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.

Atom Interferometry (AI) is a well-established quantum sensor concept based on the superposition and interference of atomic wave packets, which affords exceptionally high sensitivity, for example, to inertial/gravitational effects. AI experimental designs take advantage of features used by state-of-the-art atomic clocks in combination with established techniques for building inertial sensors.

The experimental landscape of AI projects has expanded significantly in recent years, ranging from ultra-sensitive experimental setups to portable devices and even commercially available gravimeters. Several large-scale terrestrial AI projects based on cold atom technologies are currently under construction, in planning stages, or being proposed.

Five large-scale fully funded prototype projects are currently under construction, namely, a 10 m fountain at Stanford1 and MAGIS-1002 at FNAL in the US, MIGA3 in France, VLBAI4 at Hannover in Germany, AION-105 at Oxford with possible 100 m sites at Boulby in the UK and at CERN under investigation, and a 10 m fountain6 and ZAIGA7 in China. These projects will demonstrate the feasibility of AI at large scales, paving the way for terrestrial kilometer -scale experiments as the next steps.

There have already been discussions of projects to build one or more kilometer -scale detectors, including ELGAR in Europe,8 MAGIS-km at the Sanford Underground Research facility (SURF) in the US,2 AION-km at the STFC Boulby facility in the UK,5 and advanced ZAIGA in China.7 

The goal is that by about 2035 at least 1 km-scale detector will have entered operation. These kilometer-scale experiments would not only be able to explore systematically the mid-frequency band of gravitational waves and probe possible ultralight dark matter, but would also demonstrate the readiness of key technologies ahead of a space-based AI mission such as AEDGE.9,10

The main goals of the workshop were to establish the pathway for a proto-collaboration that could develop a Roadmap for the design and technology choices for one or several kilometer-scale detectors to be ready for operation in the mid-2030s. This initiative is supported by the cold atom community and the potential user communities interested in its science goals. This Roadmap will outline technological milestones as well as refine interim and long-term scientific goals.

The Workshop brought together members of the cold atom, astrophysics, cosmology, and fundamental physics communities and built upon the Community Workshop on Cold Atoms in Space held in September 2021,11 which reviewed the cold atom experiment landscape for space and established a corresponding Roadmap for cold atoms in space.12 

One of the main goals of the workshop was to establish a pathway for forming an international TVLBAI proto-collaboration. This collaboration will bring together researchers from various institutions to plan and secure funding for terrestrial large-scale AI projects. The aim is to develop a roadmap that outlines the design and technology choices for one or several kilometer-scale detectors, which will be ready for operation in the mid-2030s.

The Workshop Summary, presented here, represents an important step toward the formation of the TVLBAI Proto-Collaboration. This will be followed by a town hall meeting in fall 2023, where interested communities, technology experts, and potential host site stakeholders will gather to discuss the formation of the collaboration. Templates from previous collaborations will be presented to guide the structured discussion and reach a consensus.

Atom interferometers have potential applications to some of the central topics in fundamental physics, namely the nature of dark matter and measurements of gravitational waves.

One of the prominent schools of thought about dark matter13–16 is that it may consist of waves of ultralight bosonic fields forming coherent waves that move non-relativistically through the Universe. Many models of such ultralight dark matter (ULDM) candidates postulate that they would have very weak interactions with the particles of the Standard Model (SM) that make up the visible matter in the Universe. As we discuss below, large-scale atom interferometers are extremely sensitive to such interactions of ULDM fields with SM particles,17 thanks to the very precise measurements of atomic properties that they enable.

Atom interferometers are also sensitive to the small distortions of space-time induced by the passage of gravitational waves (GWs).18 The LIGO and Virgo laser interferometers discovered GWs emitted during the mergers of black holes (BHs) and neutron stars (NSs)19 (and have now been joined by the KAGRA detector20) pulsar timing arrays have recently published evidence of nHz GWs that may be emitted by binary systems of supermassive BHs (SMBHs),21–24 and mergers of SMBHs are among the prime target of the planned LISA,25 TianQin26 and Taiji27 space-borne laser interferometers. As we discuss in more detail below, large-scale atom interferometers are sensitive to GWs in a frequency range intermediate between LISA and LIGO/Virgo, offering complementary measurements of GWs that might cast light on the mechanisms for forming supermassive BHs including the detection of mergers of intermediate mass BHs28—the missing link between stellar-mass and supermassive BHs.

Atom interferometers can in addition test the limits of quantum mechanics,29 probe quantum effects in gravitational fields,30 test QED and the Standard Model31 and provide stringent tests of general relativity by probing Einstein's Equivalence Principle.32 Beyond these significant contributions to fundamental physics, the unprecedented sensitivity and accuracy of atom interferometry has important applications in the measurement of many physical quantities, such as acceleration, local gravity, and rotation, which can be used to create highly sensitive sensors for navigation, gravimetry and geodesy.12 

Motivated by these many potential applications of atom interferometry to fundamental physics and beyond, it is a rapidly-developing field of research that is moving out of the laboratory and up to larger scales. Many of the technological challenges involved in the design and operation of large-scale atom interferometers are being addressed, and research into its possibilities is advancing rapidly.

The study of matter waves and their interference dates back to the early days of quantum mechanics, following the proposal of wave-particle duality by Louis de Broglie,33 which was confirmed for electrons in experiments by Davisson and Germer34 and Thomson35 and for atoms by Esterman and Stern.36 These experiments suggested that the principle of interferometry could be extended to matter waves, but the development of laser cooling and trapping techniques in the 1980s and 1990s greatly expanded the possibilities for this area of research.

The advent of lasers enabled precise control over the internal states and momenta of atoms, creating a new field of research using cold atoms and the creation of Bose–Einstein condensates (BECs), which opened different possibilities for the study of matter waves. The first complete atom interferometers37,38 were prepared using microfabricated diffraction gratings, while the first atom interferometers based on laser pulses were constructed by the group of Bordé39 and by Kasevich and Chu.40 See Refs. 41 and 42 for introductions and reviews of atom interferometry and some applications.

As illustrated in Fig. 1, the principle of atom interferometers is similar to that of optical interferometers, but with physical beamsplitters and mirrors replaced by laser–atom interactions. Clouds of cold atoms are addressed by a laser beam. In the simplest case, an initial π / 2 laser pulse splits the cloud into equal populations of ground- and excited-state atoms, the latter with a momentum kick due to the absorption of a photon, separating it from the ground-state cloud. A subsequent, longer π laser pulse inverts the populations of ground and excited states, which are brought back together, and a final π / 2 pulse then acts as another beamsplitter and the numbers of atoms in the ground and excited states are read out by, e.g., fluorescence imaging. These are sensitive to phase shifts induced by interactions with ULDM and the passage of GWs.

Fig. 1.

Left: Outline of the principle of a Mach–Zehnder laser interferometer, as described in Refs. 43 and 44. Right: Outline of an analogous atom interferometer. Atoms in the ground state, | g , are represented by solid blue lines, the dashed red lines represent atoms in the excited state, | e , and laser pulses are represented by wavy lines.

Fig. 1.

Left: Outline of the principle of a Mach–Zehnder laser interferometer, as described in Refs. 43 and 44. Right: Outline of an analogous atom interferometer. Atoms in the ground state, | g , are represented by solid blue lines, the dashed red lines represent atoms in the excited state, | e , and laser pulses are represented by wavy lines.

Close modal

Achievements of atom interferometers already include a sensitive terrestrial test of the Einstein Equivalence Principle (EEP) by observing the free fall of clouds of 85Rb and 87Rb, which verified the EEP at the 10 12 level.32 They also include a precise determination of the fine structure constant at the 10 11 level through precise determination of the recoil velocity induced by coherent scattering of a photon from a 87Rb atom.31 Another experiment used pairs of 87Rb clouds launched simultaneously to different heights in a vertical interferometer to verify a gravitational analog of the Aharonov–Bohm effect,30 namely a phase shift induced by the gravitational field of a tungsten source mass placed close to one of the clouds.

The future sensitivities of atom interferometers to ULDM and GWs will depend, in particular, on the intensities of the atom sources and the separations between their trajectories generated by the laser pulses. The former control the level of atom shot noise and the latter are increased by repeating the laser pulse sequence many times to obtain large momentum transfers (LMT). These are key R&D targets on the path toward realizing the potential of very large atom interferometers.

In Secs. III and IV of this report, we first summarize the physics case for such very large atom interferometers and then discuss the synergies between them and laser interferometers. The following sections review the cold atom technology developments that will be required for realizing this science program, describe the principal detector options that are under consideration, and review possible site options. The final sections discuss supplementary topics and summarize the contents of the report.

The Standard Model of particle physics is one of the most successful theories constructed by human beings. It is able to describe the physics of matter from sub-nuclear distances as small as 10 18 m to the scale of the cosmos 10 28 m. It has also withstood every direct experimental test that it has been subjected to over the past thirty years. Despite this unprecedented success, we know that the Standard model is not a complete theory of nature. This failure is manifest both on observational and theoretical fronts. On the observational side, the Standard model cannot account for the matter/anti-matter asymmetry. Nor can it explain the nature of dark matter. On the theoretical side, we know that the Standard model cannot describe the physics of strong gravity such as gravitational singularities encountered in black holes and the Big Bang singularity of the early universe. It is also beset with theoretical puzzles such as the hierarchy, cosmological constant and strong CP problems where estimates of values of physical parameters based on well understood calculation principles are in massive contradiction with observational data. How can we make experimental progress on these issues given the obstinate agreement between experiment and theory when we directly probe the Standard model?

Over the past century, human mastery over electromagnetism has enabled us to probe physics at higher and higher energies through a variety of particle colliders. These colliders are the right technology to probe physics at high energies that have reasonably large interactions with particles such as electrons and protons. However, they are statistically unable to probe new physics of any mass that interacts weakly with these particles. This raises the interesting possibility that there might be weakly coupled new physics at low energies, opening a hitherto under-explored frontier in the hunt for new physics beyond the Standard model. This possibility is bolstered by a compelling theoretical case for probing low-mass, weakly coupled particles.

For example, the universal nature of gravitation implies that every object in the Universe, past or present, is able to send signals to us through gravitational waves. Direct detection of these gravitational waves enables us to probe the dynamics of black hole space-times as well as the physics of the infant universe prior to era of recombination. This unique opportunity permits us to unveil the secrets of gravity in regimes that have never been observationally probed. Given the historic discoveries of gravitational waves by the LIGO/Virgo collaboration, there is a strong case for further exploration of the gravitational wave spectrum. Experiments that are able to explore other parts of this spectrum are guaranteed to make discoveries. In addition to this, there is also a strong theoretical case to look for new, weakly coupled particles. The existence of dark matter strongly suggests that the new physics likely interacts weakly with the Standard Model. Popular solutions to outstanding theoretical puzzles such as the strong CP, hierarchy, and cosmological constant problems also feature the existence of such particles. Given these exciting possibilities, what technology gives access to this space?

