Relativistic effects in atom and neutron interferometry and the differences between them

In recent years there has been enormous progress in matter wave interferometry. The Colella-Overhauser-Werner (COW) type of neutron interferometer and the Kasevich-Chu (K-C) atom interferometer are the prototypes of such devices and the issue of whether they are sensitive to relativistic effects has recently aroused much controversy. We examine the question as to what extent the gravitational redshift and the related twin paradox effect can be seen in both of these atom and neutron interferometers. We point out an asymmetry between the two types of devices. Because of this, the nonvanishing, nonrelativistic residue of both effects can be seen in the neutron interferometer, while in the K-C interferometer the effects cancel out, leaving no residue, although they could be present in other types of atom interferometers. Also, the necessary shifting of the laser frequency (chirping) in the atom interferometer effectively changes the laboratory into a free-fall system, which could be exploited for other experiments.

Figure 1

Matter wave interferometers. (a) The neutron interferometer: Coherent splitting of the beam is achieved by elastically scattering off the crystal planes at $A$, and the beams are then coherently recombined at $D$. Because the initially horizontal interferometer is rotated about the incident beam $AC$, both beams are at different heights, introducing a gravitational potential between them. (In this diagram we ignore the small bending due to gravity.) (b) The atom interferometer: The beam is directed vertically upward. It is coherently split and recombined by absorbing a photon from a pulsed laser. Gravity is always acting. The dotted beam is the original path, and the solid beams are the interferometer beams created by the laser interactions. Here, time is plotted along the horizontal axis.

Figure 2

Momentum changes at time $T$ in the matter interferometer. Atom interferometer: (a) No gravity—the atom ($k_1$) absorbs a photon (κ) and recoils ($k_2$). It thus does not conserve energy. (b) Gravity present—the atom has some extra momentum due to falling, δ$k$, which is passed on when it absorbs the photon. Neutron interferometer: (c) No gravity—the neutron bounces elastically off the crystal atoms, thus conserving energy. (d) Gravity present—the recoil off the crystal planes includes the momentum due to gravity, which is now in the opposite direction.

Figure 3

Laue scattering in the crystal ear. The neutron wave packet hits the crystal at time $T$, and scatters off the planes perpendicular to the face of the crystal, setting up standing waves inside the crystal, which convert back to traveling waves as it leaves the crystal. (This is different from hitting a single horizontal mirror at a fixed height, which would happen at different times, depending on the angle of the beam.) For simplicity we will ignore the thickness of the crystal in our analysis. See Refs. [3, 21, 33] for some more details.

Figure 4

Trajectories of the particles in the atom interferometer in the laboratory. In the laboratory after the beams separate, with the upper beam absorbing a vertical photon of momentum $\hslash \kappa$ at time 0, they both fall under the influence of gravity. Both beams absorb a photon at time $T$ and they both keep falling. They finally recombine at time 2$T$ and continue on to one of the two detectors.

Figure 5

Converting distance to phase. (a) The wave vectors of the two incoming atomic beams each have an angle θ with respect to the horizontal line, and are indicated by solid lines. The corresponding wave fronts, depicted by dashed lines, are orthogonal to these wave vectors. From where the two beams meet, the vertical distance is given by δ$y$. Every time δ$y$ changes by $\ell$, where $\ell \sin \theta =\lambda$, the phase φ changes by 2π. (b) When one beam is advanced by the distance $x$, where $x/\lambda =\delta \phi /2\pi =\delta \ell /\ell$, its phase is advanced by δφ. (c) If one beam is advanced by $x$, while the other is retarded by $x$, the maximum will still be shifted by $\delta \phi /2\pi =x/\lambda =\delta \ell /\ell$ along the screen, even though the two beams differ in phase from each other by $2\delta \phi$. [See Eq. (11).]

Figure 6

Atom interferometer in the free-fall system. There is no gravity in this system, and the atoms in each beam move as though it was an inertial system. But the lasers and detectors, which are at rest in the laboratory, are accelerating upward in this system, as are the lines of constant phase of the lasers, which are shown here by the closely dashed lines. They are shown displaced by the distance $d=g{T}^2/2$ that they have moved by time $T$. After time 2$T$, they have moved by 4$d$.

Figure 7

Neutron interferometer in the laboratory system. Without gravity acting, the beams would follow the dashed lines. With gravity, they both deflect downward. But in the elastic bounce at the center slab, the velocities, including that due to gravity, are reversed. So the beams return to their initial height at the final slab.

Figure 8

Neutron interferometer in the free-fall system. In the system freely falling with the neutrons, the neutrons travel in straight lines. However, the crystal planes that they rebound from are accelerating upward, with acceleration $g$, and this causes an asymmetrical effect between the two beams, as the crystal is accelerating toward the lower beam, and away from the upper beam. The result is a difference in proper times taken by each of the two beams before they recombine.

Figure 9

Lack of any extra phase shift due to changing trajectories. A particle that would have traveled from $P$ to $A$ in the unperturbed beam will be diverted to point $B$ by a weak perturbation (along the trajectory $D$, denoted by the dashed line), where $A$ and $B$ are arbitrary points along the respective trajectories. The solid lines represent the unperturbed trajectories while the circular arcs indicate the lines of constant phase of the unperturbed wave, one wavelength apart. The line $C$ is the unperturbed trajectory from $P$ to $B$. But the length of the path of the perturbed beam $D$ is the same as that of the unperturbed beam $C$, to lowest order in the perturbation. So while a different trajectory arrives at point $B$, the phase will still be the same at point $B$ as it was before the perturbation, even though a different beam arrives there. And the same is true at every point (including point $A$).

Figure 10

Phase shift when energy is not conserved. The incident atomic interferometer beam is hit by a laser at $t_0=0$, at point $A$, and splits into two beams, as per Fig. 1. The upper beam absorbs momentum $\hslash \kappa$ from the laser, so that $\hslash k_u=\hslash (k_{\ell }+\kappa )$, $\hslash {\omega }_u=\hslash ({\omega }_{\ell }+\omega )$, while the path lengths on the two segments of each beam are $s_1$ and $s_2$. Subsequently, at $t_1=T$, the upper beam loses its extra momentum, while the lower beam picks it up. Then at $t_3=2T$ the lower beam loses its extra momentum and the two beams recombine.

Figure 11

Stimulated Raman transition. In the K-C interferometer experiments [8, 10, 11], the momentum transfer to the cesium atom was supplied by the absorption of a photon from a hyperfine level $|b\rangle$ of an $S$ state to a virtual $P$ state, followed by an emission of a counterpropagating photon back down to another hyperfine level $|a\rangle$ of the $S$ state.