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Light-pulse atom interferometry3 is based on the wave–particle duality of quantum mechanics and transduces the acceleration experienced by atoms into a phase difference between interfering atomic matter waves. In our set-up, caesium atoms are laser-cooled and launched13 upwards into free fall. Pulses from counter-propagating laser beams transfer them from their initial quantum state |a〉 to another state |b〉. Each atom absorbs one photon, having a momentum ℏk1, from the first beam while simultaneously being stimulated to emit another photon into the second beam, gaining additional momentum ℏk2. This results in a total momentum change of ℏkeff, where keff = k1 + k2. The first interferometer pulse has a duration such that the transfer takes place with 50% probability. It acts as a coherent beam splitter for matter waves, placing the atom into a superposition of the initial state |a, p0〉 with momentum p0 and the state |b, p0 + ℏkeff〉. The two states separate spatially. After a pulse separation time T, a second laser pulse transfers the states |a, p0〉 → | b, p0 + ℏkeff〉 and |b, p0 + ℏkeff〉 → |a, p0〉, thus inverting the relative motion. After another interval T, a third pulse acts as a final beam splitter which combines the partial matter waves (Fig. 1). When combined, the matter waves add constructively or destructively, depending on their phase difference Δφ, giving a probability P ∼ cos2(Δφ/2) of finding the atom in the quantum state |a〉. The probability and thus the phase difference is found by measuring the population ratio of states |a〉 and |b〉 when many atoms undergo this process simultaneously. In the simplest case, Δφ = keffaT2, where a is the total acceleration experienced by the atoms. Using keff ∼ 107 m−1, large atom samples for sufficient phase resolution, and T of up to 1.15 s (ref. 14) creates an enormous lever arm by which even small changes of a generate measurable phase changes.

Figure 1: Cavity matter-wave interferometry.
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Left: experimental set-up. The acceleration acyl of caesium atoms towards a cylindrical tungsten source mass suspended in ultrahigh vacuum is measured. The cylinder has mass mcyl = 0.19 kg, height h and diameter d = h = 2.54 cm. The axial through-hole has radius 0.5 cm, and the slot has width 0.5 cm. Making a differential measurement isolates the effect of any interactions sourced by the tungsten mass. Right: Mach–Zehnder interferometer based on Raman transitions in an optical cavity. Three laser pulses manipulate the caesium atoms during free fall. The pulses split the atomic wavepacket along two different trajectories, reflect the two trajectories near their apex, and then recombine and interfere the matter waves to measure the phase difference accumulated between the two paths during the interferometer time of 2T = 110 ms. We obtain a measurement of the acceleration experienced by the caesium atoms ensemble-averaged over ∼105 atoms.

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Unfortunately, this lever arm is significantly shortened when it comes to measuring the gravitational force created by a small mass M. The useful free-fall time, distance between the atoms and the source mass, and the dimensions of the atom cloud are all constrained because atoms far away from the source see a reduced gravitational potential. Accordingly, the most sensitive atom interferometers15,16 use the entire Earth as a source mass. Measurements using small source masses have not been previously demonstrated, but could be useful to explore new regimes12,17.

Testing gravity in new ways may help to answer pressing questions. Cosmological measurements18,19 have firmly established that the universe is expanding at an accelerating rate which is consistent with dark energy permeating all of space. The observed dark energy density Λ04 ≈ (2.4 meV)4 is tens of orders of magnitude smaller than expected from the vacuum energy of quantum field theories. This chasm, the ‘cosmological-constant problem,’ probably requires new fields for its resolution. By Weinberg’s no-go theorem20, however, even such new fields cannot solve the problem unless they are dynamic, not in equilibrium. A new field must therefore be light if it is to address the cosmological-constant problem, (m ≤ H0 where H0 ∼ 10−33 eV is the Hubble constant) so as to remain in non-equilibrium today, 1010 years after the Big Bang. Such a light field, however, should mediate a long-range interaction, in disagreement with precision tests of gravity. Over the past decade, this has motivated a family of theories that predict significant deviations from general relativity only in the ultraweak-field regime20, where a force of gravitational strength or larger is suppressed further by ‘screening’ as a function of the environment. Existing theories do not solve the cosmological-constant problem, but screening is probably a key ingredient of any future solution. Experimental tests of gravity have focused on the short-distance regime, the post-Newtonian regime or the strong field regime21, leaving the ultraweak-field regime largely untested.

Screening arises when coupling between the field and matter hides effects of the field (such as a new force) in high-density regions like Earth. In contrast, the field is unsuppressed and most potent in low-density regions like the cosmos. Such fields can enact important astrophysical effects while remaining hidden from laboratory and solar system tests.

