Optomechanical resonator-enhanced atom interferometry

Matter-wave interferometry and spectroscopy of optomechanical resonators offer complementary advantages. Interferometry with cold atoms is employed for accurate and long-term stable measurements, yet it is challenged by its dynamic range and cyclic acquisition. Spectroscopy of optomechanical resonators features continuous signals with large dynamic range, however it is generally subject to drifts. In this work, we combine the advantages of both devices. Measuring the motion of a mirror and matter waves interferometrically with respect to a joint reference allows us to operate an atomic gravimeter in a seismically noisy environment otherwise inhibiting readout of its phase. Our method is applicable to a variety of quantum sensors and shows large potential for improvements of both elements by quantum engineering. Precise gravity sensing in dynamic environments is challenging. Here, the long-term stability of a matter-wave gravimeter and high bandwidth of an optical resonator are combined in a compact gravity sensor with high seismic noise suppression.

In recent years, developments in the quantum engineering of optomechanical resonators have yielded devices with exciting applications in fields such as quantum information, fundamental physics and, likewise, in inertial sensing 38,39 . To this end, optically reading out length variations of an optical cavity allows determination of acting accelerations with high bandwidth and resolution. In addition, interfaces between cold atoms and micromechanical cantilevers 40 or nanomembranes 41 have been demonstrated, showing the potential of these hybrid systems for fundamental physics.
Here, we combine a high-bandwidth optomechanical resonator with a long-term stable light-pulse atom interferometer, and measure the accelerations of the resonator's test mass and a freely falling cloud of atoms relative to the atom interferometer's inertial reference. Our atom interferometer measures gravity under strong seismic perturbations without loss of phase information. In contrast to previous approaches, our method merges two systems both benefiting from the large toolbox of methods usually exploited in optical spectroscopy and photonics into a highly customisable device for atom interferometry in rough environments.

