Gravitational Redshift Tests with Atomic Clocks and Atom Interferometers

Open Access

Gravitational Redshift Tests with Atomic Clocks and Atom Interferometers

Fabio Di Pumpo, Christian Ufrecht, Alexander Friedrich, Enno Giese, Wolfgang P. Schleich, and William G. Unruh

PRX Quantum 2, 040333 – Published 11 November 2021

Abstract

Atomic interference experiments can probe the gravitational redshift via the internal energy splitting of atoms and thus give direct access to test the universality of the coupling between matter-energy and gravity at different spacetime points. By including possible violations of the equivalence principle in a fully quantized treatment of all atomic degrees of freedom, we characterize how the sensitivity to gravitational redshift violations arises in atomic clocks and atom interferometers, as well as their underlying limitations. Specifically, we show that: (i) Contributions beyond linear order to trapping potentials lead to such a sensitivity of trapped atomic clocks. (ii) Bragg-type interferometers, even with a superposition of internal states, with state-independent, linear interaction potentials are at first insensitive to gravitational redshift tests. However, modified configurations, for example by relaunching the atoms, can mimic such tests under certain conditions and may constitute a competitive alternative. (iii) Guided atom interferometers are comparable to atomic clocks. (iv) Internal transitions lead to state-dependent interaction potentials through which light-pulse atom interferometers can become sensitive to gravitational redshift violations.

Received 29 April 2021
Revised 29 July 2021
Accepted 1 October 2021

DOI:https://doi.org/10.1103/PRXQuantum.2.040333

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Alternative gravity theories Atom interferometry Atomic, optical & lattice clocks Cold atoms & matter waves Gravitation Laboratory studies of gravity Light-matter interaction Quantum fields in curved spacetime Quantum metrology

Gravitation, Cosmology & AstrophysicsAtomic, Molecular & OpticalParticles & Fields

Authors & Affiliations

Fabio Di Pumpo ^1,*, Christian Ufrecht ¹, Alexander Friedrich ¹, Enno Giese ^1,2,3, Wolfgang P. Schleich ^1,4,5,6, and William G. Unruh^5,7

¹Institut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm, Albert-Einstein-Allee 11, Ulm D-89069, Germany
²Institut für Angewandte Physik, Technische Universität Darmstadt, Schlossgartenstr. 7, Darmstadt D-64289, Germany
³Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, Hannover D-30167, Germany
⁴Institute of Quantum Technologies, German Aerospace Center (DLR), Söflinger Straße 100, Ulm D-89077, Germany
⁵Hagler Institute for Advanced Study and Department of Physics and Astronomy, Institute for Quantum Science and Engineering (IQSE), Texas A&M University, College Station, Texas 77843-4242, USA
⁶Texas A&M AgriLife Research, Texas A&M University, College Station, Texas 77843-4242, USA
⁷Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada

^*fabio.di-pumpo@uni-ulm.

Popular Summary

An atom’s internal states can be used to observe that time passes differently at different heights in a gravitational field, an effect often referred to as the gravitational redshift. Such time measurements rely on the quantum nature of the atom, as it is brought into a superposition of the relevant internal energy levels, leading to atomic clocks. If general relativity holds, time measurements must not depend on the specific realization of the clock. This assumption is validated by tests of the gravitational redshift.

At the same time, the atom can also be brought into a superposition of two different trajectories, and by that being superposed at two different heights in gravity. This concept is routinely used for inertial sensing in the field of atom interferometry. Bringing an atom into a superposition of two different heights in a gravitational field triggers immediately the question of redshift tests with atom interferometers. The fundamental physical connection between redshift tests with clocks and with atom interferometers has remained an open question.

In this article, a gold standard is derived for redshift tests with atomic quantum sensors, by introducing possible violations of general relativity into a common framework. With that, we are in the position to identify the physical reasons for a sensitivity of atomic clocks and atom interferometers to violations of the universality of gravitational redshift.