The ultraweak fields φ can be characterized by their mass m(φ) and coupling to normal matter β(φ), which may both be functions of the field itself. The acceleration of an object

(in our case, an atom) caused by the field is highly sensitive to the surrounding matter geometry22. Here, MPl = (ℏc/8πG)1/2 ≈ 2.4 × 1018 GeV is the reduced Planck mass, and 0 ≤ λa ≤ 1 is a screening function that depends on m, β and the object’s mass and size. Moreover, λa → 1 for a small and light test particle but λa ≪ 1 for macroscopic objects, where only a thin, outermost layer interacts with the field. An atom in ultrahigh vacuum with a local miniature source mass minimizes screening and is well-suited as a test mass for such theories8. Prime examples of such scalar fields are chameleons and symmetrons.

A chameleon scalar field23,24 is characterized by an effective potential density

The self-interaction

is characterized by an energy scale Λ, which must be close to the cosmological-constant scale, Λ ≃ Λ0 = 2.4 meV, if the chameleon is to drive cosmic acceleration. The interaction with matter of density ρm

is characterized by an energy scale M, which is expected to be below the Planck mass. The chameleon profile due to an arbitrary static distribution of matter ρm(x) is obtained by solving the nonlinear Poisson equation:

Deep inside a large, dense object, ∇2φ ≃ 0 and φ rapidly approaches a negligible value that minimizes Veff(φ). Thus, the bulk of such an object is largely decoupled from the field, except for a thin outer shell, leading to screening. For general ρm(x), we must resort to numerical integration22. Given the resulting field profile φ(x), the chameleon-mediated acceleration on an atom is given by equation (1) with βcham = MPl/M.

A symmetron scalar field25,26 has an effective potential symmetric under φ → −φ. The simplest models have a Higgs-like quartic self-interaction

in which λ is the self-coupling, and μ is the bare potential mass scale. The field couples to matter through an explicitly density-dependent mass term,

The coupling is again characterized by an energy scale MS. We focus here on 1 MeV < MS < 1 TeV, approximately the regime in which the fifth force is screened in a typical laboratory apparatus. The acceleration equation (1) in a constant-density region is roughly characterized by βsym(φ) = φMPl/MS2. The field φ, and thus the coupling βsym, is zero in high-density regions and nonzero at low densities. A sharp transition away from the symmetric, uncoupled phase will occur only in a vacuum chamber larger than π/μ, and forces are suppressed at distances much smaller than 1/μ, so the range10,11 of μ probed by our experiment is approximately 0.01 meV < μ < 1 meV .

Our basic set-up has been described previously9. To reach the sensitivity to observe the gravitational attraction between the atoms and the source mass, we installed significant technical upgrades. Three-dimensional Raman sideband cooling reduces the atom temperature to ∼300 nK. Launching the atoms vertically upwards in a cavity-enhanced optical lattice doubles the available interrogation time and quadruples the accumulated phase. Two levels of passive vibration isolation attenuate seismic noise, and an active stabilization loop provides further attenuation. After the interferometer, performing the lattice launch in reverse ‘catches’ atoms remaining in the cavity mode while the rest fall away. This spatial selection of atoms participating in the interferometer increases contrast by an order of magnitude to over 40%.

A schematic of our apparatus is shown in Fig. 1. Laser beams inside a Fabry–Pérot cavity provide well-controlled optical wavefronts and resonant power enhancement for coherent manipulation of the atomic probe. A tungsten cylinder of mass mcyl = 0.19 kg is our source mass. We have optimized this geometry using detailed numerical calculations of screened field profiles22 in our vacuum chamber. An axial through-hole allows the cavity mode to pass through the mass unimpeded. A rectangular slot along one side of the cylinder allows the mass to be toggled between one position near the atoms and another position far away, without interrupting the cavity mode. A differential measurement between the two positions suppresses the Earth’s gravitational acceleration and isolates the acceleration arising from the cylindrical source mass, acyl = anear − afar. This acceleration should be purely gravitational in the absence of any anomalous interactions.