The key technological requirement is a sensing platform that combines high precision with superior noise cancelation capabilities, in order to be sensitive to the weak signals expected from this kind of physics. Atom interferometry offers an interesting solution to this technological challenge. It combines the pristine quantum-mechanical nature of atomic properties with noise cancelation abilities inherent to interferometry to make possible sensors that have exquisite sensitivity to accelerations, energy shifts and spin precession–the dominant ways in which new physics can affect Standard model sensors. The demonstrated performance of these sensors is sufficient to probe new regions of parameter space and foreseeable advances made possible by focused efforts will lead to the creation of advanced sensors that can expand significantly the searches for gravitational waves and new physics.

In the following, we discuss some of the physics opportunities that can be exploited by a variety of atom interferometry configurations. We begin in Sec. III B by focusing on gravitational waves in the frequency band 0.01–10 Hz (the “dHz range”) where terrestrial atom interferometers have unique capabilities for gravitational wave detection. Following this, in Sec. III C we discuss the ability of atom interferometers to probe a variety of ultralight dark matter candidates. We then discuss new tests of quantum mechanics in Sec. III D, followed by discussions of probes of new fundamental interactions in Sec. III E and probes of the charge neutrality of atoms in Sec. III F.

Figure 2 illustrates the potential sensitivities of terrestrial atom interferometers to gravitational waves based on studies of the Atom Interferometer Observatory and Network (AION) in its planned 100 m and 1 km versions.5,45 The dashed line labeled GGN indicates the potential impact of gravitational gradient noise assuming the Peterson new low-noise model (NLNM) model46 without implementing any of the mitigation strategies discussed in Sec. IV B. We also include other planned and currently running experiments starting from high frequencies with LIGO/Virgo/KAGRA and their latest O3 data47 and design sensitivity,48 as well as the planned ET detector.49 Below the optimal frequencies of atom interferometers we see the sensitivity of LISA25 and at much lower frequencies those of pulsar timing arrays (PTAs) and SKA.50 We also include gray violins indicating the fit to a stochastic gravitational wave background reported by the NANOGrav PTA in their 15-year data,21 which is corroborated by data from other PTAs,22–24,51–58 and also indicate the prospective sensitivities of the space-borne atom interferometer experiments AEDGE and AEDGE+.45,59

Fig. 2.

Sensitivities to the energy density of GWs, Ω GW h 2, using power-law integration of the proposed terrestrial atom interferometers AION-100, AION-km, as well as the space-borne incarnations of the technology AEDGE and AEDGE+, together with other existing and planned experiments LIGO/Virgo/KAGRA (LVK), ET, PTAs, and SKA. Also shown in gray are likelihood distributions in each frequency bin for the 15-year GW data reported by the NANOGrav Collaboration, as described in Ref. 21.

Fig. 2.

Sensitivities to the energy density of GWs, Ω GW h 2, using power-law integration of the proposed terrestrial atom interferometers AION-100, AION-km, as well as the space-borne incarnations of the technology AEDGE and AEDGE+, together with other existing and planned experiments LIGO/Virgo/KAGRA (LVK), ET, PTAs, and SKA. Also shown in gray are likelihood distributions in each frequency bin for the 15-year GW data reported by the NANOGrav Collaboration, as described in Ref. 21.

Close modal

The peak sensitivity of atom interferometers such as AION, positioned between the terrestrial interferometers LIGO/Virgo and the LISA space mission, makes them ideal tools for probing many astrophysical phenomena otherwise beyond our reach. It would be ideal for measuring the high-frequency tails of mergers and ringdown stages of intermediate-mass black hole binaries. It can also observe the early infall stages of mergers that subsequently merge within the LIGO/Virgo and ET frequency band. These prospects were discussed in detail in Refs. 5 and 45, for more details and updated predictions see also Sec. IV C. Proposals for space-borne laser interferometer projects beyond LISA to target similar frequency ranges include BBO60 and DECIGO.61 

Another interesting target unique to GW observatories is the direct observation of early Universe phenomena through primordial stochastic backgrounds of GWs.62 One of the possible sources at such early times is a first-order phase transition, as appears in a plethora of scenarios for physics beyond the SM (BSM). The dynamics of such transitions are largely described by their energy compared to the background, parameterized by α, and the ratio of their inverse time duration to the Hubble time, β / H, as well as the transition temperature T *. Figure 3 shows the sensitivity of AION and other planned experiments to phase transitions in terms of these parameters. We show fixed values of β / H = 10, 102, and 103 using, for simplicity, the sound-wave spectra calculated in Refs. 63–65. Note that the latter are most appropriate for relatively weak phase transitions, i.e., α 0.1. We see that AION-km has good sensitivity for transition temperatures between T * = 10 3 and 105 GeV, provided α 0.1 and β / H 10 2. It is important to point out that probing the spectrum in a large range of frequencies would be necessary to measure well the shape of the spectrum and identify a phase transition as the source behind the signal.

Fig. 3.

Sensitivities in the ( T * , α ) plane of AION-100 and -km, as well as other planned experiments, to the SGWB spectrum from sound waves in the plasma that could be formed in the aftermath of bubble collisions. Dashed lines show SNR = 1 while solid lines SNR = 10 except for AION-km GGN for which SNR = 10 is depicted by a thick dashed line while the dotted line corresponds to SNR = 1. Figure reused with permission from Badurina et al., Philos. Trans. R. Soc., A 380, 035 (2022). Copyright 2022 Royal Society Publishing.45 

Fig. 3.

Sensitivities in the ( T * , α ) plane of AION-100 and -km, as well as other planned experiments, to the SGWB spectrum from sound waves in the plasma that could be formed in the aftermath of bubble collisions. Dashed lines show SNR = 1 while solid lines SNR = 10 except for AION-km GGN for which SNR = 10 is depicted by a thick dashed line while the dotted line corresponds to SNR = 1. Figure reused with permission from Badurina et al., Philos. Trans. R. Soc., A 380, 035 (2022). Copyright 2022 Royal Society Publishing.45 

Close modal

The electroweak symmetry breaking (EWSB) transition, which occurs at an energy scale T * 100 GeV, offers a particularly interesting test-bed in this context, since it certainly took place in the early Universe, and represents the frontier between “known” high-energy physics—the Standard model of particle physics that has been tested on Earth in particle experiments—and its untested higher-energy extensions. Searching for a SGWB signal from such a phase transition therefore provides a probe of possible physics beyond the Standard Model (BSM). BSM scenarios for which the electroweak phase transition is of first order typically predict weakly first-order (and consequently brief) phase transitions, characterized by 10 2 < β / H * < 10 4.66,67 The corresponding SGWBs are outside the sensitivity range of LISA, but could be detected by an interferometer operating at higher frequency. Consequently, the detection frequency range of an atom interferometer such as AION-km has the advantage of probing favored regions in the EWSB parameter space. Furthermore, as can be appreciated from Fig. 3, AION-km would be sensitive to first-order phase transitions occurring also at higher temperatures than EWSB, providing the opportunity to test the fundamental high-energy theory beyond the reach of any present or near-future particle collider.

Another potential source of gravitational waves from the Early Universe is a cosmic string network. If produced in a phase transition at very high energies, it would continue emitting GWs until today, producing a spectrum featuring a relatively flat plateau over a large range of frequencies.68–70 The recent GW signal in the 15-year data from NANOGrav21 could, in fact, be fitted very well with such a signal52,71 (for earlier analysis of this potential source, see Refs. 72–74) provided the mass per unit length of the network lies within G μ × 10 11 10 12 with intercommutation probability p 10 3 10 1.75 This interpretation would indicate that the signal could also be measured in AION-km as well as LISA, ET, and AEDGE, although not necessarily in upcoming runs of the LIGO, Virgo, and KAGRA experiments.75 Should this interpretation prevail, the measurement of the spectrum over a wide range of frequencies would also enable the mapping of the expansion rate of the Universe, as any modification would leave its imprint on the spectrum. We show examples of modifications of the spectrum coming from an early period of matter domination and kination in the left panel of Fig. 4. However, it is important to point out that even much smaller modifications such as a change in the number of relativistic degrees of freedom could be measured. The right panel of the same figure shows the reach of experiments in terms of the temperature at which the expansion rate is modified.

Fig. 4.

Left panel: cosmic super string spectrum with G μ = 10 11.75 and intercommutation probability p = 10 2.25 in standard cosmology together with its possible modifications by a period of kination or matter domination (MD) ending at temperatures T > 5 MeV and 5 GeV. The gray violins indicate the spectra capable of explaining the NANOGrav 15 y data. Right panel: Sensitivity of various experiments to a modification of the expansion rate at a temperature T Δ for a given value of the string tension G μ with p = 1. The gray bands indicate values favored by the NANOGrav 12.5 y data.72,76 The right panel is reused with permission from Badurina et al., Philos. Trans. R. Soc., A 380, 035 (2022). Copyright 2022 Royal Society Publishing.45 

Fig. 4.

Left panel: cosmic super string spectrum with G μ = 10 11.75 and intercommutation probability p = 10 2.25 in standard cosmology together with its possible modifications by a period of kination or matter domination (MD) ending at temperatures T > 5 MeV and 5 GeV. The gray violins indicate the spectra capable of explaining the NANOGrav 15 y data. Right panel: Sensitivity of various experiments to a modification of the expansion rate at a temperature T Δ for a given value of the string tension G μ with p = 1. The gray bands indicate values favored by the NANOGrav 12.5 y data.72,76 The right panel is reused with permission from Badurina et al., Philos. Trans. R. Soc., A 380, 035 (2022). Copyright 2022 Royal Society Publishing.45 

Close modal

For masses below approximately 1   eV, a bosonic ultralight dark matter (ULDM) field within our galaxy could be effectively described as a superposition of classical waves.80 The coherent oscillations of these ULDM waves would give rise to a diverse range of time-dependent signals that could be explored using atom interferometers. These signals encompass various phenomena, including the time-dependent oscillations of fundamental “constants” in the context of scalar ULDM candidates,45,77,81,82 the time-dependent differences in accelerations between atoms in theories involving vector candidates,17 and the time-dependent precession of nuclear spins in the case of pseudoscalar candidates.83 In general, these signals have a frequency determined by the ULDM mass, an amplitude proportional to the local ULDM density and mass, and a coherence time that depends on both the ULDM mass and the ULDM virial velocity within our galaxy.84 

The left panel of Fig. 5 shows sensitivity projections for 100 m and 1 km baseline vertical gradiometers to the linear coupling of a scalar ULDM field to electrons ( d m e) as a function of the ULDM mass ( m ϕ). This linear coupling induces an effective time-dependent correction to the electron mass, which, in turn, induces small time-dependences in the energy levels of atoms.85 (There are similar sensitivities for scalar ULDM couplings to the photon and to quarks.) The projections in the left panel of Fig. 5 have been calculated for the “clock” transition in 87 Sr and follow the procedure outlined in Ref. 5. For the 100 m baseline, we assume 1000   k atom optics and 10 4   rad / Hz phase resolution; while for the 1 km baseline, we assume 2500   k atom optics and 10 5   rad / Hz phase resolution. The atom interferometer sensitivity oscillates as a function of the ULDM mass, as illustrated by the light-blue 100 m curve in the left panel of Fig. 5. However, the peaks and troughs can be shifted by running with slightly different interrogation times,86 so it is usually only the envelope of the oscillations that is plotted (darker blue and green lines). It is important to note that these projections are atom-shot-noise limited and, therefore, do not take into account gravity gradient noise (GGN), which is expected to impact the 1 km baseline projections below approximately 1 Hz in the absence of mitigation strategies: see the discussion in Sec. IV B.

