Optical clocks based on atoms and ions probe relativistic effects with unprecedented sensitivity. They resolve time dilation due to atom motion or different positions in the gravitational potential through frequency shifts. However, all measurements of time dilation so far can be explained effectively as the result of dynamics with respect to a classical proper time parameter. Here we show that atomic clocks can probe effects where a classical description of the proper time dynamics is insufficient as superpositions of proper time emerge. We apply a Hamiltonian formalism to derive time dilation effects in harmonically trapped clock atoms and show how second-order Doppler shifts due to the vacuum energy, squeezing, and quantum corrections to the dynamics arise. We also demonstrate that time-dilation-induced entanglement between motion and clock evolution can become observable in state-of-the-art clocks when the motion of the atoms is strongly squeezed, realizing proper time interferometry. Our results show that experiments with trapped ion clocks are within reach of probing relativistic evolution of clocks for which a quantum description of proper time becomes necessary.

In a recent work we showed that the detection of the exchange of a single graviton between a massive quantum resonator and a gravitational wave can be achieved. Key to this ability are the experimental progress in preparing and measuring massive resonators in the quantum regime, and the correlation with independent detections of gravitational waves (GWs) that induce stimulated absorption, such as by the Laser Interferometer Gravitational Wave Observatory (LIGO). But can stimulated single-graviton processes shed light on the quantization of gravity? Here we analyze this question and make a historic analogy to the early days of quantum theory. We discuss in what ways such experiments can indeed probe key features of the quantized interaction between gravity and matter. We outline five experimental tests utilizing stimulated emission and absorption processes of single gravitons, opening the first window into experimental exploration of the fundamentals of quantum gravity.

We propose a multi-mode bar consisting of mass elements of decreasing size for the implementation of a gravitational version of the photo-electric effect through the stimulated absorption of up to kHz gravitons from a binary neutron star merger and post-merger. We find that the multi-mode detector has normal modes that retain the coupling strength to the gravitational wave of the largest mass-element, while only having an effective mass comparable to the mass of the smallest element. This allows the normal modes to have graviton absorption rates due to the tonne-scale largest mass, while the single graviton absorption process in the normal mode could be resolved through energy measurements of a mass-element in-principle smaller than pico-gram scale. We argue the feasibility of directly counting gravito-phonons in the bar through energy measurements of the end mass. This improves the transduction of the single-graviton signal, enhancing the feasibility of detecting single gravitons.

Quantum dynamics on curved spacetime has never been directly probed beyond the Newtonian limit. Although we can describe such dynamics theoretically, experiments would provide empirical evidence that quantum theory holds even in this extreme limit. The practical challenge is the minute spacetime curvature difference over the length scale of the typical extent of quantum effects. Here, we propose a quantum network of alkaline earth (like) atomic processors for constructing a distributed quantum state that is sensitive to the differential proper time between its constituent atomic processor nodes, implementing a quantum observable that is affected by post-Newtonian curved spacetime. Conceptually, we propose to delocalize one clock between three locations by encoding the presence or absence of a clock into the state of the local atoms. By separating three atomic nodes over approximately kilometer-scale elevation differences and distributing one clock between them via a 𝑊 state, we demonstrate that the curvature of spacetime is manifest in the interference of the three different proper times that give rise to three distinct beat notes in our nonlocal observable. We further demonstrate that 𝑁-atom entanglement within each node enhances the interrogation bandwidth by a factor of 𝑁. We discuss how our proposed system can probe new facets of fundamental physics, such as the linearity, unitarity, and probabilistic nature of quantum theory on curved spacetime. Our protocol combines several recent advances with neutral atom and trapped ions to realize a novel quantum probe of gravity uniquely enabled by quantum networks.

Quantum technologies rely on the control of quantum systems at the level of individual quanta. Mathematically, this control is described by Hamiltonian or Liouvillian evolution, requiring the application of various techniques to solve the resulting dynamic equations. Here, we present a tutorial for how the quantum dynamics of systems can be solved using a Lie-algebra decoupling method. The approach involves identifying a Lie algebra that governs the dynamics of the system, enabling the derivation of differential equations to solve the Schrödinger equation. As background, we include an overview of Lie groups and Lie algebras aimed at a general-physicist audience. We then prove the Lie-algebra decoupling theorem and apply it to both closed and open dynamics. The results represent a broad methodology to find the dynamics of quantum systems with applications across many fields of modern quantum research.

