Gravitational Redshift in Quantum-Clock Interferometry

The creation of delocalized coherent superpositions of quantum systems experiencing different relativistic effects is an important milestone in future research at the interface of gravity and quantum mechanics. This milestone could be achieved by generating a superposition of quantum clocks that follow paths with different gravitational time dilation and investigating the consequences on the interference signal when they are eventually recombined. Light-pulse atom interferometry with elements employed in optical atomic clocks is a promising candidate for that purpose, but it suffers from major challenges including its insensitivity to the gravitational redshift in a uniform field. All of these difficulties can be overcome with the novel scheme presented here, which is based on initializing the clock when the spatially separate superposition has already been generated and performing a doubly differential measurement where the differential phase shift between the two internal states is compared for different initialization times. This scheme can be exploited to test the universality of the gravitational redshift with delocalized coherent superpositions of quantum clocks, and it is argued that its experimental implementation should be feasible with a new generation of 10-meter atomic fountains that will soon become available. Interestingly, the approach also offers significant advantages for more compact setups based on guided interferometry or hybrid configurations. Furthermore, in order to provide a solid foundation for the analysis of the various interferometry schemes and the effects that can be measured with them, a general formalism for a relativistic description of atom interferometry in curved spacetime is developed. It can describe freely falling atoms as well as the effects of external forces and guiding potentials, and it can be applied to a very wide range of situations. As an important ingredient for quantum-clock interferometry, suitable diffraction mechanisms for atoms in internal-state superpositions are investigated too. Finally, the relation of the proposed doubly differential measurement scheme to other experimental approaches and to tests of the universality of free fall is discussed in detail.

Popular Summary

According to Einstein’s theory of general relativity, two clocks at different heights in a gravitational field tick at different rates. The best atomic clocks to date are sensitive to this effect even for a 1-cm height difference in Earth’s gravitational field. However, these measurements are based on comparing two independent clocks. We propose creating a quantum superposition of a single clock at two different heights and measuring the effect of the gravitational time dilation on the interference signal when the two atomic wave packets in the superposition are eventually recombined. This will allow testing, for the first time, a regime where relativistic effects and macroscopically delocalized quantum superpositions simultaneously play an essential role.

The atom interferometers capable of reaching the required sensitivities employ laser pulses applied for short times to split, redirect, and recombine the atomic wave packets, but the atoms are otherwise freely falling the whole time. This means that such interferometers are not sensitive to gravitational time-dilation effects, as can be seen by considering a freely falling frame. Our proposed scheme overcomes this hurdle by simultaneously initializing the clock in the two interferometer arms after the spatially separate superposition has been generated. Because of the relativity of simultaneity pointed out by Einstein, the spatially separated initialization events are simultaneous in the laboratory frame but not in the freely falling one.

When combined with other methods presented in this paper, an experimental implementation of our proposal should be feasible with new facilities that are currently becoming available in Stanford and Hannover.

Figure 1

Central trajectories for a reversed Ramsey-Bordé interferometer in a uniform gravitational field. A doubly differential measurement comparing the outcomes for different initialization times ($t_{\mathrm{i}}$ and $t_{\mathrm{i}}^{\prime }$) is directly related to the proper-time difference between the two dashed worldline segments and is sensitive to gravitational time-dilation effects.

Figure 2

Spacetime trajectories in a uniform gravitational field for a two-level atom trapped at constant height ($a$) and for a freely falling one ($b$). They correspond to the central trajectories of the atomic wave packets.

Figure 3

Central trajectories for the interfering wave packets of a quantum clock at exit port I. Nontrivial effects arise when the proper times along the two interferometer branches ($a$ and $b$) differ. Analogous results hold for exit port II.

Figure 4

Central trajectories for a Ramsey-Bordé [27] interferometer in a uniform gravitational field as seen in the freely falling frame. In this frame, the spacetime trajectories are straight lines, and the propagation phases are manifestly independent of the gravitational acceleration $g$. Laser pulses are depicted by grey dashed lines.

Figure 5

The mass difference $\mathrm{\Delta }m$ leads to a differential recoil for the two internal states. As a consequence, the central trajectories of the diffracted wave packets (orange lines) differ slightly for the ground state (continuous line) and the excited one (dashed line).