Data was taken for more than 170 h through three quiet weekends in October 2016, resulting in ∼4.3 × 105 experimental runs (see Fig. 2). Averaging the measurements of the acceleration acyl weighted by the standard error over these three datasets results in acyl = (74 ± 19stat ± 15syst) nm s−2, where the first error bar (one standard deviation) is statistical and the second arises from systematic uncertainties (see Methods). The positive acceleration indicates a force toward the source mass. This agrees well with the expected gravitational pull of the cylinder agrav = (65 ± 5) nm s−2. We obtain an anomalous acceleration aanomaly = acyl − agrav = (9 ± 24) nm s−2, giving a 95% confidence interval of −39 nm s−2 < aanomaly < 57 nm s−2. Using a one-tailed test to bound fifth force interactions (which must be attractive for scalar fields with a universal matter coupling), we constrain anomalous accelerations aanomaly < 49 nm s−2 at the 95% confidence level. We note that the 24 nm s−2 (1σ) accuracy required to resolve the gravity signature from our miniature mass corresponds to just 2.4 ppb in the Earth’s acceleration of free fall. Although spatial constraints set by the miniature size of the mass preclude using the beam sizes, and free-fall distances used in the best absolute gravimeters15,16, we can resolve such a small acceleration by toggling the mass position, thereby suppressing systematic effects.

Figure 2: Experimental data.
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A differential measurement is performed by toggling the source mass between a near and far position. a, Two interferometer fringes, taken for the wavevector direction, up (k+) and down (k−) (see Methods). Combining these two measurements gives one data point out of 3,215 taken over 68 h. b, A 3-hour section of data. Four measurements are taken at each source mass position, which are then averaged. Blue points indicate that the source mass is in the far position; red points indicate the near position. An overall offset g0 is subtracted for clarity. The difference between subsequent measurements after toggling the source mass position gives one measurement of acyl. Error bars represent 1-σ standard deviation (statistical). c, A single determination of acyl takes ∼500 s. The full dataset (one of three) is shown here. d, The set of individual acyl measurements are fitted to the cumulative distribution function of a normal distribution with mean μ and standard deviation σ. N is the number of individual measurements. This least-squares fit (solid black line) gives acyl for the dataset.

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Specializing to chameleon and symmetron fields, following Burrage et al. 8, we improve previous limits9,10,11 on these models by more than two orders of magnitude. Figure 3 shows excluded parameter ranges for these models. For chameleon fields with Λ at the dark energy value Λ0 = 2.4 meV and n = 1, we exclude up to M < 2.8 × 10−3 MPl, narrowing the gap to torsion pendulum constraints1,27. One can see that these fields are nearly ruled out, with only a one order of magnitude range left for the coupling strength M. Furthermore, for all Λ > 5.1 meV, this gap is fully closed, ruling out all such models. Our symmetron limits are complementary to torsion pendula1,10,11 as well. We improve previous constraints on λ by two orders of magnitude throughout the entire range of MS and μ probed by our experiment. Our constraint is strongest in the regime where the atom is screened, where for μ = 0.1 meV we rule out λ < 1.

Figure 3: Constraints on screened scalar fields.
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a, Chameleon field: the shaded areas in the M–Λ plane are ruled out at the 95% confidence level. MPl/M gives the coupling strength to normal matter in relation to gravity; Λ = Λ0 ≈ 2.4 meV (indicated by the black line) could drive cosmic acceleration today. A comparison is made to previous experiments: neutron interferometry28/neutron gravity resonance29, microsphere force sensing30, and torsion balance1,27. b, Chameleon limits in the n–βcham plane with Λ = Λ0, showing the narrowing gap in which basic chameleon theories could remain viable. n is the power law index describing the shape of the chameleon potential; βcham ≡ MPl/M is the strength of the matter coupling. c, Symmetron fields: constraints by atom interferometry complement those from torsion pendulum experiments10,11 (shown with μ = 0.1 meV) for the range of μ considered. For μ < 10−1.5 meV, the field vanishes entirely inside the vacuum (see Methods), leaving this parameter space unconstrained. The same effect produces the sharp cutoff in our limits at low MS.

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Tests of gravity in the ultraweak-field regime with a miniature, in-vacuum source mass probe screened field theories with the potential to explain the accelerated expansion of our universe. In the future, technologies such as lattice interferometry13 in our optical cavity and large momentum transfer Bragg beam splitters will allow us to hold quantum probe particles in proximity to a miniature source mass, evading geometric constraints from the source mass’ small size, and boosting sensitivity. With modest improvements, chameleon fields at the cosmological energy density will be either discovered or completely ruled out. This also will enable study of novel quantum phenomena such as the gravitational Aharonov–Bohm effect12, and provide even better resolution of atom–source mass interactions.

Methods

Setup.