Fig. 5.

Left panel: projections for sensitivities to scalar ULDM linearly coupled to electrons (shot noise limited and assuming SNR = 1). The lighter-blue 100 m baseline curve shows the oscillatory nature of the sensitivity projections, while the darker-blue and green curves show the envelope of the oscillations. Right panel: parameter reconstruction, adapted from with permission from Badurina et al., “Super-Nyquist ultralight dark matter searches with broadband atom gradiometers,” arXiv:2306.16477. Copyright 2023 Authors, licensed under a Creative Commons Attribution Unported License,77 of an injected signal with f ϕ = 9.1   Hz and d m e = 3.7 × 10 5 (green cross) for a 1 km baseline assuming a constant sampling frequency of 0.3 Hz. The purple contours show the islands of parameter space compatible with the signal at 95.4% CL. In both panels, the shaded orange region shows constraints from MICROSCOPE as described in Refs. 78 and 79.

Fig. 5.

Left panel: projections for sensitivities to scalar ULDM linearly coupled to electrons (shot noise limited and assuming SNR = 1). The lighter-blue 100 m baseline curve shows the oscillatory nature of the sensitivity projections, while the darker-blue and green curves show the envelope of the oscillations. Right panel: parameter reconstruction, adapted from with permission from Badurina et al., “Super-Nyquist ultralight dark matter searches with broadband atom gradiometers,” arXiv:2306.16477. Copyright 2023 Authors, licensed under a Creative Commons Attribution Unported License,77 of an injected signal with f ϕ = 9.1   Hz and d m e = 3.7 × 10 5 (green cross) for a 1 km baseline assuming a constant sampling frequency of 0.3 Hz. The purple contours show the islands of parameter space compatible with the signal at 95.4% CL. In both panels, the shaded orange region shows constraints from MICROSCOPE as described in Refs. 78 and 79.

Close modal

The right panel of Fig. 5 shows an example of ULDM parameter reconstruction in the event of a (simulated) 5 σ discovery assuming a 1 km baseline, 2500   k atom optics, 10 5   rad / Hz phase resolution, 108 s integration time, and a constant sampling frequency of 0.3 Hz.77 The green cross shows the parameters of the injected signal, while the purple contours show the islands of parameter space compatible with the signal at 95.4% CL. The injected signal falls within the 95.4% CL contour of one island. The multiple islands arise from aliasing, since the injected signal is above the Nyquist frequency.77 As the signal strength increases, or if the sampling frequency is increased, the number of islands decreases. Each island has a width δ f ϕ 10 6 Hz, which is determined by the integration time and the degree to which the time-series data are stacked.77 

Next, we consider the shot-noise-limited sensitivity to a BL coupled vector ULDM candidate. The projected sensitivity, shown in the left panel of Fig. 6 in terms of the vector mass (mA) and coupling ( g B L),87,88 arises from comparing the accelerometer signals from two simultaneous atom interferometers run with dual species (here, 87 Sr and 88 Sr). The sensitivity is quoted in terms of the acceleration sensitivity, where the blue line could be achieved with a 100 m baseline, 100   k atom optics and 10 3   rad / Hz phase resolution, while the green line could be achieved with a 100 m baseline, 1000   k atom optics and 10 4   rad / Hz phase resolution.

Fig. 6.

Left panel: shot noise limited projection, adapted with permission from Abe et al., Quantum Sci. Technol. 6, 044003 (2021). Copyright 2021 IOP Publishing,2 to B–L coupled vector ULDM for a dual-species interferometer ( 87 Sr and 88 Sr). The projections are given in terms of the acceleration sensitivities achievable with VLBAI (see the text). The shaded orange region shows constraints from MICROSCOPE as described in Ref. 78. Right panel: Shot noise limited projection, adapted with permission from Graham et al., Phys. Rev. D 97, 055006 (2018). Copyright 2018 Authors, licensed under a Creative Commons Attribution Unported License, to the spin coupling of pseudoscalar ULDM to atoms. The projections are given in terms of the interrogation time. The shaded yellow region shows bounds from supernova cooling.

Fig. 6.

Left panel: shot noise limited projection, adapted with permission from Abe et al., Quantum Sci. Technol. 6, 044003 (2021). Copyright 2021 IOP Publishing,2 to B–L coupled vector ULDM for a dual-species interferometer ( 87 Sr and 88 Sr). The projections are given in terms of the acceleration sensitivities achievable with VLBAI (see the text). The shaded orange region shows constraints from MICROSCOPE as described in Ref. 78. Right panel: Shot noise limited projection, adapted with permission from Graham et al., Phys. Rev. D 97, 055006 (2018). Copyright 2018 Authors, licensed under a Creative Commons Attribution Unported License, to the spin coupling of pseudoscalar ULDM to atoms. The projections are given in terms of the interrogation time. The shaded yellow region shows bounds from supernova cooling.

Close modal

Finally, the right panel of Fig. 6 shows the shot-noise-limited sensitivity to a pseudoscalar ULDM candidate. The spin coupling of the pseudoscalar field to atoms creates a spurious phase shift due to the precession of the spin induced by the pseudoscalar field, which can be measured by interfering two atoms in different nuclear spin states. The projections shown in Fig. 6 in terms of the pseudoscalar mass (ma) and coupling (gaNN) assume the use of a resonant sequence that amplifies the phase shift at the pseudoscalar frequency89 and a phase resolution of 10 4   rad / Hz. The sensitivity is shown for two values of the interrogation time. While 1 s can be achieved with a O ( 10 )   m baseline, 10 s is expected to be achievable with a 1 km baseline.

Quantum mechanics is the bedrock of physics. However, its axioms were derived phenomenologically, leading to the enticing possibility that these axioms are an approximation to a more complete theory. There is a strong need to develop formalisms for consistent deviations from these axioms and probe them experimentally.

1. Linearity

One of the central axioms of quantum mechanics is the linearity of time evolution. In every other system, linearity is an approximation. Why should quantum mechanics be perfectly linear? Following the failure of prior attempts,90,91 it has been widely believed that linearity is necessary for causality. However, it was shown in Ref. 92 that this belief is false. Furthermore, this work showed that causal nonlinear evolution that preserves other symmetries of nature (such as gauge invariance and conservation laws) was possible in quantum mechanics. The key point of Ref. 92 is as follows. In quantum mechanics, when we consider a particle like an electron moving from one point to another, the particle takes all possible paths in physical space in going between these points. In linear quantum mechanics, when we consider the motion of two electrons, the paths taken by one electron affects the paths taken by the other electron via the electromagnetic force. This occurs causally. However, in linear quantum mechanics, the paths of a single electron do not influence each other via the electromagnetic force. Such an interaction would be nonlinear. Given that the paths of distinct electrons are able to affect each other causally, why should causality preclude nonlinearity, i.e., the paths of the single electron influencing each other? In linear quantum mechanics, the natural way to describe interactions in a causal way is quantum field theory. Thus, in Ref. 92, nonlinearities were directly introduced into quantum field theory and it was shown that a wide class of causal and consistent nonlinear time evolution was possible in quantum mechanics. This work also naturally realized the general structure that91 showed must be present in any causal nonlinear quantum mechanical theory.

Excitingly, this work also showed that prior experimental limits on causal nonlinear quantum mechanical systems from probes of pristine quantum mechanical systems such as atomic and nuclear physics were weak. However, this physics can be readily probed in dedicated experiments. A preliminary suite of experiments was performed to test the electromagnetic aspects of this framework.93,94 Present and future atom interferometers will be able to test the gravitational aspects of this framework with significant sensitivity, well beyond present limits on such non-linearities in the gravitational sector.

In linear quantum mechanics, the time evolution of a quantum state | χ is described by a convolution of the state | χ with a path integral that describes the evolution of the basis elements of the Hilbert space of the theory. This path integral is independent of the state | χ . In Ref. 92, nonlinear evolution is introduced by making this path integral depend on the state | χ by changing the Lagrangian in a state-dependent way. For example, in linear quantum mechanics, the interaction of the gravitational field g μ ν with a scalar field ϕ is described by the term g μ ν μ ϕ ν ϕ. In nonlinear quantum mechanics, the interaction is instead described by the term ( g μ ν + ϵ χ | g ̂ μ ν | χ ) μ ϕ ν ϕ, where χ | g ̂ μ ν | χ is the expectation value of the metric operator g ̂ μ ν in the quantum state | χ . How can these terms be probed experimentally? Nonlinearities permit different arms of a superposition to interact with each other. In linear quantum mechanics, the effects of a quantum superposition can only be measured if the system is quantum coherent. Bizarrely, in nonlinear quantum mechanics, the effects of a quantum superposition can be detected even in the presence of decoherence.91,92 This leads to the following experimental concept.

First, one creates a macroscopic quantum superposition. This superposition need not be quantum coherent and is, thus, easily created by performing a measurement on a quantum system such as a spin 1/2 system. When such a measurement is made, the Schrodinger equation predicts that this creates a macroscopic superposition of two “worlds”: one in which the spin was measured to be up and in another where the spin was measured to be down. At a fixed location in the laboratory, we place a test mass when spin up is obtained. At the same location in the laboratory, if we get spin down, we place an accelerometer such as an atom interferometer. In this quantum state, even in linear quantum mechanics, the expectation value χ | g ̂ μ ν | χ at this particular point in the laboratory is non-zero. However, since the evolution is independent of χ | g ̂ μ ν | χ , this cannot be observed—the accelerometer that is entangled with spin down will not respond to the test mass that is entangled with spin up.

This is not the case in nonlinear quantum mechanics where χ | g ̂ μ ν | χ directly affects the evolution of the physical states. Thus, the accelerometer that is entangled with spin down will register an acceleration arising from the test mass that is entangled with spin up. Due to this nonlinearity, there is interaction between the two arms (i.e., “worlds”) of the superposition. This phenomenon was dubbed the “Everett phone” by Polchinski,91 and it is a requirement of any causal nonlinear modification of quantum mechanics.

An experiment of this kind can be naturally incorporated into the science program envisaged for a long baseline interferometry setup. The atom interferometers are natural accelerometers and, when arranged in a gradiometer configuration, they are naturally insensitive to a variety of systematics such as vibrational noise that affects accelerometry measurements. Test masses are also envisaged in these setups in order to search for new interactions (see Sec. III E). All that is missing is a quantum spin whose outcome determines where the mass and the accelerometer are placed. This can be readily obtained by either accessing a variety of publicly available quantum randomizers such as the IBM quantum processor or through in-house quantum randomizers constructed either with atomic systems or simple, low-activity radioactive sources. Conservatively, an accelerometer with sensitivity 10 13 g / Hz and a 100 kg test mass, operating for about 10 6 s can probe the nonlinear parameter ϵ down to ϵ 10 6, a million-fold improvement over the current ϵ O ( 1 ) on this parameter from.95 A null result in this experiment would also be hugely important, since it would prove the quantum nature of gravity.95 

Atom interferometers also have a unique ability to probe another key aspect of nonlinear quantum mechanics. Unlike linear quantum mechanics, the observational phenomenology of nonlinear quantum mechanics is fundamentally tied to the full quantum state,92 including its cosmological history. In conventional inflationary cosmology, where our observed universe is a tiny part of the entire wave-function of the universe, laboratory experiments of the kind described above will give null results even if the fundamental nonlinear term ϵ is large simply because of the small overlap of our universe with the full quantum state of the cosmos. However, in such a scenario, nonlinear terms can still be tested since these terms naturally violate the equivalence principle. Tests of the equivalence principle are one of the key goals of long baseline atom interferometers (see Sec. III E), and these tests are directly applicable in constraining these aspects of nonlinear quantum mechanics.

