }-m_{+}g{T}_1-\hslash k$. Consequently, we observe the additional recoil $\mathrm{\Delta }{p}^{\prime }=-\mathrm{\Delta }mg{T}_1$ in the freely falling frame resulting in a small gravity-dependent velocity, which results in slightly inclined solid lines in the middle segment. It is this residual recoil which introduces the coupling between $\mathrm{\Omega }$ and $g$ in the freely falling frame.