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Vol. 2, Iss. 4 — November - December 2021

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Figure 1
Comparison of two clocks moving along different configurations in spacetime: two freely falling atoms (a); two clocks trapped at different heights (b); as well as two clocks starting at the same position, guided by two trapping potentials to different heights, and back to the initial height (c). The center of each trap is bordered by red potential barriers, while the unperturbed trajectories $z (t)$ of the atoms are represented by black lines. The initialization and readout of the clocks is marked by purple pulses. To show that the detectors project on the internal state of the atom, each detector is colored in blue and green, representing both internal states. The two trajectories caused by a perturbation through the dilaton field with an effective acceleration $(1 + β_{j}) g$ are also drawn with these colors. The differential phases $Φ_{C}$ listed for each geometry in the table all show a sensitivity to UGR tests for a height difference of $δ ζ_{0}$ between the positions of the trap centers.
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Figure 2
Spacetime diagrams of different quantum-clock atom interferometers: a conventional Mach-Zehnder interferometer (a); a scheme relying on pulsed optical relaunching (b); and an interferometer guided by two trapping potentials to different heights, and back to the initial height (c). The center of each trap is bordered by red potential barriers, while the unperturbed trajectories $z (t)$ of the atoms are represented by dashed black lines, highlighting the superposition between the arms of the interferometer. The diffracting Bragg pulses that act differently on each branch are denoted by red pulses. The yellow pulses are used for pulsed optical relaunching that is independent of the state and branch. One can measure the exit-port population for each internal state independently. To indicate this measurement scheme, we introduced two separate detectors colored in blue and green. Accordingly, the trajectories of each internal state caused by a perturbation through the dilaton field with an effective acceleration $(1 + β_{j}) g$ are also drawn with these colors. The guided atom interferometer in panel (c) takes the same form as for two guided clocks with recoilless initialization and readout pulses, with the only difference that the atom is in a superposition of the branches (indicated by the dashed unperturbed trajectory). The differential phases $Φ_{AI}$ are listed for each geometry in the table below. While the Mach-Zehnder interferometer in panel (a) is insensitive to the gravitational redshift, the pulsed optical relaunching in panel (b) mimics a UGR test for the special choice $a = g$ , which corresponds to a vanishing effective acceleration and perfect levitation in the central segment. For other choices of $a$ , the resulting effective acceleration leads on average to curved trajectories of the atoms, indicated by the grayed-out configuration. In this case one observes an accelerational redshift. The table also includes the respective height differences $δ z_{0}$ or $δ ζ_{0}$ .
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Figure 3
Spacetime diagrams of two UGR-sensitive atom interferometer geometries with internal transitions: a Ramsey-Bordé-like geometry used in a doubly differential scheme (a) and a scheme relying on successive symmetrical transitions of the internal state (b). The unperturbed trajectories $z (t)$ of the atoms are represented by dashed black lines, highlighting their superposition. To indicate that the exit-port population for such interferometers can be measured independently for each internal state, we introduce two separate detectors that are colored blue and green. Accordingly, the trajectories of each internal state caused by a perturbation through the dilaton field with an effective acceleration $(1 + β_{j}) g$ are also drawn in these colors. The redshift sensitivity of the Ramsey-Bordé-like geometry in panel (a) relies on a doubly differential scheme where the first two Bragg pulses denoted by red lines open the interferometer. While the atom drops in superposition of a height difference $δ z_{0}$ , it is brought into an internal superposition shown by a purple pulse at a time $t_{1}$ initializing the clock. The differential phase $Φ_{AI} (t_{1})$ between the interference patterns for each individual state depends on the initialization time. By subtracting the differential phase of another initialization time $t_{2}$ , one measures the gravitational redshift between the two realizations, marked by the purple shaded area. In contrast, the configuration of panel (b) relies on symmetrical internal transitions that are combined with momentum transfers, symbolized by pictograms next to the pulse. Similar to panel (a), the atom travels during a central time segment in a superposition of parallel dropping trajectories. The phase $φ_{b / a}$ differs for two different input states (only one situation is depicted) and is already sensitive to UGR violations, even though additional terms arise so that the interferometer has to be performed with a differential measurement. The table below the figures lists the respective phases as well as the height differences $δ z_{0}$ .
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