General relativistic effects in atom interferometry

Atom interferometry is now reaching sufficient precision to motivate laboratory tests of general relativity. We begin by explaining the nonrelativistic calculation of the phase shift in an atom interferometer and deriving its range of validity. From this, we develop a method for calculating the phase shift in general relativity. Both the atoms and the light are treated relativistically and all coordinate dependencies are removed, thus revealing novel terms, cancellations, and new origins for previously calculated terms. This formalism is then used to find the relativistic effects in an atom interferometer in a weak gravitational field for application to laboratory tests of general relativity. The potentially testable relativistic effects include the nonlinear three-graviton coupling, the gravity of kinetic energy, and the falling of light. We propose specific experiments, one currently under construction, to measure each of these effects. These experiments could provide a test of the principle of equivalence to 1 part in ${10}^{15}$ (300 times better than the present limit), and general relativity at the 10% level, with many potential future improvements. We also consider applications to other metrics including the Lense-Thirring effect, the expansion of the Universe, and preferred frame and location effects.

Figure 1

A space-time diagram of a light-pulse atom interferometer. The black lines indicate the motion of a single atom. Laser light used to manipulate the atom is incident from above (light gray) and below (dark gray) and travels along null geodesics. Here the lasers’ worldlines are taken to be the two vertical lines on the left and right edges of the graph.

Figure 2

(a) shows an energy level diagram for a stimulated Raman transition between atomic states $|1⟩$ and $|2⟩$ through a virtual excited state using lasers of frequency ${\omega }_1$ and ${\omega }_2$. (b) shows the probability that the atom is in states $|1⟩$ and $|2⟩$ in the presence of these lasers as a function of the time the lasers are on. A $\frac{\pi }{2}$ pulse is a beam splitter since the atom ends up in a superposition of states $|1⟩$ and $|2⟩$ while a $\pi$ pulse is a mirror since the atom’s state is changed completely.

Figure 3

The atomic energy level diagram for a series of sequential two-photon Bragg transitions plotted as energy versus momentum. The horizontal lines indicate the states through which the atom is transitioned. The diagonal lines connecting the states represent the laser frequencies used in the transition. The result of this transition is to give the atom a large momentum.

Figure 4

Separation phase. This is a magnified view of the end of the interferometer which shows the upper and lower arms converging at the final beam splitter at time $t_0$, and the resulting interference. The dashed and solid lines designate the components of the wave function in internal states $|A_1⟩$ and $|A_2⟩$, respectively. After the beam splitter, each output port consists of a superposition of wave packets from the upper and lower arm. Any offset $\mathbf{\Delta }\mathbf{x}={\mathbf{x}}_l-{\mathbf{x}}_u$ between the centers of the wave packet contributions to a given output port results in a separation phase shift.

Figure 5

Projected limits to be set by the principle of equivalence measurement. Limits are shown (labeled solid lines) for several possible new Yukawa forces arising from a coupling to baryon number ($B$), baryon minus lepton number ($B-L$) and the dilaton. These assume a uniform Earth density. Previous limits are shown in solid shading [78]. The dashed line is the current limit on a force coupling to $B$ from an equivalence principle experiment using a realistic Earth model [56].

Figure 6

The response function $T_{gz}$ as defined in Eq. (87). It gives the phase shift response of the atom interferometer to a Fourier component of the local gravitational field with wavelength $\lambda$ and amplitude ${10}^{-9}{g}_{\mathrm{earth}}$. All curves assume the example 10 m atom interferometer. The top curve assumes it is run in a “symmetric” configuration (see text), the next lower curve assumes the atoms are dropped from rest at the top of the device and the following two curves assume a downward launch velocity of $1\frac{\mathrm{m}}{\mathrm{s}}$ and $10\frac{\mathrm{m}}{\mathrm{s}}$ respectively.

Figure 7

The magnitude of the power spectra of the local gravitational field ${\widetilde{\delta g}}_z(\lambda )$ from Eq. (87) for several example sources. The solid curve is a ${10}^{-2}\mathrm{kg}$ point source, 10 cm from the center of the interferometer. Similarly, the dotted curve is a 1 kg source at 1 m and the dash-dotted curve is 1000 kg at 10 m. The long-dashed curve is a thin 10 m long rod of mass 10 kg, parallel to the interferometer, whose center is 1 m from the interferometer.