It is exciting that there is a consistent parameterized deviation from quantum mechanics.92 If nonlinear quantum mechanics is discovered in the experiments proposed above, it could lead to transformative effects on science and society. For example, if nonlinear quantum mechanics is discovered using the “Everett-phone” setup described above,91 it will lead to a technological revolution. This discovery would enable us to parallelize readily-available classical resources to solve a wide variety of problems, including computing problems. Moreover, any discovery of nonlinear quantum mechanics is likely to open new avenues for solving the black hole information problem in an experimentally testable manner. There is thus a very strong scientific and technological case for testing these theories.

2. Superposition

Among the tests of the foundations of Quantum Mechanics, it is also of fundamental importance to the limits of validity of the quantum superposition principle for larger systems. The reason why quantum properties of microscopic systems (in particular, the possibility of being in the superposition of two states at once) do not carry over to macroscopic objects has been subject of intense debate during recent decades.96–102 Its possible resolution could be a progressive breakdown of the superposition principle when moving from the microscopic to the macroscopic regime. The most important consequence would be to change fundamentally our understanding of Quantum Mechanics—now commonly considered as the fundamental theory of Nature—as an effective theory appearing only as the limiting case of a more general one.103 Several models have been proposed to account for such a breakdown of the quantum superposition principle. They go under the common name of (wavefunction) collapse models,103–105 and modify the standard Schrödinger dynamics by adding collapse terms whose action leads to the localization of the wavefunction in a chosen basis.

Another suggested motivation for collapse models, beyond having a universal theory whose validity stretches from the microscopic world to the macroscopic world, comes from a cosmological perspective. Collapse models have been proposed to justify the emergence of cosmic structures in the Universe, whose signatures are imprinted in the Cosmic Microwave Background (CMB) in the form of temperature anisotropies.106–108 Moreover, collapse models were also proposed as possible candidates to implement an effective cosmological constant, thus explaining the acceleration of the expansion of the Universe.109 The application of collapse models to cosmology is however not straightforward, as it requires a relativistic generalization of the non-relativistic models discussed below. How to build these relativistic generalizations of collapse models is still not clear: several proposals have been suggested,110–114 but each has limitations and the debate in the theoretical community is still open.

The most studied collapse model is the Continuous Spontaneous Localization (CSL) model,115,116 a phenomenological model that treats the system under scrutiny as fundamentally quantum but subject to the weak and continuous action of some measurement-like dynamics that occurs universally. The CSL model is characterized by two free parameters: the collapse rate λ, which characterizes the strength of the collapse, and the correlation length of the collapse noise r C, which is the length-scale defining the spatial resolution of the collapse and, thus, characterizing the transition between the micro and macro domains. Although extensive research over the past 20 years has set ever stronger upper bounds on these parameters,117,118 there is still a wide unexplored region in the parameter space. Since the structure of the CSL dynamics resembles that of a weak continuous Gaussian measurement (at zero efficiency, since the outcome of the measurement is not recorded)—which is a quite general framework—one typically regards it as a figure of merit for a wide class of collapse models.

Also worthy of mention is the Diósi–Penrose (DP) model,99,119 which is also considered among the most important collapse models. The DP model predicts the breakdown of the superposition principle when gravitational effects are strong enough. Penrose provided several arguments why there is a fundamental tension between the principle of general covariance in General Relativity and the superposition principle of Quantum Mechanics,119,120 suggesting that systems in spatial superposition should collapse spontaneously to localized states and that this effect should get stronger the larger the mass of the system. A model that describes this effect was introduced by Diósi in Ref. 121 and is known as the DP model. It is fully characterized by the Newtonian kernel G / 1 / | x y | , where G is the gravitational constant, so that the model is free from any fitting parameter. However, due to the standard divergences of the Newtonian potential at small distances, the collapse rate for a point-like particle diverges, irrespective of its mass. This implies an instantaneous collapse even for microscopic particles, in contrast to the requirements of the model. To avoid this divergence, one takes a Gaussian smearing of the Newtonian kernel of width R0, which becomes the free parameter of the DP model. Several experiments set lower bounds on R0,118,122 and the strongest bound is given currently by a search for spontaneous radiation emission from germanium.123 

The direct way to test collapse models is to quantify the loss of quantum coherence in interferometric experiments with particles as massive as possible, so as to magnify the collapse effects on the superposition.117 Currently, the most massive particle that has been placed in a superposition has had a mass around 2.5 × 10 4 amu.124 The corresponding bound is, however, around 9 orders of magnitude away from ruling out the CSL model. With the aim of testing such values, one would need to prepare superpositions with masses around 109 amu on a time-scale of 10 s,125 which is far beyond the current capabilities of the state-of-the-art and near-future technology.

In parallel to the interferometric approach, alternative strategies have been developed, which provide stronger bounds, without necessarily requiring the creation of a superposition state. They are based on indirect effects of the modifications collapse models introduce into quantum dynamics,118 such as extra heating and diffusion or spontaneous radiation emission. Among them, the measurement of the variance in position σ t 2 of a non-interacting BEC in free fall can be considered. It may be expressed as
σ t 2 = σ QM , t 2 + 2 6 m 0 2 r C 2 λ t 3 .
(3.1)
The variance is enhanced by the action of collapse models on the BEC with respect to that predicted by quantum mechanics σ QM , t 2 t 2, exhibiting a different scaling that is proportional to the cube of the free evolution time. This test can be implemented directly without requiring additional instrumentation beyond what is already envisioned for the interferometric experiment.

A study of BEC expansion has already set a competitive bound on CSL,29 which provides bounds four orders of magnitude stronger than interferometric experiments. This experiment was performed on the ground,126 where the major limitation was provided by gravity, which constrains the total duration of the experiments to a few seconds. In such an experiment, a BEC is created in a vertically-oriented quadrupole trap, allowed to evolve freely and cooled down through the use of a delta-kick technique to make σ QM , t as small as possible. Finally, it is again allowed to evolve freely, and eventually its position variance is measured.

As discussed in Sec. III C, dark matter candidates may induce a signal in atom interferometers,81 where two spatially separated devices are operated by common laser beams.82 The differential measurement suppresses common mode noise and compares the signal induced at two different points in space-time. Dark matter may induce signatures in both the motion and the internal (electronic) energy states of the atom.127 The working principle of atom-based GW detectors is similar, but the signature of a GW is encoded in the phase of the light pulse and read out by the atom interferometer.18,128,129 These examples highlight the point that light-pulse atom interferometers are based on the propagation of atoms and light, as well as on their interaction at different points in space time. Different approaches to model atom interferometers have mainly focused on the propagation of atoms,128,130–135 but also detailed studies of diffraction induced by atom-light interactions have been performed.136–140 These efforts have to be re-evaluated when designing detectors for BSM physics.

To include such fundamental interactions, one starts by introducing additional fields that give rise to novel physics. The hypothetical interactions with all known constituents of the Standard Model need to be described, that is, the interactions with elementary particles such as electrons, photons, gluons, quarks, and others. Typically, a complementary approach to high-energy or particle physics is chosen: classical light and weak BSM fields are assumed instead of second-quantized particles. The next step is to solve perturbatively the equation of motion of the field and its interaction with other particles. When the coupling between the new field and known particles is introduced, light-pulse atom interferometers require the treatment of light and atoms. The latter are composite particles, as are their nuclei, and require an effective description. One should consider both the center-of-mass motion of the composite particle and its internal states, as both are manipulated by light and are central to atom interferometry. One finds an effective coupling of the novel field to the center-of-mass motion as well as to the internal states, which in general differs from that to the individual elementary particles. Simultaneously, new fields may directly couple the photons that make up the light pulses through and modify Maxwell's equations.

In principle, the described procedure is generic and can be applied to various types of fields with different symmetries and properties, such as scalar fields like dilatons,141 but also to pseudoscalar fields142,143 such as axions,144,145 or dark photons,146,147 which could all in principle account for dark matter. There is a variety of possible extensions of the Standard Model, and a comprehensive overview is beyond the scope of this article. Instead, we focus on the example of a dilaton field.

In general, the dilaton couples non-trivially to all constituents of the Standard Model, and its couplings can be linearized since it is a weakly coupled field. Via this procedure, all coupling constants are modified linearly by the dilaton, including the fine structure constant or the electron and nucleon masses.141,148 In particular, both the atom's mass and its internal energy structure are modified and depend, to lowest order, linearly on the dilaton. Maxwell's equations are modified similarly, whereas no modification of Einstein's equations is found to lowest order, since the field is ultralight and couples weakly. The specific form of the dilaton field depends in general on the local environment and may be modified, e.g., by source masses such as the Earth. It obeys an inhomogeneous wave equation, where the homogeneous part gives rise to plane waves with a specific wave vector that corresponds to its momentum.127 This contribution serves as a model for dark matter, where the velocity distribution inferred from galactic observations is matched to the plane waves. The inhomogeneous solution arises from the metric and depends on the local mass-energy density. The modified Maxwell's equations lead only to an effect on the amplitude of a classical propagating light ray, not on its phase to leading order, so that no signature arises from light propagation in dark-matter backgrounds.127 In contrast, the atomic motion and the resulting interferometric phase are sensitive to the dilaton, and the internal energies and the mass of the atom depend on it.81,82 The inhomogeneous solution sourced by a local mass reflects itself in an apparent violation of the equivalence principle.32,149–152 It induces a mass- and internal-state-dependent acceleration. In contrast, the plane-wave solution that models dark matter gives rise to oscillations of internal energies, and also the total mass of the atom. Consequently, there are two read-out strategies. To measure an effect on the center of mass, the atom remains in the same internal state3,8,81 using Bragg diffraction.138–140,153 Such an experiment needs two counterpropagating beams so that laser phase noise couples for large spatial separations. In contrast, single photon transitions18,154 change the internal state, whose energy difference depend on the dilaton field. Similar to atomic clocks,155–157 this contribution is dominant,158 as it originates from the rest energy of the atom.2,82,127 The treatment sketched above has to be reviewed for other possible dark matter candidates, that might differ in their fundamental interactions with atom interferometers.

Although atom neutrality is commonly accepted, it raises the fundamental question of charge quantization in the framework of the SM,159,160 and therefore relies mainly on experimental observations. Several measurements of the electrical neutrality of matter have been performed using different laboratory approaches. All experimental evidence to date is consistent with atoms being electrically neutral, i.e., there is an exact matching between the charge of the electron (qe) and the proton (qp), and the neutron charge (qn) is zero. The best limits for the electron-proton charge asymmetry ( q p + q e ) / q e and the residual neutron charge q n / q e are near 10 21.161–163 Most of the methods used so far are measurements based on macroscopic dilute or bulk ensembles that suffer from difficulties in the modeling of systematic effects related to spurious charging effects or the inhomogeneity of electric fields.