This summary of the second Terrestrial Very-Long-Baseline Atom Interferometry (TVLBAI) Workshop provides a comprehensive overview of our meeting held in London in April 2024 (Second Terrestrial Very-Long-Baseline Atom Interferometry Workshop, Imperial College, April 2024), building on the initial discussions during the inaugural workshop held at CERN in March 2023 (First Terrestrial Very-Long-Baseline Atom Interferometry Workshop, CERN, March 2023). Like the summary of the first workshop (Abend et al. in AVS Quantum Sci. 6:024701, 2024), this document records a critical milestone for the international atom interferometry community. It documents our concerted efforts to evaluate progress, address emerging challenges, and refine strategic directions for future large-scale atom interferometry projects. Our commitment to collaboration is manifested by the integration of diverse expertise and the coordination of international resources, all aimed at advancing the frontiers of atom interferometry physics and technology, as set out in a Memorandum of Understanding signed by over 50 institutions (Memorandum of Understanding for the Terrestrial Very Long Baseline Atom Interferometer Study).

Quantum technologies present new opportunities for fundamental tests of nature. One potential application is to probe the interplay between quantum physics and general relativity - a field of physics with no empirical evidence yet. Here we show that quantum networks open a new window to test this interface. We demonstrate how photon mediated entanglement between atomic or atomlike systems can be used to probe time-dilation-induced entanglement and interference modulation. Key are nonlocal measurements between clocks in a gravitational field, which can be achieved either through direct photon interference or by using auxiliary entanglement. The resulting observable depends on the interference between different proper times, and can only be explained if both quantum theory and general relativity are taken into account. The proposed protocol enables clock interferometry on kilometer-scale separations and beyond. Our work thus shows a realistic experimental route for a first test of quantum theory on curved spacetime, opening up new scientific opportunities for quantum networks.

The interplay between quantum theory and general relativity remains one of the main challenges of modern physics. A renewed interest in the low -energy limit is driven by the prospect of new experiments that could probe this interface. Here we develop a covariant framework for expressing post -Newtonian corrections to Schr & ouml;dinger's equation on arbitrary gravitational backgrounds based on a 1/c2 expansion of Lorentzian geometry, where c is the speed of light. Our framework provides a generic coupling prescription of quantum systems to gravity that is valid in the intermediate regime between Newtonian gravity and General Relativity, and that retains the focus on geometry. At each order in 1/c2 this produces a nonrelativistic geometry to which quantum systems at that order couple. By considering the gauge symmetries of both the nonrelativistic geometries and the 1/c2 expansion of the complex Klein-Gordon field, we devise a prescription that allows us to derive the Schr & ouml;dinger equation and its post -Newtonian corrections on a gravitational background order -by -order in 1/c2. We also demonstrate that these results can be obtained from a 1/c2 expansion of the complex Klein-Gordon Lagrangian. We illustrate our methods by performing the 1/c2 expansion of the Kerr metric up to O(c-2), which leads to a special case of the Hartle-Thorne metric. The associated Schr & ouml;dinger equation captures novel and potentially measurable effects.

The quantization of gravity is widely believed to result in gravitons – particles of discrete energy that form gravitational waves. But their detection has so far been considered impossible. Here we show that signatures of single graviton exchange can be observed in laboratory experiments. We show that stimulated and spontaneous single-graviton processes can become relevant for massive quantum acoustic resonators and that stimulated absorption can be resolved through continuous sensing of quantum jumps. We analyze the feasibility of observing the exchange of single energy quanta between matter and gravitational waves. Our results show that single graviton signatures are within reach of experiments. In analogy to the discovery of the photo-electric effect for photons, such signatures can provide the first experimental clue of the quantization of gravity.

This document presents a summary of the 2023 Terrestrial Very-Long-Baseline Atom Interferometry Workshop hosted by CERN. The workshop brought together experts from around the world to discuss the exciting developments in large-scale atom interferometer (AI) prototypes and their potential for detecting ultralight dark matter and gravitational waves. The primary objective of the workshop was to lay the groundwork for an international TVLBAI proto-collaboration. This collaboration aims to unite researchers from different institutions to strategize and secure funding for terrestrial large-scale AI projects. The ultimate goal is to create a roadmap detailing the design and technology choices for one or more kilometer–scale detectors, which will be operational in the mid-2030s. The key sections of this report present the physics case and technical challenges, together with a comprehensive overview of the discussions at the workshop together with the main conclusions.