The phase of matter waves depends on proper time and is therefore susceptible to special-relativistic (kinematic) and gravitational (redshift) time dilation. Hence, it is conceivable that atom interferometers measure general-relativistic time-dilation effects. In contrast to this intuition, we show that (i) closed light-pulse interferometers without clock transitions during the pulse sequence are not sensitive to gravitational time dilation in a linear potential. (ii) They can constitute a quantum version of the special-relativistic twin paradox. (iii) Our proposed experimental geometry for a quantum-clock interferometer isolates this effect.

Light-pulse interferometers ( 12 ) use this concept of pulsed optical gratings to manipulate matter waves. In case of interferometers closed in phase space ( 25 ) and for potentials up to the second order in z, the phase difference Δφ can be calculated from Eq. 2 by integrating along the classical trajectories.

In general relativity, the proper time along a world line z = z(t) is invariant under coordinate transformations and can be approximated as τ = ∫ d τ ≅ ∫ d t [ 1 − ( z ̇ / c ) 2 / 2 + U / c 2 ] (1)where c denotes the speed of light. Here, z ˙ = d z / d t is the velocity of the particle and U(z) is the Newtonian gravitational potential along the trajectory. This classical quantity is connected to the phase φ = − ω C τ + S em / ℏ (2)acquired by a first-quantized matter wave, assuming that it is sufficiently localized such that it can be associated with this trajectory. Here, ω C = mc 2 /ℏ denotes the Compton frequency of a particle of mass m and S em = − ∫ d t V em (3)is the classical action arising from the interaction of the matter wave with electromagnetic fields described by the potential V em (z, t) evaluated along the trajectory. For instance, if the electromagnetic fields generate optical gratings, then this potential can transfer momentum to the matter wave, thus changing its trajectory, which, in turn, affects proper time.

In this work, we study a quantum version of the twin paradox, where a single twin is in a superposition of two different world lines, aging simultaneously at different rates, illustrated in Fig. 1B . We show that light-pulse atom interferometers can implement the scenario where time dilation is due to special-relativistic effects but are insensitive to gravitational time dilation. To this end, we establish a relation between special-relativistic time dilation and kinematic asymmetry of closed atom interferometers, taking the form of recoil measurements ( 15 , 21 , 23 , 24 ). For these geometries, a single atomic clock in a superposition of two different trajectories undergoes special-relativistic time dilation. The induced distinguishability leads to a loss of visibility upon interference such that the proposed experiment represents a realization of the twin paradox in quantum-clock interferometry.

Atom interferometry, in conjunction with atomic clocks, has led to the idea of using time dilation between two branches of an atom interferometer as a which-way marker to measure effects like the gravitational redshift through the visibility of the interference signal ( 5 , 6 ). However, no specific geometry for an atom interferometer was proposed and no physical process for the manipulation of the matter waves was discussed. The geometry as well as the protocols used for coherent manipulation crucially determine whether and how the interferometer phase depends on proper time ( 22 ). Therefore, the question of whether the effects connected to time dilation can be observed in light-pulse atom interferometers is still missing a conclusive answer.

In analogy to optical interferometry, atom interferometers measure the relative phase of a matter wave accumulated during the propagation by interfering different modes. Although it is possible to generate these interferometers through different techniques, we focus here on light-pulse atom interferometers like the one of Kasevich and Chu ( 12 ) with two distinct spatially separated branches, where the matter waves are manipulated through absorption and emission of photons that induce a recoil to the atom. Conventionally, these interferometers consist of a series of light pulses that coherently drive atoms into a superposition of motional states, leading to the spatial separation. The branches are then redirected and finally recombined such that the probability to find atoms in a specific momentum state displays an interference pattern and depends on the phase difference Δφ accumulated between the branches that is susceptible to inertial forces. Hence, light-pulse atom interferometers do not only provide high-precision inertial sensors ( 13 , 14 ) with applications in tests of the foundations of physics ( 15 – 21 ) but also constitute a powerful technique to manipulate atoms and generate spatial superpositions.

( A ) As a consequence of relativity, two initially co-located twins experience time dilation when traveling along different world lines. Upon reunion, they find that they aged differently due to the relative motion between them. ( B ) In a quantum version of this gedankenexperiment, a single individual is traveling along two paths in superposition, serving as his own twin and aging at two different rates simultaneously.

The astonishing consequences of time dilation can be illustrated by the story of two twins ( 3 ), depicted in Fig. 1A : Initially at the same position, one of them decides to go on a journey through space and leaves his brother behind. Because of their relative motion, he experiences time dilation and, upon meeting his twin again after the voyage, has aged slower than his brother who remained at the same position. Although this difference in age is notable by itself, the twin who traveled could argue that, from his perspective, his brother has moved away and returned, making the same argument. This twin paradox can be resolved in the context of relativity, where it becomes apparent that not both twins are in an inertial system for the whole duration. In the presence of gravity, two twins that separate and reunite experience additional time dilation depending on the gravitational potential during their travel. The experimental verifications of the effect that leads to the difference in age, namely, special-relativistic and gravitational time dilation, were milestones in the development of modern physics and have, for instance, been performed by the comparison of two atomic clocks ( 8 – 10 ). Atomic clocks, as used in these experiments, are based on microwave and optical transitions between electronic states and define the state of the art in time keeping ( 11 ).