Matter-wave interferometers with a macroscopic separation between interferometer arms allow one to shape electromagnetic and gravitational potentials,30,164,165 opening the way to new measurements in fundamental physics based on geometrical phase shifts. In particular, a new test of atom neutrality with an atom interferometer based on the scalar Aharonov–Bohm effect has been proposed.166,167 The scalar Aharonov–Bohm effect168 appears when opposite electric potentials ± V are applied on the two interferometer arms during a time τ. The electric potentials are turned on when the atoms are inside the electrode assembly where the electric field is vanishing. If the atom carries a non-zero electric charge δq, a phase shift proportional to δq is induced: Δ ϕ = δ qV τ / . Therefore, one can infer a limit on the atom neutrality and the charge per nucleon from the uncertainty in the Aharonov-Bohm phase shift. In addition, by performing the measurements on the two atomic species, one can place independent bounds on the neutron charge qn and on the electron-proton charge asymmetry q e + q p. Then, assuming charge conservation in neutron β-decay ( n p + e + ν ¯ e), it is possible to infer a limit on the neutrino ( ν ¯ e) electric charge with the same accuracy as q p + q e and qn.

The Aharonov–Bohm method can potentially surpass by orders of magnitude the current bounds on atom neutrality. In particular, very large scale atom interferometers offer opportunities for efficient implementation of such tests, as they provide large separations between the interferometer arms leading to well-separated interaction zones and very long duration of the voltage pulse τ. In addition, the gravitational antennas studied here anticipate an extremely high signal-to-noise ratio and a very low-noise environment,2 opening up the prospect of electrical neutrality tests below 10 28 q e.

Astrophysical methods can also provide bounds on the neutrality of atoms169,170 and neutrinos.171,172 However, these limits depend on specific assumptions and models. Therefore, the new bounds from atom interferometry could help to refine such astrophysical models, and might stimulate new studies in astrophysics. In addition, pushing the limits on atom neutrality is of great interest in particle physics, as the origin of the extreme fine-tuning between the charges of fundamental particles can be considered as a hint for new physics beyond the SM.159 Indeed, some specific models propose de-quantization of the electric charge.167,173 We close by recalling the comment that most of the experiments “were done decades ago, and at the time were rather one-[person] shows. This is a pity in view of the effort invested in other searches beyond the standard model”.174 

In this section we summarize three aspects of the synergies between terrestrial atom and laser interferometer experiments. Both types of experiment must confront and mitigate the effects of the Earth's seismic activity. Laser interferometers are particularly vulnerable to vibrations of their mirrors, which can be mitigated by the design of their mounts, whereas the motions of clouds in atom interferometers are sensitive to fluctuations in the Earth's gravitational field as it vibrates, called gravity gradient noise (GGN), which cannot be shielded. The synergies between the two types of experiment depend on the degree to which GGN effects can be minimized, allowing atom interferometers to explore frequency ranges that are inaccessible to laser interferometers and maximize their complementarity, as discussed in the first part of this section. Prospective synergies in studies of astrophysical GWs are discussed in the second part of this section. These include the observations of inspiral stages of stellar-mass black holes that will be observed later by laser interferometers such as LIGO/Virgo/KAGRA, ET and CE. These will provide tests of gravity including constraints on the graviton mass and Lorentz violation175 as well as predicting the times and directions of subsequent mergers, thereby facilitating multimessenger observations. Furthermore, observing the long inspiral phases of stellar-mass black hole binaries also allows important tests of their, yet mysterious, formation processes and of their environments, e.g., through by detecting the eccentricities of their orbits176–179 and/or of the peculiar velocities and accelerations of their centers of mass.180–182 Atom interferometers could also observe the mergers of intermediate-mass black holes (IMBHs), which could probe the strong gravity regime as well as probe the hierarchical merger history of SMBHs. Other aspects of the complementarity between atom and laser interferometers are discussed in the final part of this section.

Due to their exquisite sensitivity to the propagation of atoms and changes in their structure, terrestrial very long-baseline atom gradiometers will be exceptionally powerful probes of gravitational waves (GW) in the unexplored “mid-frequency band”18,45,175,183 and linearly-coupled scalar ultralight dark matter (ULDM) with mass between × 10 17 and × 10 11 eV.82,86

The projected reach of these experiments is ultimately limited by fundamental noise sources. (See Sec. IX B for a more complete discussion of possible noise sources.) For instance, vertical single-photon atom gradiometers such as AION5 and MAGIS-1002 are designed to reach the atom shot-noise limit above 1 Hz, but would suffer from GGN at lower frequencies.184–187 This type of phase noise arises as a result of mass density fluctuations of the ground and atmosphere,188 which perturb the local gravitational potential around the atom clouds and imprint a noisy phase shift in a differential measurement.

For the experimental configurations and frequency range of interest, the dominant source of GGN is expected to consist of ground density perturbations induced by horizontally propagating seismic waves, in particular fundamental Rayleigh modes, that are confined near the Earth's surface by horizontal geological strata and are generated at strata interfaces, such as the Earth's surface.188,189 Representing the seismic field as an incoherent superposition of monochromatic plane waves propagating isotropically at the Earth's surface, following previous studies performed for LIGO,188,189 and using the Peterson low and high noise models (NLNM NHNM) driven by seismic-noise data from different terrestrial locations, Ref. 190 showed that GGN could significantly impact the projected reach of experiments with baselines L O ( 100   m ) in the sub-Hz regime.

Several passive noise mitigation strategies could be implemented to recover large swathes of parameter space accessible to a purely shot-noise-limited device, especially in the crucial 0.1–10 Hz band. We focus here on ULDM searches, although the same applies in the context of GW searches with these large-scale quantum sensors, assuming the parameters listed in Table I. As illustrated in the left panel of Fig. 7, by choosing quiet sites (i.e., with GGN-limited sensitivity curves close to the NLNM curve), it may be possible to regain up to three orders of magnitude in sensitivity.

Table I.

List of experimental parameters used for the computation of the sensitivity plots shown in this Section, which could be implemented in future vertical gradiometers, such as AION-km. For reference, L is the length of the baseline, T is the interrogation time, 4 n 1 is the total number of LMT kicks transferred during a single cycle, δ ϕ is the shot noise-limited phase resolution, T int is the integration time, and Δ z max is the maximum gradiometer length given the choice of interferometer parameters. We also consider scenarios where Δ z is shorter than the maximum value. The set of geological parameters is taken from Ref. 190.

L (m) T (s) n Δ z max (m) δ ϕ   ( 1 / Hz ) T int (s)
1000  1.7  2500  970  10 5  108 
L (m) T (s) n Δ z max (m) δ ϕ   ( 1 / Hz ) T int (s)
1000  1.7  2500  970  10 5  108 
Fig. 7.

Impact of GGN on the projected 95% CL exclusion sensitivity to the ULDM-electron coupling of a single atom gradiometer with the design parameters defined in Table I. Left panel: comparison between the atom shot noise (ASN) (gray) and ASN-plus-GGN-limited sensitivities assuming that the GGN background is described by the Peterson NHNM (orange) or NLNM (blue). The solid and dotted lines are for Rayleigh wave velocities c H = 205   and c H = 3232   m   s 1, respectively. Right panel: projected 95% CL exclusion sensitivities for different values of Δ z and different atom interferometer positions, where we assume the NHNM and c H = 205   m   s 1. We show exclusion curves for interferometers located toward the Earth's surface (green) and toward the bottom of the shaft (purple), assuming Δ z = 100 m but keeping all other experimental parameters unchanged. In both panels, the orange shaded region is excluded by MICROSCOPE, as described in Ref. 78.

Fig. 7.

Impact of GGN on the projected 95% CL exclusion sensitivity to the ULDM-electron coupling of a single atom gradiometer with the design parameters defined in Table I. Left panel: comparison between the atom shot noise (ASN) (gray) and ASN-plus-GGN-limited sensitivities assuming that the GGN background is described by the Peterson NHNM (orange) or NLNM (blue). The solid and dotted lines are for Rayleigh wave velocities c H = 205   and c H = 3232   m   s 1, respectively. Right panel: projected 95% CL exclusion sensitivities for different values of Δ z and different atom interferometer positions, where we assume the NHNM and c H = 205   m   s 1. We show exclusion curves for interferometers located toward the Earth's surface (green) and toward the bottom of the shaft (purple), assuming Δ z = 100 m but keeping all other experimental parameters unchanged. In both panels, the orange shaded region is excluded by MICROSCOPE, as described in Ref. 78.

Close modal

It may also be possible to suppress GGN in the crucial mid-frequency band significantly by carefully choosing the experimental parameters, such as the vertical positions of a string of interferometers along the baseline. Since the GGN differential phase shift decays exponentially with depth,190 experiments that are deep underground and far from sources of fundamental Rayleigh modes are expected to be more powerful probes of GW and ULDM. As shown in the right panel of Fig. 7, where we assume the NHNM, experiments that are deeper underground would be characterized by a sensitivity enhancement of up to two orders of magnitude in the sub-Hz regime, but would be less sensitive to a signal above 1 Hz.

It would be advantageous to select sites whose geological composition supports Rayleigh waves with high horizontal propagation speed cH. As shown in the left panel of Fig. 7, below 1 Hz where the decay length exceeds the separation between the interferometers, the GGN-limited background is suppressed by several orders of magnitude at high cH.190 This is because the GGN-induced gradiometer phase shift at low frequencies is proportional to cH.

In geological media where Rayleigh modes propagate with a low horizontal speed, it may instead be advantageous to employ a network of N 3 atom interferometers along the same vertical baseline, i.e., a multigradiometer configuration.190 Since the GGN and ULDM signals scale differently with the gradiometer length, this design would facilitate the mitigation of GGN by up to two orders of magnitude in the mid-frequency band for different spatial configurations. This is clearly visible in Fig. 8, which shows the sensitivity reach in scalar ULDM parameter space of several illustrative configurations employing N = 5 interferometers and assuming the NHNM.

Fig. 8.

GGN mitigation using a multigradiometer configuration. Left panel: projected 95% CL exclusion sensitivities for an atom multigradiometer with the experimental parameters listed in Table I and N = 5 interferometers, assuming that GGN is modeled by the NHNM. The red dot-dashed, purple dotted and green solid lines show the atom multigradiometer exclusion curves for equally spaced, unequally spaced (ends), and unequally spaced (center) configurations. The orange shaded region is excluded by MICROSCOPE, as described in Ref. 78. For comparison, the gray and orange lines show the exclusion sensitivities for a single atom gradiometer ( N = 2) with ASN-only and ASN-and-GGN backgrounds,, respectively. Right panel: schematic representations of the three interferometer configurations with N = 5. The purple dots show the positions of the interferometers in the “unequal spacing (ends)” configuration, the red dots show their positions in the “equal spacing” configurations, and the green dots show the “unequal spacing (center)” configuration.

Fig. 8.

