Phase Shift in an Atom Interferometer due to Spacetime Curvature across its Wave Function

Spacetime curvature induces tidal forces on the wave function of a single quantum system. Using a dual light-pulse atom interferometer, we measure a phase shift associated with such tidal forces. The macroscopic spatial superposition state in each interferometer (extending over 16 cm) acts as a nonlocal probe of the spacetime manifold. Additionally, we utilize the dual atom interferometer as a gradiometer for precise gravitational measurements.

Figure 1

(a) Spacetime diagram of the centers of the wave packets for a dual interferometer sequence, neglecting gravitational acceleration. An initial atom source is split into two clouds that drift apart with momentum difference $N_1\hslash k$ for time $\tau$, ultimately separating by the baseline $L$. Each cloud is used as the source for a Mach-Zehnder interferometer with momentum splitting $n\hslash k$ and pulse spacing $T$. The path separation between the two interferometer arms reaches a distance of $\mathrm{\Delta }z=(n\hslash k/m)T$. The numerical values in parentheses indicate typical experimental parameters. In this work, $k$ is negative for the initial splitting and interferometer pulses; i.e., the interferometer trajectories are below the unperturbed launch height. Data plots: Fluorescence images of spatial interference fringes for an interferometer with (b) $\mathrm{\Delta }z=4\mathrm{cm}$ ($n=10$) and (c) $\mathrm{\Delta }z=12\mathrm{cm}$ ($n=30$).

Figure 2

(a) Schematic representation of the experimental setup for measuring the gravity gradient of seven lead bricks (total mass 84 kg). (b) Measured gradiometer phase of a sequence with $h=8.45\mathrm{m}$, $L=32\mathrm{cm}$, $n=20$, and $T=600\mathrm{ms}$ ($\mathrm{\Delta }z=7\mathrm{cm}$), with (solid circles) and without (open circles) the bricks present. (c) Gradiometer phase difference (with and without bricks present) as a function of the launch height with $L=10\mathrm{cm}$, $n=30$, and $T=900\mathrm{ms}$ ($\mathrm{\Delta }z=16\mathrm{cm}$). The black, solid curve represents the full phase shift calculation.

Figure 3

(a) Spacetime diagram for a gradiometer with $h=8.25\mathrm{m}$, $L=32\mathrm{cm}$, $T=550\mathrm{ms}$, and $n=30$. (b) Calculated gravitational acceleration induced by the bricks as a function of the height along the interferometer axis. (c) Schematic of the interferometer trajectories, neglecting gravitational acceleration: upper (solid red lines) and lower (solid blue lines) arms, midpoint line (dashed gray line), and the trajectory without recoil from interferometer atom optics (dashed black line). (d) Gradiometer phase difference of a sequence with $h=8.25\mathrm{m}$, $L=32\mathrm{cm}$, and $T=550\mathrm{ms}$ as a function of $n$ ($\mathrm{\Delta }z=2\mathrm{cm}$ to $\mathrm{\Delta }z=10\mathrm{cm}$). The black, dashed curve is the phase calculated without including the tidal phase shift. The solid red and blue curves are $\mathrm{\Delta }{\varphi }_u$ and $\mathrm{\Delta }{\varphi }_l$, respectively (see the main text). The gray, dashed (black, solid) curve is the phase predicted by the midpoint line (full phase shift) calculation. Each data point is the average of 15 shots.

Figure 4

(a) Schematic of the interferometer and surrounding mass, to scale, with indicated launch heights. (b) Measured gravity gradient (blue points) as a function of launch height $h$ for a gradiometer with $L=10\mathrm{cm}$, $n=10$, and $T=500\mathrm{ms}$. The red, dashed curve is a model of the interferometer response to the gravity gradient produced by a spherical Earth, including the effects of the cylindrical pit and basement in which the interferometer is located. The model has one free parameter, the density of Earth’s surface near the interferometer, which fits to $\rho =2.3\mathrm{g}/{\mathrm{cm}}^3$. The black, solid curve is an interpolation (logistic fit, all parameters free). Each data point is the average of 50 shots. Statistical uncertainties are $\approx 6E$ (error bars smaller than data points).