## Trapped atoms to probe gravity

Exploiting the wave nature of atoms in an interferometer setup can be used to provide a precise measure of gravity. The precision is limited to the time scale of the interferometric measurement, which in turn is limited to the distance that the atoms drop, typically just over a couple of seconds for a 10-meter drop tower. Xu *et al.* describe a trapped atom interferometer in which the interrogation time of the interferometric measurements can be extended to 20 seconds. The new interferometer design and subsequent improved precision can be used to make fundamental tests of general relativity as well as precision measurements of other potentials.

*Science*, this issue p. 745

## Abstract

Atom interferometers are powerful tools for both measurements in fundamental physics and inertial sensing applications. Their performance, however, has been limited by the available interrogation time of freely falling atoms in a gravitational field. By suspending the spatially separated atomic wave packets in a lattice formed by the mode of an optical cavity, we realize an interrogation time of 20 seconds. Our approach allows gravitational potentials to be measured by holding, rather than dropping, atoms. After seconds of hold time, gravitational potential energy differences from as little as micrometers of vertical separation generate megaradians of interferometer phase. This trapped geometry suppresses the phase variance due to vibrations by three to four orders of magnitude, overcoming the dominant noise source in atom-interferometric gravimeters.

Matter-wave interferometers with freely falling atoms have demonstrated the ability to precisely measure, e.g., gravity (*1*) and fundamental constants (*2*, *3*), to test general relativity (*4*–*6*), and to search for new forces (*7*, *8*). A major obstacle to increasing their sensitivity, however, has been the limited time during which coherent, spatially separated superpositions of atomic wave packets can be interrogated. Up to 2.3 s of interrogation time has been realized in a 10-m atomic fountain (*9*), and several seconds of interrogation time are the target of experiments in fountains measuring hundreds of meters (*10*, *11*), zero-gravity planes (*12*), drop towers (*13*), sounding rockets (*14*), and the International Space Station (*15*–*17*). Geometries that use Bloch oscillations to periodically bounce (*18*) or trap the interferometer (*19*–*21*) have been limited to interrogation times of 1 s, despite the long coherence times of Bloch oscillations in an optical lattice (*22*).

We demonstrate 20 s of coherence in an atom interferometer held in an optical lattice, overcoming trap dephasing by using an optical cavity as a spatial mode filter. After 20 s, the phase variance from our laboratory vibrations is suppressed by up to 10^{4} relative to traditional Mach-Zehnder atomic gravimeters at the same dc sensitivity, owing to the continuous accumulation of free evolution phase in the trapped wave packets. This contrasts with Mach-Zehnder geometries, which are limited by the substantial excess noise caused by aliasing vibrations between the laser pulses. Such aliasing is nearly absent in a lattice-hold geometry. Differential measurements performed by free-space atom interferometers can cancel vibrations, but their sensitivity is ultimately limited by the available free-fall time in a gravitational field. Trapping the interferometer allows the sensitivity to be increased by extending interrogation times rather than wave-packet separations or free-fall distances, reducing experimental complexity and potentially minimizing systematics.

Our matter-wave interferometer builds upon the setup described previously (*7*, *23*). Cesium atoms are laser-cooled to ~300 nK, prepared in the magnetically insensitive *m _{F }*= 0 state, and launched millimeters upward into free fall [see materials and methods for details (

*24*)]. In free fall, counterpropagating laser beams in the cavity manipulate the atomic trajectories. We stimulate two-photon Raman transitions between the hyperfine ground states of cesium,

*F*= 3 and

*F*= 4, imparting two photons’ momenta to the atoms with each laser pulse. The pulse intensities are tuned to kick atoms with 50% probability (“π/2 pulses”), enacting coherent matter-wave beam splitters that separate the two partial wave packets with a relative velocity of 2

*v*

_{rec}= ℏ

*k*

_{eff}/

*m*

_{Cs}= 7 mm/s, where

*v*

_{rec}is the recoil velocity of a cesium atom absorbing a photon on its D2 line.