Figure 6

Central trajectories in the laboratory frame for a guided interferometer where the wave packets are held at different constant heights for some time $T$. The effect of the gravitational redshift on the differential phase shift can be identified by comparing the outcome with another interferometer, where the holding time $T$ is extended, while leaving everything else unchanged (dashed lines).

Figure 7

In the freely falling frame, the proper-time difference between the dashed segments in the doubly differential measurement is a consequence of the lack of simultaneity for the spatial hypersurfaces associated with the initialization pulses.

Figure 8

Central trajectories for an open interferometer in the laboratory frame. The proper-time difference between the extended wordlines as the detection time is delayed (dashed lines) increases due to the gravitational time dilation, but this is exactly compensated by the growth of the separation phase as explained in Appendix pp3-s3.

Figure 9

Central trajectories in the freely falling frame showing how the residual recoil from the initialization pulse leads to slightly modified trajectories for the excited state. Nevertheless, the phase shift, as well as the interpretation of the doubly differential measurement in terms of proper-time differences, remains unaffected.

Figure 10

Central trajectories for a guided interferometer in the laboratory frame. Performing a doubly differential measurement for different initialization times has a number of practical advantages and directly reveals the differences of gravitational time dilation between the two branches.

Figure 11

Central trajectories in the laboratory frame for a hybrid interferometer involving a reversed Ramsey-Bordé sequence combined with an optical lattice applied between the two pairs of pulses, which holds the atomic wave packets at constant height and where they undergo Bloch oscillations (BO). Doubly differential measurements are sensitive to the gravitational redshift in this case. too.

Figure 12

Schematic representation of the central trajectories of the atomic wave packets in a light-pulse atom interferometer, where branches $a$ and $b$ correspond to the two interfering wave packets in exit port I. The example displayed here is a Mach-Zehnder interferometer with different times between the mirror pulse and the initial and final beam-splitter pulses.

Figure 13

Central trajectories at the exit port of an open interferometer. Simultaneous detection for nonvanishing central velocities (a) corresponds to nonsimultaneous spacetime events in the comoving frame, where the interfering wave packets are at rest (b). The resulting proper-time difference, corresponding to the dashed segment in panel (b), is exactly compensated by the nonvanishing separation phase $\delta {\varphi }_{\mathrm{sep}}$ in panel (a).

Figure 14

Single-photon transitions between the clock states shown in an energy-momentum diagram that takes into account the internal energy and the c.m. motion (as described in the freely falling frame where the atoms are initially at rest). The frequencies of the two counterpropagating beams are shifted by ${\omega }_{\mathrm{rec}}$ and $-{\omega }_{\mathrm{rec}}$ from the clock frequency ${\omega }_0$ so that they drive resonant transitions of atoms initially at rest through absorption (blue arrow) and stimulated emission (red arrow), respectively. By applying simultaneous $\pi /2$ pulses, one can generate an equal-amplitude superposition involving an undiffracted wave packet plus a diffracted one with swapped internal states and a single-photon momentum transfer. Relevant off-resonant transitions are additionally indicated with dashed lines.

Figure 15

If one shifts the frequencies of the two counterpropagating beams by $3{\omega }_{\mathrm{rec}}$ and $-3{\omega }_{\mathrm{rec}}$, one can resonantly address the diffracted atoms in Fig. 14 with an additional single-photon momentum transfer in the same direction while swapping the internal states again. When simultaneous $\pi$ pulses with these frequencies are applied right after the simultaneous $\pi /2$ pulses of Fig. 14, the net result is to leave the internal states unchanged but generate an equal-amplitude superposition, as far as the c.m. motion is concerned, of an undiffracted wave packet and a diffracted one with a total momentum transfer $\hslash {\mathbf{k}}_{\mathrm{eff}}$ corresponding to twice the single-photon momentum.

Figure 16

Central trajectories for a Mach-Zehnder interferometer in the freely falling frame where the atomic wave packet is initially at rest at $z=0$. They are no longer straight lines (except for segment $A$) due to the tidal forces associated with the gravity gradient. From the figure, it is clear that the propagation phases for segments $B$ and $C$ are the same and so are the spatially dependent parts of the laser phases for branches $a$ and $b$.