GGN mitigation using a multigradiometer configuration. Left panel: projected 95% CL exclusion sensitivities for an atom multigradiometer with the experimental parameters listed in Table I and N = 5 interferometers, assuming that GGN is modeled by the NHNM. The red dot-dashed, purple dotted and green solid lines show the atom multigradiometer exclusion curves for equally spaced, unequally spaced (ends), and unequally spaced (center) configurations. The orange shaded region is excluded by MICROSCOPE, as described in Ref. 78. For comparison, the gray and orange lines show the exclusion sensitivities for a single atom gradiometer ( N = 2) with ASN-only and ASN-and-GGN backgrounds,, respectively. Right panel: schematic representations of the three interferometer configurations with N = 5. The purple dots show the positions of the interferometers in the “unequal spacing (ends)” configuration, the red dots show their positions in the “equal spacing” configurations, and the green dots show the “unequal spacing (center)” configuration.

Close modal

These results show that more detailed and site-specific modeling will be needed to improve our understanding of the projected reach of these experiments in well-motivated parts of parameter space. For example, it would be desirable to assess whether specific sites have GGN levels closer to the NHNM or the NLNM, and the speeds of Rayleigh waves should be measured. Furthermore, in order to model correctly the GGN at realistic sites, the model presented here should be extended to anisotropic environments with different geological strata, which would give rise to a much richer spectrum of Rayleigh modes.188,189 In addition, beyond the simple noise mitigation strategies presented here, active GGN mitigation techniques could also be implemented. For example, it may be possible to model the GGN phase shift imprinted onto the atoms by using an array of seismic sensors, whose data would then be subtracted from the interferometer's data stream through Wiener filtering.191 

The analysis summarized here should be interpreted as a first stepping stone toward understanding how to characterize and ultimately suppress the GGN background within the context of terrestrial very-long baseline atom gradiometers. A discussion of possible mitigation strategies can be found in Sec. IX B 3.

Atom interferometers have the potential to bridge the frequency gap between LISA and ground-based laser interferometers, as shown in Fig. 9. As seen there, this frequency range is optimal for probes of inspiralling stellar-mass BH binaries and observing directly the mergers of IMBH binaries.

Fig. 9.

The GW strain sensitivities and benchmark signals from BH binaries of different masses at different redshifts. The colored dots indicate the times before mergers at which inspirals could be measured.

Fig. 9.

The GW strain sensitivities and benchmark signals from BH binaries of different masses at different redshifts. The colored dots indicate the times before mergers at which inspirals could be measured.

Close modal

As seen in Fig. 9, a space-based atom interferometer such as AEDGE could probe the inspirals of stellar mass BH binaries for several months. Depending on the level at which GGN can be suppressed, this may be possible also with very-long-baseline terrestrial atom interferometers. Over such a long observation time, the reorientation and movement of the detector with respect to the GW source become relevant. Because of these effects, the sky location of the binary can be measured very accurately.192 (For comparison, the LIGO/Virgo network sees only the last seconds of the signals, limiting its sky-localization accuracy without additional detectors. LISA will have similar accuracy to AION/AEDGE, but for heavier binaries.) In addition, as demonstrated in Table II, the long observation time enables a very accurate measurement of the binary chirp mass that controls how the frequency of the signal changes with time. Toward the end of the inspiral, hours or minutes before the merger and ringdown stages of the event, the GW signal leaves the sensitive frequency window of the atom interferometers. Since the time of the merger can be accurately predicted from their observations, together with the sky location, the atom interferometers provide an excellent early warning system for measurements of the GW signal from the binary merger as well as for searches for possible electromagnetic counterparts.

Table II.

Estimated accuracies of AION 1 km and AEDGE measurements of the chirp mass M c, the coalescence time tc, the luminosity distance DL and the sky location, specified by the angles θ and ϕ, assuming a GW150914-like benchmark source. As described in Ref. 175.

σ M c / M c σ t c / s σ D L / D L σ θ / ° σ ϕ / °
AION 1 km  2.2 × 10 5  12  0.7  2.7  2.7 
AEDGE  3.6 × 10 7  0.38  0.03  0.15  0.077 
σ M c / M c σ t c / s σ D L / D L σ θ / ° σ ϕ / °
AION 1 km  2.2 × 10 5  12  0.7  2.7  2.7 
AEDGE  3.6 × 10 7  0.38  0.03  0.15  0.077 

The long observation time of the inspiral phase enables also searches for modifications of general relativity that could alter the GW signal. For example, a modified GW dispersion relation E 2 = p 2 + A p α would introduce an extra phase for the GW signal, Ψ ( f ) Ψ ( f ) + δ Ψ ( f ), whose frequency dependence is determined by the magnitude A and the parameter α, δ Ψ A f α 1. Modifications with α < 1 change the phase more at small frequencies and are easier to probe from the inspiral phase than from the merger. Therefore, as illustrated in Fig. 10, the measurements of the inspiral signals of stellar mass BH binaries with atom interferometers will complement the measurements done with the ground-based laser interferometers, providing a better probe of modified dispersion relations with α < 1. In addition, the low-frequency sensitivity of AI gravitational-wave detectors is beneficial for probing extra radiation channels of gravitational waves, in particular the possibility of dipolar radiation,193 which is ultimately related to the strong equivalence principle.

Fig. 10.

Prospective sensitivities to modified GW dispersion relations of AION 1 km and AEDGE, compared with the constraints from LVK and gravitational Cherenkov radiation. Figure reused with permission from Ellis and Vaskonen, Phys. Rev. D 101, 124013 (2020). Copyright 2020 Authors, licensed under a Creative Commons Attribution Unported License.

Fig. 10.

Prospective sensitivities to modified GW dispersion relations of AION 1 km and AEDGE, compared with the constraints from LVK and gravitational Cherenkov radiation. Figure reused with permission from Ellis and Vaskonen, Phys. Rev. D 101, 124013 (2020). Copyright 2020 Authors, licensed under a Creative Commons Attribution Unported License.

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The population of BHs with masses intermediate between those whose mergers have been observed by LIGO and Virgo and those known to be present in the centers of galaxies has not been explored to the same extent, though insight into their potential importance for the assembly of SMBHs is now being provided by JWST observations of 107–108 solar-mass black holes at redshifts up to z = 10.194 Very large terrestrial atom interferometers offer prospects for detecting IMBH mergers that are inaccessible to laser interferometers, as seen in Fig. 9. We see there that the early inspiral stages of IMBH binaries could be measured by LISA25 and TianQin,195 enabling the timings and sky locations of subsequent IMBH mergers to be predicted and facilitating multimessenger studies in combination with atom inteferometer observations, complementing the observations of heavier binaries that will be made with LISA and providing opportunities to probe strong gravity in a novel regime. However, the prospects for observing IMBH mergers depend on the extent to which GGN can be suppressed or mitigated. The left panel of Fig. 11 displays the IMBH merger sensitivities at the SNR = 8 level in three GGN scenarios: the NHNM with cH = 205 m/s and no mitigation as in Fig. 8, the same scenario with 5 gradiometers as also seen in Fig. 8, and assuming total mitigation or suppression of the GGN. We see that in these scenarios AION 1 km could detect O ( 10 4 M ) binaries up to redshift z 2 , 10 and 70, respectively. Since the first astrophysical BHs are thought to form at z 20, we would not expect to observe mergers of IMBHs beyond this redshift unless they are primordial, which would be of major interest.

Fig. 11.

Left panel: The sensitivities of AION 1 km to GWs from equal mass BH binaries of total mass M at redshift z, calculated assuming a level of GGN close to the NHNM and assuming that Rayleigh waves propagate with a speed of 205 m/s. The contours compare estimates made assuming either no mitigation of GGN, or the level of suppression discussed in the previous subsection, or complete suppression/mitigation of GGN. Right panel: The mean GW energy density spectrum from massive BH mergers compared with the sensitivities of the indicated experiments. The colored bands correspond to different BH mass bands and are obtained assuming a constant merger efficiency factor 0.3 < p BH < 1, adapted with permission from Ellis et al., “Cosmic superstrings revisited in light of NANOGrav 15-Year Data,” arXiv:2306.17147. Copyright 2023 Authors, licensed under a Creative Commons Attribution Unported License:75 plot adapted with permission from, Ellis et al., Astron. Astrophys. 676, A38 (2023). Copyright 2023 Authors, licensed under a Creative Commons Attribution Unported License.196 

Fig. 11.

Left panel: The sensitivities of AION 1 km to GWs from equal mass BH binaries of total mass M at redshift z, calculated assuming a level of GGN close to the NHNM and assuming that Rayleigh waves propagate with a speed of 205 m/s. The contours compare estimates made assuming either no mitigation of GGN, or the level of suppression discussed in the previous subsection, or complete suppression/mitigation of GGN. Right panel: The mean GW energy density spectrum from massive BH mergers compared with the sensitivities of the indicated experiments. The colored bands correspond to different BH mass bands and are obtained assuming a constant merger efficiency factor 0.3 < p BH < 1, adapted with permission from Ellis et al., “Cosmic superstrings revisited in light of NANOGrav 15-Year Data,” arXiv:2306.17147. Copyright 2023 Authors, licensed under a Creative Commons Attribution Unported License:75 plot adapted with permission from, Ellis et al., Astron. Astrophys. 676, A38 (2023). Copyright 2023 Authors, licensed under a Creative Commons Attribution Unported License.196 

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The detection of IMBH mergers would enable studies of models for SMBH formation. In the absence of primordial SMBHs, they could have been assembled hierarchically from seeds such as BHs formed by the deaths of the first stars or by the direct collapses of gas clouds (for a review, see, e.g., Ref. 197), combined with accretion. Astrophysical seeds would have been much lighter than the SMBHs we detect in galactic centers today, with masses ranging from O ( 100   M ) in the stellar scenario to O ( 10 4   M ) in the gas-cloud scenario. The hierarchical merging of BHs originating from such seeds provides a potential probe of the SMBH formation through GW observations. Atom interferometers could probe the mergers of the seeds, potentially distinguishing between the seed scenarios.

The existence of SMBH binaries is currently being probed by pulsar timing arrays (PTAs), and first detections of a GW background signal that may come from such binaries have recently been reported.76,198–200 The right panel of Fig. 11 shows the mean energy of the GW background from BH binaries estimated by extrapolating the PTA results to lower BH masses and higher frequencies, assuming for simplicity a constant merger efficiency 0.3 < p BH < 175,196 and neglecting a possible additional contribution from low-mass seeds. Most of the GW signal expected in the LISA and atom interferometer frequency ranges is expected to be due to resolvable binaries, and it has been estimated that a space-based atom interferometer such as AEDGE could observe O ( 100 ) O ( 1000 ) IMBH mergers per year, while a terrestrial detector such as AION-km might observe as many as O ( 10 ) IMBH mergers per year,75,196 see also Ref. 201.

The results summarized here show that terrestrial very-long-baseline atom interferometers could complement laser interferometers by exploring an intermediate frequency range where measurements could be used to probe general relativity, provide “early warnings” of future mergers of stellar mass BHs, and observe for the first time mergers of IMBHs. We note finally that the sensitivity to GWs from BH mergers would be closer to the “without GGN” curve in Fig. 11 at a site where the GGN level is closer to the NLNM and/or the speed of Rayleigh waves is higher, emphasizing the importance of future site studies (see the discussion in Sec. IV A).