Our lattice interferometer employs two pairs of π/2 pulses (Fig. 1). The first pair, separated by a time *T*, splits the matter-waves into four paths. We select two of these paths, in which the wave packets are vertically separated by a distance Δ*z* = 2*v*_{rec}*T* while sharing the same hyperfine ground state and external momentum. At the apex, atoms are adiabatically loaded into the ground band of a far-detuned optical lattice with a period of *d* = λ_{latt}/2, where the laser wavelength is λ_{latt} = 866 nm. The atoms are suspended in the lattice for a time τ, undergoing Bloch oscillations (*25*–*27*) with period τ_{B} = (*m*_{Cs}*gd/h*)^{−1} ≈ 707.5 μs, where *m*_{Cs} is the atomic mass of cesium and *g* is the local gravitational acceleration. Next, the atoms are adiabatically unloaded from the lattice and the wave packets are recombined using the final pair of π/2 pulses. At the last pulse, the atomic matter-waves interfere according to the phase difference Δϕ = ϕ_{upper} − ϕ_{lower} accumulated between the upper and lower arms. As a result, the probabilities *P*_{3,4} of detecting an atom in the output ports corresponding to *F *= 3 and *F *= 4 oscillate with this phase difference *P*_{3,4} = [1 ± *C* cos(Δϕ)]/2, where *C* is the fringe contrast. Because only two of the four spatially unresolved output ports interfere, our maximum contrast is 0.5 in this geometry. The atom number in each port (*N*_{3,4}) is measured through fluorescence imaging, and the total interferometer phase, Δϕ, is extracted from oscillations in the population asymmetry 𝒜 = (*N*_{3} − *N*_{4})/(*N*_{3} + *N*_{4}) = *C* cos (Δϕ).

For traditional atomic gravimeters operating in free space, the total phase Δϕ = Δϕ_{L} + Δϕ_{FE} is dominated by the atom-light interaction phase Δϕ_{L} (“laser phase”), whereas the free evolution phase Δϕ_{FE} is zero. Each laser pulse contributes a phase ϕ* _{i}* proportional to the atoms’ position. In this pulse sequence, the beam-splitter pulses imprint an overall atom-light interaction phase (

*20*,

*21*) of Δϕ

_{L}= (ϕ

_{1}− ϕ

_{2}) − (ϕ

_{3}− ϕ

_{4}) =

*k*

_{eff}

*gT*(

*T*+

*T*

_{α}). For our experiment (

*24*), the time

*T*

_{α}= 2

*t*

_{A}is given by the total free-fall time between pulses 2 and 3 (Fig. 1). For atoms in free fall, Δϕ

_{L}can provide a sensitive measurement of accelerations such as gravity (

*1*), which influence the atoms’ position at each laser pulse.

This lattice-based interferometer realizes nearly the opposite scenario in that the free evolution phase Δϕ_{FE} constitutes more than 99% of the total phase alongside only a small contribution from Δϕ_{L}. Between the beam-splitter pulses, each arm accumulates a phase ϕ_{FE}, which can be calculated by integrating the Lagrangian ℒ over the classical trajectory (*28*). Suspending the interferometer causes a phase difference*U* = *m*_{Cs}*g*Δ*z*, across a vertical separation Δ*z*. There is zero net contribution to the free evolution phase outside the lattice hold.

Figure 2A shows interference fringes due to the gravitational potential energy difference from a vertical separation of Δ*z* = 3.9 μm, corresponding to nine lattice spacings. Fringes remain visible as the interferometer is trapped for up to τ = 20 s, at which point Δϕ_{FE} = 1.6 Mrad. Without the lattice to hold atoms against Earth’s gravity, interrogating atoms in free fall for 20 s would require a vacuum system about 0.5 km tall. In our interferometer, atoms travel less than 2 mm. This allows highly sensitive yet very compact atomic setups, which help suppress spatially dependent systematic effects such as gravitational and magnetic field gradients. Moreover, differential measurement of Δϕ_{FE} between short (~0.2 s) and long (20 s) holds substantially suppresses phases independent of the hold time, isolating the gravitational signal.

