Terrestrial very-long-baseline atom interferometry: Workshop summary

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 matter^13–16 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 interferometers^37,38 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.

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

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.

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 signal^52,71 (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.

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 $1000 ℏ k$ atom optics and $10 − 4 rad / Hz$ phase resolution; while for the 1 km baseline, we assume $2500 ℏ k$ 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.

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, $2500 ℏ k$ 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 B–L coupled vector ULDM candidate. The projected sensitivity, shown in the left panel of Fig. 6 in terms of the vector mass (m_A) 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, $100 ℏ k$ atom optics and $10 − 3 rad / Hz$ phase resolution, while the green line could be achieved with a 100 m baseline, $1000 ℏ k$ atom optics and $10 − 4 rad / Hz$ phase resolution.

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 (m_a) and coupling (g_aNN) 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.

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.

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 $G / ℏ ⌊ 1 / | x − y | ⌋$⁠, 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.

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 fields^142,143 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 state^3,8,81 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 transitions^18,154 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 (q_e) and the proton (q_p), and the neutron charge (q_n) 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: $Δ ϕ = δ qV τ / ℏ$⁠. 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 q_n 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 q_n.

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 atoms^169,170 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 orbits^176–179 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.

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 c_H. 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 c_H.¹⁹⁰ This is because the GGN-induced gradiometer phase shift at low frequencies is proportional to c_H.

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.