The spectacular technological advances in atom and laser interferometers are having a significant impact on defining the current landscape of searches of new physics. This brief contribution describes some of the “new physics” cases that have not been discussed in Secs. IV A to IV C, emphasizing how large atom interferometers and laser space interferometers can give complementary information on these problems. Finally, another direction that is gaining momentum among the community of physicists interested in precision measurements and fundamental physics is also mentioned briefly: the search for ultrahigh-frequency gravitational waves.

The LISA mission25 is expected to be launched in the mid 2030s. The final configuration and specifications are to be decided, but LISA is expected to be able to detect gravitational waves in a band 10 4 1 Hz, which has an enormous physics potential. In addition to the anticipated astrophysical signals, one of the best studied fundamental physics signals that could be measured by LISA is a stochastic background of GWs from cosmological sources.62 The physics of inflation, the existence and evolution of primordial black holes, the existence and dynamics of topological defects such as cosmic strings or the possibility of first-order phase transitions in the early Universe, especially at the electroweak scale, will be intensively explored by LISA. In all these cases it will be essential to make complementary observations in other frequency bands to establish the nature of any possible signal. Large atom interferometers offer a unique opportunity for exploring the frequencies between LISA and Earth-based laser interferometers, see, e.g., Figs. 4 and 5.

However, the topics above are far from being a complete list of the fundamental physics horizons that LISA will explore, see, e.g., Refs. 202 and 203. In order to assess the main scientific outputs of the LISA mission, and to plan and develop the work needed to ensure their delivery, the LISA Consortium has established a Science Investigation work Package (SIWP) within the LISA Science Group, which gives advice on how to optimize the LISA mission to maximize its scientific return. Furthermore, several Working Groups containing interested members of the scientific community also operate. Among these, the Cosmology Working Group and the Fundamental Physics Working Group are particularly focused on the ability of LISA to test “new” physics and explore novel ideas. The efforts within these Working Groups aim at exploring the vast potential of GWs to test both the late and the early Universe, and thereby probe fundamental questions such as the nature of gravity and high-energy particle theory beyond the Standard Model, and yet mysterious phenomena such as inflation, dark energy and the late-time expansion of the Universe, dark matter, primordial phase transitions and the unification of forces, baryogengesis, and so on. In the following, we focus mainly on three directions for new physics: dark matter, tests of black holes and tests of general relativity.

The dark matter studies are mostly devoted to two of the most significant ways in which it can impact the GW signals to be detected in LISA. (The possibility that part of the dark matter is made of primordial black holes may leave a signal in LISA, see, e.g., Ref. 202). The first is related to GWs from ultralight dark matter, and is based on rotational superradiance, the phenomenon behind the fact that in the presence of ultra-light bosonic fields with mass mb, a black hole of mass M 1 / m b (in G = c = 1 units) with large spin will develop a non-spherical cloud of the bosonic field.204 This cloud could have a mass that is a significant fraction of M, and would generate GWs as it rotates.205 Calculations show that LISA may be sensitive to any new boson with mass in the range 10 19 10 17 eV (many uncertainties fall into this projection, in particular there are different models for the population of highly spinning black holes of different mass), while LIGO, Virgo and KAGRA are sensitive to masses in the range 10 13 10 12 eV.206 Such particles are currently being explored as dark matter candidates, as discussed in Secs. III C and IV B, and atom interferometers can explore the mass range between the LISA, LIGO, Virgo and KAGRA ranges in two ways: by searching directly for ULDM as discussed in Subsection III C or by discovering mergers of IMBHs and measuring their spins, as illustrated in Fig. 12. Note, however, that these constraints are are model-dependent, and those shown in Fig. 12 assume negligible bosonic self-interactions.

Fig. 12.

Exclusions of weakly-interacting ultra-light bosonic fields from the measured spins of SMBHs and LIGO/Virgo/KAGRA BHs compared with the prospective sensitivity of a large atom interferometer, which could also exclude the intermediate mass range by measuring spins of IMBHs. These constraints assume negligible bosonic self-interactions.

Fig. 12.

Exclusions of weakly-interacting ultra-light bosonic fields from the measured spins of SMBHs and LIGO/Virgo/KAGRA BHs compared with the prospective sensitivity of a large atom interferometer, which could also exclude the intermediate mass range by measuring spins of IMBHs. These constraints assume negligible bosonic self-interactions.

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The second way to explore dark matter is via its effects on GW emission from binary systems. The orbital motion of a binary BHs may be affected by dark matter, modifying the general relativity predictions for GWs emitted during inspiral and infall, for which different possibilities have been explored in the literature.203,207–210 The observability of this effect with LISA may require the density of dark matter in the binary to be much higher than that reached in minimal models, but it is nevertheless an interesting target for LISA and for large atom interferometers.

Regarding tests of black holes, one possibility is that (some) of the compact objects producing GWs may not be BHs, but other exotic compact objects (ECOs). Different candidates are described in Ref. 211, and may correspond to new sectors of matter or to modifications of General Relativity (that may be inspired from quantum gravity or from agnostic modifications of the theory). The possible tests of this hypothesis from GWs are multiple and include new channels of emission, deviations from the unique properties and no-hair theorem of the Kerr metric (that uniquely describes a rotating black hole in General Relativity), changes in the tidal deformation and quasi-normal modes of the object, and even the presence of echoes of the original GW signal.202 The impact of the LISA mission duration on this possibility has been studied in Ref. 203. See Fig. 13 for results from a study of possible ECO signals in LIGO, LISA and atom interferometers, as well the background to be expected from the mergers of stellar mass BHs.212 

Fig. 13.

Sensitivities of LVK, LISA and large atom interferometers to GWs from mergers of ECOs weighing between 20 and 200 solar masses, compared with the backgrounds from BH-BH and BH-neutron star binaries, reused with permission from Banks et al., Phys. Rev. D 108, 035017 (2023). Copyright 2023 Authors, licensed under a Creative Commons Attribution Unported License.212 

Fig. 13.

Sensitivities of LVK, LISA and large atom interferometers to GWs from mergers of ECOs weighing between 20 and 200 solar masses, compared with the backgrounds from BH-BH and BH-neutron star binaries, reused with permission from Banks et al., Phys. Rev. D 108, 035017 (2023). Copyright 2023 Authors, licensed under a Creative Commons Attribution Unported License.212 

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As concerns tests of general relativity (GR), the approach followed by the SIWP is an agnostic one. The current and future probes of Lorentz-violating deviations from GR in the GW waveform203,213 using laser interferometers can be found in Refs. 214 and 215, and the sensitivities of atom interferometers have been studied in Ref. 175, see Sec. IV.

Some other new directions that have been discussed within the AION Collaboration. One is the possible influence of dark matter (DM) scattering on the atoms. (A background that we know is present is the one produced by neutrinos of cosmic origin, whose detection remains a challenge.216) Since the nature of DM is unknown, this search should consider all possible interactions to see if its scattering with the atoms may be substantial for some models. There may be detectable effects even at zero-momentum transfer that could fill a gap in searches for DM models below those explored by more traditional detectors, with possible spectacular bounds.217,218 If one considers a possible transfer of momentum, the trajectories of the atomic samples may be modified, something that will also impact the patterns of interferometry, see, e.g., Ref. 219 for related ideas with LISA.

Another direction to explore with atom interferometers is the possibility of improving the measurement of the gravitational (Newton) constant GN. This fundamental constant is not measured to high accuracy, and its value is the subject of some controversy.220 The measurement of GN would require a movable test mass located next to the interferometer, so that its effect on the atomic states may be measurable, see, e.g., Refs. 221–223. The presence of such a moveable mass would also allow for probes of a possible fifth force, see, e.g., Refs. 224 and 225.

It should also be noted that the dynamics of the lasers operating in atom interferometers may also be impacted by new physics: some inspiration from light interferometers can be found in Refs. 226 and 227.

For completeness, we mention another fundamental physics topic that is beyond the range of laser interferometers, namely the search for ultrahigh frequency GWs (UHFGWs).228 It seems extremely hard to generate a substantial GW signal in this region with standard astrophysical processes, but this provides an opportunity, since there are primordial or nonstandard processes that can do it. Detecting these high-frequency GWs requires small, laboratory-size experiments. The passage of a GW will slightly deform a solid, which was suggested long ago as a way to detect them.229 It will also generate photons out of a stationary magnetic field.230 These effects will, for instance, affect the dynamics of a loaded cavity (they may generate mode mixing).231 We have recently revisited these ideas and found them very promising for future searches of UHFGWs232,233 (see also Refs. 234 and 235) There are still many opportunities for synergies between cutting-edge precision devices and UHFGWs, and we hope that this contribution triggers new ideas in this direction.

The synergies between observations using atom interferometers and laser interferometers depend on the complementarity between the frequency ranges they can cover. Atom interferometers are potentially more sensitive in the mid-frequency band between terrestrial and space-borne laser interferometers, but realizing this potential gain depends on the extent to which atom interferometers can overcome GGN. Techniques for mitigating GGN in atom interferometers were discussed in Sec. IV B, and the resulting synergistic tests of gravity and probes of supermassive astrophysical black holes were discussed in Sec. IV C, while Sec. IV D discussed possible synergies in searches for new physics. As examples of the possible synergies, Fig. 11 illustrates the potential sensitivity of a 1-km atom interferometer to the mergers of intermediate-mass black holes that would have played key roles in the assembly of supermassive black holes, Fig. 12 illustrates the complementarity of atom interferometer searches for ultralight dark matter to current constraints from measurements of black hole spins, and Fig. 13 illustrates the sensitivities of large atom interferometers to exotic compact objects.

In this section, we summarize the status of the core atomic physics technologies needed to reach the target sensitivities for TVLBAI science. As described below, reaching the sensitivity required for gravitational wave detection will require a number of technological advances, chiefly associated with atom optics and the reduction of the noise associated with atom detection. Table III lists the current state-of-the-art for key performance metrics, along with the targets needed in each area to move toward a full-scale terrestrial gravitational wave detector. The technology development path will involve improving the pulse efficiency of large momentum transfer (LMT) atom optics, developing atom sources with increased flux at low temperatures, and integrating spin squeezing to reduce atom shot noise.

Table III.

State-of-the art performance of key sensor technologies and their improvement targets for a full-scale terrestrial detector. The sensitivity enhancement is stated relative to current instruments.

Sensor technology State-of-the-art Target Enhancement
LMT atom optics  10 2   k  10 4   k  100 
Matter-wave lensing  50 pK  5 pK  ⋯ 
Laser Power  10 W  100 W  ⋯ 
Spin squeezing  20 dB (Rb), 0 dB (Sr)  20 dB (Sr)  10 
Atom flux  105 atoms/s (Rb)  107 atoms/s (Sr)  10 
Baseline length  10 m  1000 m  100 
Sensor technology State-of-the-art Target Enhancement
LMT atom optics  10 2   k  10 4   k  100 
Matter-wave lensing  50 pK  5 pK  ⋯ 
Laser Power  10 W  100 W  ⋯ 
Spin squeezing  20 dB (Rb), 0 dB (Sr)  20 dB (Sr)  10 
Atom flux  105 atoms/s (Rb)  107 atoms/s (Sr)  10 
Baseline length  10 m  1000 m  100 

The first part of this section (Sec. V A) describes the use of LMT atom optics using additional laser pulses to enhance the beam separation and thus the sensitivity of atom interferometers. Clock atom interferometry, in particular, is highlighted as a technique that takes advantage of narrow transitions commonly used in atomic clocks. The use of these single photon transitions for atom optics enables improved common-mode suppression of laser frequency noise over long baselines and supports high pulse efficiencies, making it valuable for gravitational wave detection and dark matter searches.