Proper time is operationally defined ( 1 ) as the quantity measured by an ideal clock ( 2 ) moving through spacetime. As the passage of time itself is relative, the comparison of two clocks that traveled along different world lines gives rise to the twin paradox ( 3 ). Whereas this key feature of relativity relies on clocks localized on world lines, today’s clocks are based on atoms that can be in a superposition of different trajectories. This nature of quantum objects is exploited by matter-wave interferometers, which create superpositions at macroscopic spatial separations ( 4 ). One can therefore envision a single quantum clock such as a two-level atom in a superposition of two different world lines, suggesting a twin paradox, in principle susceptible to any form of time dilation ( 5 – 7 ). We demonstrate which atom interferometers implement a quantum twin paradox, how quantum clocks interfere, and their sensitivity to different types of time dilation.

RESULTS

Time dilation and gravito-kick action Because the light pulses act differently on the two branches of the interferometer, we add superscripts α = 1,2 to the potential V em ( α ) . Moreover, we separate V em ( α ) = V k ( α ) + V p ( α ) into a contribution V k ( α ) causing momentum transfer and V p ( α ) imprinting the phase of the light pulse without affecting the motional state (26). Consequently, we find that the motion z ( α ) = z g + z k ( α ) along one branch can also be divided into two contributions: z g caused by the gravitational potential and z k ( α ) determined by the momentum transferred by the light pulses on branch α. For a linear gravitational potential, the proper-time difference between both branches takes the form Δ τ = ∫ d t [ z ¨ k ( 1 ) z k ( 1 ) − z ¨ k ( 2 ) z k ( 2 ) ] / ( 2 c 2 ) (4)(see Materials and Methods). It is explicitly independent of z g as well as of the particular interferometer geometry, which is a consequence of the phase of a matter wave being invariant under coordinate transformations. When transforming to a freely falling frame, both trajectories reduce to the kick-dependent contribution z k ( α ) and the proper-time difference Δτ is thus independent of gravity (22). Accordingly, closed light-pulse interferometers are insensitive to gravitational time dilation. Our result implies that time dilation in these interferometer configurations constitutes a purely special-relativistic effect caused by the momentum transferred through the light pulses. Our model of atom-light interaction assumes instantaneous momentum transfer and neglects the propagation time of the light pulses. A potential V k linear in z, where the temporal pulse shape of the light is described by a delta function, that is z ¨ k ∝ δ ( t − t ℓ ) , reflects exactly such a transfer. For such a potential, we find the differential action Δ S em = 2 ℏ ω C Δ τ + Δ S gk + Δ S p (5)(see Materials and Methods), which can be interpreted (27) as the laser pulses sampling the position of the atoms z = z k + z g . The first contribution has the form of the proper-time difference, which highlights that the action of the laser can never be separated from proper time in a phase measurement in the limit given by Eq. 2. It arises solely from the interaction with the laser, and in the case of instantaneous acceleration z ¨ k , these kicks read out the recoil part of the motion z k according to Eq. 4. Similarly, the second contribution in Eq. 5 is the action that arises from the acceleration z ¨ k measuring the gravitational part z g of the motion and takes the form Δ S gk = m ∫ d t Δ z ¨ k z g (6)where we define the difference Δ z ¨ k = z ¨ k ( 1 ) − z ¨ k ( 2 ) between branch-dependent accelerations. Although this contribution is caused by the interaction with the light, the position of the atom still depends on gravity and is caused by the combination of both the momentum transfers and gravity. Hence, we refer to it as gravito-kick action. Last, the lasers imprint the laser phase action Δ S p = − ∫ d t Δ V p (7)with Δ V p = V p ( 1 ) − V p ( 2 ) . So far, we have not specified the interaction with the light but merely assumed that the potential V k is linear in z. In the context of our discussion, beam splitters and mirrors are generated through optical gratings made from two counter-propagating light beams that diffract the atoms (12). In a series of light pulses, the periodicity of the ℓth grating is parameterized by an effective wave vector k ℓ . Depending on the branch and the momentum of the incoming atom, the latter receives a recoil ±ℏk ℓ in agreement with momentum and energy conservation. At the same time, the phase difference of the light beams is imprinted to the diffracted atoms. To describe this process, we use the branch-dependent potential V k ( α ) = − ∑ ℓ ℏ k ℓ ( α ) z ( α ) δ ( t − t ℓ ) for the momentum transfer ℏ k ℓ ( α ) of the ℓth laser pulse at time t ℓ and the potential V p ( α ) = − ∑ ℓ ℏ ϕ ℓ ( α ) δ ( t − t ℓ ) to describe the phase ϕ ℓ ( α ) imprinted by the light pulses (26). Because the phases imprinted by the lasers can be evaluated trivially and are independent of z, we exclude the discussion of V p ( α ) from the study of different interferometer geometries and set it to zero in the following.