Interferometer sensitivity increases with longer hold times τ and larger wave-packet separations Δ*z*, with a signal-to-noise proportional to the contrast. At the smallest separation (Δ*z* = 3.9 μm), we measure a 1/*e* contrast lifetime in the lattice of

The cavity is instrumental in enabling these long coherence times. After 20 s in a lattice depth of *V*_{0} = 8*E*_{rec} (where *E*_{rec} is the recoil energy), each interferometer arm accumulates a ~2 Mrad lattice light shift. To observe contrast, the difference in the lattice light shift phase between the interferometer arms can only vary across the atomic sample by ≲π radians. This means that the difference in the lattice intensity profile between the partial wave packets must be below the 10^{−6} level. The fundamental cavity mode has a Rayleigh range (*z*_{R} = 1.9 m) much larger than the wave packet separations (Δ*z* ~ 100 μm), defining a highly uniform beam geometry between the interferometer arms. The cavity spatially filters the beam; with a transverse mode spacing of 8.3 cavity linewidths, only the fundamental Gaussian mode is resonant, whereas the first higher-order mode is suppressed nearly 70-fold, and higher-order modes even more strongly. The cavity’s spectral selectivity suppresses light at other frequencies, such as from the broadband emission of diode lasers. Next, the intracavity power enhancement reduces the required input power ~40-fold. This reduces the amount of light on other optical elements such as external lenses, mirrors, or vacuum viewports, reducing the stray light generated at such optical elements.

In general, the optical lattice modifies the spatial structure of the atomic wave packets. Operating the lattice interferometer requires an understanding of these dynamics. In a typical atom interferometer, contrast can be observed as long as the wave-packet positions at the time of the final pulse are within the coherence length of the sample, as determined by the thermal de Broglie wavelength λ_{T} of the atomic wave packet. In our setup, this condition is met when the distance traveled by the partial wave packets after the third pulse Δ*z*_{2} = 2*v*_{rec}*T*_{2} is within λ_{T} of the initial wave-packet separation Δ*z*_{1} = 2*v*_{rec}*T*_{1}, where *T*_{1} and *T*_{2} refer to the pulse separation times before and after the lattice hold, respectively (Fig. 3A). When matching the free-space wave-packet separations Δ*z*_{1,2} to an integer number Δ*n* of lattice spacings, contrast is observed within λ_{T} of Δ*z*_{2} = Δ*z*_{1}. This can be seen in Fig. 3B by the peak in contrast around Δ*z*_{2} = Δ*z*_{1} = 9*d.*

The finite spatial extent of the atomic wave packet can coherently distribute across multiple adjacent lattice sites, leading them to acquire a finer spatial substructure at the lattice spacing *d*. For atoms with λ_{T} ≥ *d*, the partial wave packet along each arm (Fig. 3A) necessarily distributes into adjacent lattice sites, effectively creating additional interferometer arms with a vertical separation of Δ*n* = 1. Coherence between the added paths is verified in two ways. First, we measure a peak in the contrast envelope near Δ*z*_{2} ≈ *d*, showing that components of the wave function separated by one lattice site interfere. Accordingly, Fig. 3, B and D, show spatial overlap within λ_{T} of Δ*z*_{2} = *d* for various choices of Δ*z*_{1}. Second, we vary τ to obtain the free evolution fringe shown in Fig. 3E. The oscillation frequency, *n* = 1, verifies that Δϕ_{FE} accumulates from the gravitational potential energy difference across a vertical arm separation of Δ*z* = *d*.

When the initial separation is a non-integer number of lattice spacings Δ*z*_{1} ≠ Δ*nd*, lattice loading can cause the wave packets’ spatial distributions to differ between the upper and lower arms. Figure 3C shows an example with Δ*z*_{1} = 9.5 *d*: One arm splits between two adjacent lattice sites, becoming two new interferometer arms that are separated by Δ*n* = 1 from one another, and Δ*n* = 9 and 10 from the distant arm. We verify coherence by changing Δ*z*_{2} to interfere different combinations of the three arms; Fig. 3D shows the corresponding contrast envelopes within λ_{T} of Δ*z*_{2}/*d* = 1, 9, and 10. We vary τ to obtain free evolution fringes for each combination of arms (Fig. 3 E, F, and H, respectively). The observed oscillation frequencies ω_{FE}(Δ*n*) show the quantization of arm separations in multiples of the lattice period *d*.

At Δ*z*_{2}/*d* = 9.5, the lower arm is partially spatially overlapped with both upper arms, and the final pulse closes the two interferometers with separations of Δ*n* = 9 and 10 simultaneously. As a result, we observe the adjacent wave packets separated by Δ*n* = 1 coming into and out of phase at their difference frequency ω_{FE}(Δ*n* = 1), while interfering with the distant arm constructively after integer Bloch periods τ = *N*τ_{B}, or destructively after half-integer Bloch periods τ = (*N*+1/2)τ_{B}. This results in a beating interference in the free evolution phase, which we observe by varying τ (Fig. 3G).