Here, we describe the basic outline and new features of our set-up9,31. Caesium atoms are loaded into a three-dimensional magneto-optical trap (3D-MOT) from a 2D-MOT. After sub-Doppler cooling in an optical molasses, we perform Raman sideband cooling32 in a 3D lattice which leaves ∼5 × 106 atoms in the |F = 3, mF = 3〉 state at a temperature < 300 nK. After release from the lattice, adiabatic rapid passage and a state selection pulse with microwaves transfer the cold atoms into the magnetically insensitive |F = 3, mF = 0〉 state. About 20% of the atoms are then launched upwards with a chirped optical lattice13 in the optical cavity mode at a velocity of 59.1 cm s−1. Launching the atoms moves them upwards towards the source mass, doubles the available interrogation time, and provides both spatial and velocity selection. After the launch, we perform the interferometry pulse sequence. The cavity dictates that all beams counter-propagate. Close to the apex, the Doppler shift δDopp due to atom motion is small. The frequencies driving Raman transitions imparting upward momentum (k+) therefore become degenerate with the ones imparting downward momentum (k−). Since this would cause atom loss, we make our interferometer asymmetric with respect to the apex of the atomic trajectory. To increase signal to noise, we preferentially detect atoms at the centre of the Raman beam, suppressing the signal from atoms that have not participated in the interferometer. To this end, after the interferometer we reverse the launch procedure to catch the atoms by decelerating them into a lattice at zero velocity. This selects only the atoms in the centre of the cavity mode, while nonparticipating atoms (for example, that have left the cavity mode due to thermal expansion) fall away. A pushing beam separates the two output ports of the interferometer, where they are counted by fluorescence detection to determine their relative population.

Error budget.

We discuss and quantify sources of uncertainty in the Supplementary Information. Systematic uncertainties for the individual datasets are combined, weighted by the statistical uncertainties of the datasets. Supplementary Table 1 shows the resulting error budget.

Vibration isolation.

Vibrations of the retroreflecting cavity mirrors are a leading noise source. We mount the entire vacuum chamber on two layers of passive vibration isolation: a pneumatic benchtop isolation system (Thorlabs PWA 090), which is mounted on top of a floated optical table. For active isolation, a seismometer (Kinemetrics, SS-1) sits on top of the vacuum chamber to measure residual vibrations. The seismometer signal enters an analog feedback loop which actively stabilizes against vibrations using a voice coil actuator. The seismometer is magnetically shielded from switching experimental magnetic fields with a cylindrical pipe of low-carbon steel, reducing synchronous accelerations induced by the servo loop.

The passive isolation attenuates ground vibrations by up to two orders of magnitude. Closing the servo loop reduces the seismometer error signal a further factor of ∼5–400 from 1–20 Hz, the most problematic frequency range for our 2T = 110 ms interferometer. This active servo loop reduces the expected interferometer phase noise by a factor of 16.

Numerical simulations.

Computing the force from the scalar field on the atom involves solving equation (5), a nonlinear Poisson equation, for a matter distribution ρm(x) that corresponds to the experimental set-up. This includes the source mass, as well as the walls of the vacuum chamber. We do not compute the contribution from the atoms themselves in the calculation of φ; this effect is captured by the screening factor λa.

Our approach is to use a Gauss–Seidel finite-difference relaxation scheme on a three-dimensional grid that covers the entire experiment. An initial guess for the field inside the vacuum chamber is iteratively corrected until the field value converges everywhere. We have previously used this technique in the context of chameleons22, which we have repeated for the new chameleon constraints and also extended to symmetrons. Once the field profile is known, equation (1) can be used to calculate the acceleration. We compute the average acceleration due to the scalar field as a time-weighted average over the trajectory of the atoms during the measurement.

Since the calculation is being done in near-vacuum, it is reasonable to expect the field profile to be roughly independent of M (for the chameleon) and MS (for the symmetron). This is because those parameters only appear in their equation of motion along with ρm, which is very small. Supplementary Fig. 5a demonstrates this for the chameleon field, showing that the field’s gradient is unchanged over five orders of magnitude in M.

Similarly, the vacuum value of φ for the symmetron field is inversely proportional to the square root of λ, so we might expect to be independent of λ. Indeed, Supplementary Fig. 5b shows λφ ∇ φ to be independent of both MS and λ over six and ten orders of magnitude, respectively. This finding greatly expedites the numerics, as only a single simulation need be performed for a given value of μ.

As with refs 10,11, we find a measurable acceleration only for a relatively narrow range of μ: roughly 10−1.5 meV < μ < 10−1 meV . In fact, the lower end of our range in μ is an order of magnitude higher than that of ref. 10. This is because our 3D numerical code more accurately accounts for the presence of the vacuum chamber walls, which generically causes the field to vanish below a certain value of μ (or MS, as seen in Fig. 3c). The upper end of μ is unchanged, and is due to the symmetron field becoming too short-ranged for the atoms to feel any appreciable force from the source mass.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Additional Information

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