The second part of this section (Sec. V B) focuses on LMT atom interferometers based on Bragg diffraction and Bloch oscillations. It emphasizes the importance of understanding and controlling diffraction phases and inefficiencies in these processes, and references the recent developments in theoretically modeling and characterizing these effects in order to suppress them. The final part of the section addresses the atom source technologies relevant to gravitational wave detectors based on atom interferometry. It discusses the increased atomic flux requirements needed to reach the targeted strain sensitivity, and how this compares to the current state of the art. Additionally, the control of external degrees of freedom of the atomic ensemble is highlighted as crucial for minimizing statistical and systematic errors.

Atom interferometry makes use of the wave-like properties of matter that become evident at very low energy scales.236 The concept is analogous to an optical interferometer that splits a coherent source of light into separate beams following different paths, which are then recombined and undergo interference. The interferometric measurement of the path length difference is the basis of current gravitational wave detectors such as Advanced LIGO48 and Virgo. Similarly, the wave-function of an atom can be split into a superposition of two states evolving along different paths. This is accomplished through absorption and stimulated emission of photons and the associated momentum transfers (recoils) between light and atoms.40 The sensitivity of an atom interferometer can be enhanced by additional laser pulses, a technique called large momentum transfer (LMT) atom optics.237 In a gravitational wave detector configuration, two clouds of atoms separated by a baseline are interrogated by common laser pulses. The differential phase signal measures the light travel time across the baseline and each additional pair of pulses adds another measurement of the baseline length. This is analogous to the enhancement provided by Fabry–Perot cavities in Advanced LIGO and Virgo, where the laser beams in the interferometer arms are retro-reflected many times to enhance the signal and create a much longer effective baseline. Increasing the number of laser pulses in the atom interferometer through LMT atom optics is a key area of technology development on a par with increasing the physical detector baseline by building a larger instrument (see Table III).

It is desirable for the two atomic states used in an atom interferometer to be long-lived to minimize spontaneous decay, which diminishes the output signal. Conventionally, a set of counter-propagating laser beams is used to couple either two electronic ground states (Raman atom optics40) or two momentum states of the same electronic ground state (Bragg atom optics153) via a far-detuned excited state. However, one can also resonantly drive transitions between an electronic ground state and a metastable state using a single laser beam.18,238 Such a clock atom interferometer makes use of the narrow transitions commonly used in atomic clocks.155–157 Here, the phase response is proportional to the transition frequency ωa instead of the effective wave vector k eff (frequency difference) of two lasers (see Fig. 14). Thus, clock atom interferometers support common-mode suppression of laser frequency noise, a dominant noise source for long-baseline interferometers.18 Due to the long lifetime of the metastable state, spontaneous decay can be substantially suppressed compared to two-photon atom optics on a broad optical transition involving a far-detuned excited state. Therefore, clock atom interferometers can support much higher pulse efficiencies and many thousands of sequential laser pulses. They enable the substantial sensitivity enhancement that is crucial for reaching the ambitious sensitivity targets for gravitational wave detection and dark matter searches with very-long-baseline sensors.2 

Fig. 14.

(a) Comparison of the laser frequencies involved in conventional and clock atom optics as well as the leading order phase response of the associated interferometer. (b) Space-time diagram of a relativistic Mach–Zehnder interferometer using clock atom optics (dark lines) and conventional two-photon atom optics (dark and light lines). In a clock atom interferometer, the same laser pulse addresses the entire atomic superposition, imprinting the same laser phase and allowing for common-mode noise suppression.

Fig. 14.

(a) Comparison of the laser frequencies involved in conventional and clock atom optics as well as the leading order phase response of the associated interferometer. (b) Space-time diagram of a relativistic Mach–Zehnder interferometer using clock atom optics (dark lines) and conventional two-photon atom optics (dark and light lines). In a clock atom interferometer, the same laser pulse addresses the entire atomic superposition, imprinting the same laser phase and allowing for common-mode noise suppression.

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Figure 15 illustrates a sequence of light pulses generating a pair of clock atom interferometers, one on each end of the baseline. The timing of the atomic transitions, and thus the time the atoms spend in a superposition of the ground and excited states, depends on the light travel time across the baseline.18 The resulting differential interferometer phase Δ ϕ is then proportional to the baseline length L:
Δ ϕ n ω a   L / c ,
where   ω a is the energy splitting of the clock transition, and n the order of LMT enhancement. As a result, the differential phase measurement between the two atom interferometers is sensitive to variations in both the baseline L and the clock frequency ωa that arise during the light-pulse sequence. A passing gravitational wave modulates the baseline length, while coupling to an ultralight dark matter field can cause a modulation in the clock frequency. Thus, a differential clock atom interferometer combines the prospects for both gravitational wave detection and dark matter searches into a single detector design, and both science signals are measured concurrently.2 In both cases, additional laser pulses linearly enhance the output signal and thus the sensitivity of the detector.
Fig. 15.

Space-time diagram of the interferometer trajectories based on single-photon transitions between ground (blue) and excited (red) states driven by laser pulses from both directions (dark and light gray). The pulse sequence shown here features an additional series of pulses (light gray) traveling in the opposite direction to illustrate the implementation of LMT atom optics (here n = 2).

Fig. 15.

Space-time diagram of the interferometer trajectories based on single-photon transitions between ground (blue) and excited (red) states driven by laser pulses from both directions (dark and light gray). The pulse sequence shown here features an additional series of pulses (light gray) traveling in the opposite direction to illustrate the implementation of LMT atom optics (here n = 2).

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Proof-of-principle experiments with LMT clock atom interferometry have demonstrated enhanced interferometer and gradiometer sequences,154 and a record momentum separation of over 600   k using 1200 sequential laser pulses.239,240 These results have been achieved on an intermediately narrow transition with moderate lifetime and strong Rabi coupling. Ultra-narrow clock transitions can support even higher pulse efficiencies but necessitate longer pulse durations and colder atoms. Quantum degenerate ensembles and matter-wave lensing techniques are required to make full use of these efficiency gains. Ultimately, the LMT enhancement in any terrestrial clock atom interferometer is limited by the available laser power at the target wavelength and the free fall time of the atom i.e. the size of the instrument.

Large momentum transfer (LMT) atom interferometers take advantage of the improved scaling of their sensitivity with the momentum separation of the coherent superposition of matter waves. In Very-Long-Baseline Atom Interferometry, momentum transfers of hundreds and thousands of photon recoils will be realized to achieve the sensitivity needed to detect gravitational waves or test physics beyond the Standard Model. For LMT interferometer realizations, a number of mechanisms are currently being investigated, based on one- or two-photon processes, the latter of which can induce inelastic Raman transitions or elastic scattering processes. The highest metrological sensitivity is currently achieved by means of elastic transitions. This is accomplished in the form of (i) Bragg diffraction or (ii) Bloch oscillations of atoms in optical lattices. In both processes, a very high efficiency per transmitted photon pulse k can be achieved. Bragg diffraction and Bloch oscillations underlie in particular the atom interferometric measurements of the fine structure constant31,241 and are also envisaged for the architecture of gravitational wave detectors.8,242 Limitations in atomic interferometers based on these processes are currently determined by systematic effects due to so-called diffraction phases associated with atom–light interactions, as well as losses and inefficiencies of the beam splitters and mirrors. A key to future applications of atom interferometers with large scale factors, requiring the transmission of hundreds or thousands of photon pulses, will therefore be a detailed understanding and accurate control of diffraction phases and inefficiencies of elastic scattering processes. To this end, important achievements have been made recently,139,140,243,244 providing a promising basis for the further development of very-long-baseline atom interferometry.

With respect to Bragg diffraction, a model for efficient beam splitting and mirror pulses was developed in Ref. 139 that covers the so-called quasi-Bragg regime. This regime describes pulses whose duration and intensity lie between the parameters of the deep-Bragg regime of very long, weak pulses and the Raman–Nath regime of very short, intense pulses that have been treated in the literature to date. The quasi-Bragg regime is characterized by a compromise between suppression of the velocity filter due to Doppler detunings and suppression of scattering to undesirable momentum orders. In Ref. 139, it was shown that Bragg pulses with Gaussian intensity modulations achieve efficient beam splitters and mirrors exactly when the underlying quantum dynamics corresponds to an adiabatic process in the Bloch states of the optical lattice. With this insight, a number of important properties of Bragg operations can be quantitatively and, in some cases, even analytically characterized in a simple way. For example, the pulse area condition for beam splitters or mirror pulses can be formulated as a condition on energetic phases of Bloch states, and losses in parasitic scattering orders can be understood as non-adiabatic Landau–Zener transitions. All this was combined in Ref. 139 in the form of a scattering matrix that describes individual beam splitter or mirror operations with high accuracy, accounting correctly for losses and phases.

Based on this, the description of complete interferometer sequences was developed in Ref. 140. Using the example of a Mach–Zehnder interferometer, the effects of Landau–Zener transitions into parasitic scattering orders were treated in detail. In a measurement of the relative atom number in the two main output channels of the final beam splitter, two essential effects come to light: first, parasitic output channels are inevitably also populated in a phase-dependent manner in the final beam splitter, so that the total number of atoms in the main output channels itself becomes phase-dependent. If the absolute number of atoms measured in an output channel is now related to this phase-dependent total number, a subtle and complicated phase dependence of the interferometer results. Second, this is further complicated because populations of parasitic momentum states in the first beam splitter can lead to further, parasitic paths closing in the last beam splitter in addition to the two main paths of the interferometer. This leads to intricate phase dependencies due to additional Mach–Zehnder and Ramsey–Borde interferometers. For generic Bragg pulses, this results in an overall interferometer signal that formally corresponds to an infinite Fourier series in the metrological phase (and its harmonics). Even though the description developed in Refs. 139 and 140, correctly predicts the amplitudes and phases of this Fourier series in principle, this model cannot serve as a signal template due to its complexity. However, the detailed microscopic model from Ref. 139 allows the Bragg pulses to be designed in an optimal way so that parasitic interferometer paths are strongly suppressed, which greatly simplifies the signal shape. In Ref. 140, it was shown that such suppression can achieve accuracy in the micro-radian range. Furthermore, with the analytical modeling of the transfer matrix of a Bragg interferometer, it was also possible to determine the fundamental bounds on the accuracy of such an interferometer in the form of the Cramer–Rao and the quantum Cramer–Rao bounds and thus to determine parameter ranges of optimal accuracy.

Remarkably, an analogous description is possible for LMT operations based on Bloch oscillations. In the literature, Bloch oscillations have long been considered as adiabatic processes in Bloch states