Reaching seconds of phase coherence in the lattice allows this trapped interferometer to overcome the limiting noise source in state-of-the-art atomic gravimeters: the phase noise from vibrations (*29*, *30*). Instead of reducing vibrations themselves, seconds of hold time cut off the interferometer phase sensitivity to vibrations rapidly above, e.g., ~50 mHz after holding for τ = 20 s. This suppresses the phase variance from vibrations by three to four orders of magnitude across the problematic ~0.1- to 100-Hz range, where vibrations are difficult to suppress (*31*). Because gravitational signals are predominantly around dc, this orders-of-magnitude reduction in the interferometer noise bandwidth substantially increases the signal-to-noise ratio of the measurement. In materials and methods (*24*), we derive the interferometer phase response to the acceleration noise caused by vibrations, i.e., the acceleration transfer function *H*(2π*f*), accounting for the vibration sensitivity of both Δϕ_{L} and Δϕ_{FE}.

We directly measure |*H*(2π*f*)|^{2} by using a voice coil to apply accelerations to the vacuum chamber at a frequency *f*, and record the mid-fringe phase variance as the drive frequency *f* is varied. Figure 4 shows the agreement between the measured and calculated transfer functions, |*H*(2π*f*)|^{2}/|*H*(0)|^{2}, for a lattice interferometer with *T* = 1.066 ms and τ = 5.0 s. This measurement confirms that |*H*(2π*f*)|^{2} oscillates at a frequency commensurate with the hold time, *f*_{0}^{latt} = 1/τ = 0.2 Hz. For comparable dc acceleration sensitivity to this lattice interferometer, a Mach-Zehnder (MZ) interferometer would use a pulse separation time of *T*_{MZ} = 73.5 ms, setting the first zero in its transfer function at *f*_{0}^{MZ} = 13.6 Hz, nearly 70 times higher than *f*_{0}^{latt}. For a target dc sensitivity, the vibration immunity in this lattice geometry can be further enhanced by increasing τ and decreasing *T*.

More broadly, by demonstrating the longest quantum coherence to date of a massive object in a well-separated spatial superposition, this interferometer may constrain extensions of quantum mechanics (*32*) and provide a promising technical pathway for quantum simulations (*33*) using weakly interacting ultracold atoms in lattices. In atom interferometry, this lattice scheme presents an alternative approach to inertial measurement: holding atoms to directly probe the potential energy difference, rather than dropping atoms to measure accelerations. This approach strongly suppresses vibration noise while extending interrogation times in a compact volume, overcoming the two major limitations in conventional atomic interferometry. This lattice geometry is therefore well-suited for precision gravimetry (*29*), with exciting prospects for geophysics (*34*), and fundamental tests of short-ranged forces such as dark energy (*7*), Casimir forces (*35*), or short-ranged gravity (*36*). Additionally, measuring the phase due to a potential without subjecting the atoms to an acceleration represents a milestone toward observing a gravitational analog of the Aharonov-Bohm effect (*37*), which can provide a novel measurement of Newton’s constant *G* through the gravitational potential.

## Supplementary Materials

science.sciencemag.org/content/366/6466/745/suppl/DC1

Materials and Methods

Figs. S1 to S4

Table S1

Equations S1 to S28

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank P. Haslinger for collaboration in the lab; Z. Pagel, O. Schwartz, J. Axelrod, S. Campbell, X. Wu, and P. Hamilton for valuable discussions; and W. Zhong for experimental assistance.

**Funding:**This material is based on work supported by the David and Lucile Packard Foundation, the National Science Foundation under grant no. 1708160, and the National Aeronautics and Space Administration grants 1629914 and 1612859.

**Author contributions:**V.X., M.J., and H.M. built the apparatus, collected measurements, analyzed data, and wrote the manuscript. C.D.P. contributed to data collection and analysis. S.L.K. contributed to building the apparatus. L.W.C. contributed to data analysis. All authors contributed to the manuscript.

**Competing interests:**V.X., M.J., and H.M. are inventors on U.S. patent application no. 62/832,049.

**Data and materials availability:**All data presented in this paper are deposited in Zenodo (

*38*).