Atom-interferometric twin paradox The Kasevich-Chu–type (12) Mach-Zehnder interferometer (MZI) has been at the center of a vivid discussion about gravitational redshift in atom interferometers (26–28). It has been demonstrated that its sensitivity to the gravitational acceleration g stems entirely from the interaction with the light, i.e., ΔS gk , while the proper time vanishes (26, 27). It is hence insensitive to gravitational time dilation, which, a priori, is not necessarily true for arbitrary interferometer geometries. Such an MZI consists of a sequence of pulses coherently creating, redirecting, and finally recombining the two branches. The three pulses are separated by equal time intervals of duration T. We show the spacetime diagram of the two branches z k ( α ) , the light-pulse–induced acceleration z ¨ k ( α ) as a sequence in time, and the gravitationally induced trajectory z g in Fig. 2 on the left. The contributions z k ( α ) are branch dependent, while z g is common for both arms of the interferometer. From these quantities and with the help of Eqs. 4 and 6, we obtain the phase contributions shown at the bottom of Fig. 2 (see Materials and Methods). The phase takes the familiar form Δφ = −kgT2 and has no proper-time contribution, but is solely determined by the gravito-kick action originating in the interaction with the light pulses (26, 27). Fig. 2 Time dilation in different interferometer geometries. Spacetime diagrams for the light-pulse and gravitationally induced trajectories z k and z g , as well as the accelerations z ¨ k caused by the light pulses, together with the proper-time difference Δτ, the gravito-kick action ΔS gk , the electromagnetic contribution ΔS em /ℏ, and the total phase difference Δφ of an MZI (left), a symmetric RBI (center), and an asymmetric RBI (right). The first two geometries display a symmetric momentum transfer between the two branches, leading to vanishing proper-time differences. However, the asymmetric RBI features a proper-time difference that has the form of a recoil term. The spacetime diagrams also illustrate the connection to the twin paradox by displaying ticking rates (the dashes) of the two twins traveling along the two branches. Both quantum twins in the MZI and symmetric RBI experience the same time dilation, whereas in the asymmetric RBI, one twin stays at rest and the other one leaves and returns so that their proper times are different. The arrows in the plot of z ¨ k denote the amplitude of the delta functions that scale with ±ℏk/m. Because of the instantaneous nature of z ¨ k , the integration over time in Eqs. 4 and 6 reduces to a sampling of the positions z k and z g at the time of the pulses such that the respective phase contributions can be inferred directly from the figure. The vanishing proper-time difference can be explained by the light-pulse–induced acceleration z ¨ k ( α ) that acts symmetrically on both branches. We draw on the classical twin paradox to illustrate the effect: At some time, one twin starts to move away from his brother and undergoes special-relativistic time dilation, as shown by hypothetical ticking rates in the spacetime diagram (the dashing periods in Fig. 2). After a time T, he stops and his brother starts moving toward him. Because his velocity corresponds to the one that caused the separation, he undergoes exactly the same time dilation his brother experienced previously. Hence, when both twins meet after another time interval T, their clocks are synchronized and no proper-time difference arises. In an MZI, we find the quantum analog of this configuration, where a single atom moves in a superposition of two different world lines like the quantum twin of Fig. 1B. However, because of the symmetry of the light-pulse–induced acceleration, no proper-time difference is accumulated between the branches of the interferometer. A similar observation is made for the symmetric Ramsey-Bordé interferometer (RBI), where the atom separates for a time T, then stops on one branch for a time T′ before the other branch is redirected. We show the spacetime diagrams and the light-pulse–induced acceleration z ¨ k in the center of Fig. 2 with the phase contributions below. The two light pulses in the middle of the symmetric RBI are also beam-splitting pulses that introduce a symmetric loss of atoms. As for the MZI, the proper-time difference between both branches vanishes and the phase is determined solely by the laser contribution and the gravito-kick phase, as shown by the ticking rates in the spacetime diagram. The only difference with respect to the MZI is that the two branches travel in parallel for a time T′ during which proper time elapses identically for both of them. The situation changes substantially when we consider an asymmetric RBI, where one branch is completely unaffected by the two central pulses, as shown on the right of Fig. 2. Specifically, the twin that moved away from his initial position experiences a second time dilation on his way back so that there is a proper-time difference when both twins meet at the final pulse. It is therefore the kinematic asymmetry that causes a nonvanishing proper-time difference, as indicated in the figure by the ticking rates. The proper-time difference Δ τ aRBI = − ( ℏ k / mc ) 2 T (8)is proportional to a kinetic term (23) that depends on the momentum transfer ℏk, as already implied by Eq. 4. With the light-pulse–induced acceleration z ¨ k ( α ) as well as the gravitationally induced trajectory z g also shown on the right of the figure, we find the same contribution for ΔS gk given in Fig. 2 as for the symmetric RBI. The other contribution of ΔS em /ℏ has the form 2ω C Δτ aRBI , and all of them together contribute to the phase difference Δφ.