Results and discussion
We operate a Kasevich-Chu interferometer 17 combined with an optomechanical resonator attached to the mirror providing the phase reference for the interferometer as a gravimeter (Fig. 1a, b and details in the "Methods" section). Ambient vibrational noise couples to the retroreflector at a weighted acceleration level of 3 mm s −2 per cycle. This leads to phase excursions exceeding a single fringe during one interferometric measurement with the readout appearing to be random due to the underlying 2π phase ambiguity (Fig. 1c). Accordingly, the atom interferometer signal, i.e., the relative population of the two ports, features a bimodal distribution visible in Fig. 2a. However, ambient vibrational noise also results in a displacement of the resonator test mass, which is recorded by the signal retroreflected from the optomechanical resonator (Fig. 1d). The records of the optomechanical resonator make it possible to reconstruct the atomic interference pattern (Fig. 2b). The signal from the optomechanical resonator is constrained to the band of interest. We apply high-pass filters at 0.8 Hz to suppress low-frequency drifts, as well as a digital lowpass filter at 50 Hz, the atom interferometer's corner frequency (see "Methods" section). We subsequently sample it digitally over 60 ms centred around the central light pulse of each interferometer cycle. The phase correction is finally calculated from the signal utilising the acceleration sensitivity function describing the atom interferometer's phase response 42,43 .
Residual systematic biases can be experimentally analysed as shown for commercial sensors in the ref. 31 .
Using our method, we measure the local gravitational acceleration g in an approximately 22 h-long, interruption-free, measurement series otherwise impossible when operating both sensors alone. By suppressing vibrational noise, our sensor fusion method improves the overall short-term stability by a factor 8 ( Fig. 3). Figure 4 illustrates the present and projected features of the atom interferometer, the optomechanical resonator, and the combination of both.
Our optomechanical resonator features a sensitivity comparable to commercial accelerometers, and we expect a large potential for improvements for both quantum-optical devices. The sensor fusion performance is nevertheless limited by low frequency noise (Fig. 4, solid blue trace). It displays a RMS white acceleration noise of 1 × 10 −5 m s −2 Hz −1/2 between 10 and 50 Hz. Pink noise ( ∝ f −1 ) dominates from 1 to 10 Hz. Below 1 Hz, Brownian noise ( ∝ f −2 ) processes mainly caused by the optical fibre employed for interrogating the resonator prevail. Since the resonator's sensitivity to accelerations increases quadratically with decreasing mechanical resonance frequency and linearly with optical finesse, there is room for improvements by trading sensitivity against larger bandwidth and dynamic range. We foresee an optimisation of the hybrid sensor ( Fig. 4, solid orange trace) by tuning the resonance frequency to 1500 Hz to increase the bandwidth, improving the optical finesse to 1600 by high-reflectivity coating 38 , and the readout by an order of magnitude as compared to ref. 38 by means of spectroscopy techniques developed for ultrastable resonators, e.g., Pound-Drever-Hall locking 44,45 . Millimetre-sized optomechanical resonators have already demonstrated sensitivities of 1 × 10 −6 m s −2 Hz −1/2 over bandwidths up to 12 kHz.
Additionally, pathways exist for future atom interferometers customised for gravimetry, as discussed in ref. 46 . The sensitivity can be enhanced by operating the device with T = 35 ms, a cycle rate of 1 Hz, higher-order Bragg processes transferring 4 Á k ! eff and a reduced phase noise of 3 mrad. By improving the atom interferometer and tuning the optomechanical resonator, it is plausible that the intrinsic noise can be lowered to 6 × 10 −8 m s −2 Hz −1/2 under seismic noise as described in the ref. 47 . This target performance is comparable to the noise obtained in a quiet environment with an active vibration isolation 19 and outperforms transportable, commercial devices 23 .
Many atomic gravimeters employ rubidium and generate the light for manipulating the atoms by second harmonic generation from fibre lasers in the telecom C-band 48,49 . The inclusion of the optomechanical resonator therefore requires only minor hardware changes and can be performed with an all-fibred setup. The resonator can be implemented directly into inertial reference mirrors under vacuum, thus improving the mechanical quality factor while supporting miniaturisation of the overall setup. It does not emit notable heat, and is nonmagnetic. Consequently, it does not induce systematic errors due to black body radiation 50 or due to spurious magnetic fields coupling to the matter waves 30,51 , and neither do external magnetic fields couple to the resonator test mass. Moreover it can be easily merged with the retroreflection mirror of the atom interferometer. In addition, devices with different resonance frequencies or different orientations will grant access to larger bandwidth and multiple sensitive axes. Last but not least, the small volume of cubic millimetres offers great prospects for being integrated on atom chip sensors 46 , and, hence, a large potential for miniaturisation of the sensor head.
Our method shares analogies with atomic clocks by hybridisation of long-term and short-term references. Beyond this, our optical sensor might be used for compensating inertial noise in the resonators of optical clocks, e.g., in transportable setups 52 .

Conclusion.
In conclusion, we have demonstrated an atom interferometer enhanced by an optomechanical resonator. We show operation of the atom interferometer under circumstances otherwise impeding phase measurements. Inertial forces on the atoms and on the resonator mirror are measured to the same reference permitting a direct comparison, and high common mode noise suppression in the differential signal. Our method is not restricted to atomic gravimeters and could be beneficial to nearly all atom interferometric sensors and even improve laser interferometers 53 in environments with large inertial noise, thus replacing bulky vibration isolation and motion sensors. In particular, the achievable large dynamic range opens up great perspectives for the use of atomic sensors for inertial navigation 35,54 and airborne gravimetry 55 . Finally, a possible modification of our setup's topology would ensure that the atom-optics light field is   reflected directly off a micromechanical test mass. In future experiments, we envisage exciting research on coherent lightmediated coupling of matter waves and mechanical systems 41,56 with pulsed instead of cw-interaction.

Methods
Atom interferometer. Our setup (Fig. 1a, b), which was employed as a differential gravimeter in the refs. 6,57 , comprises a Kasevich-Chu interferometer 17 . In a π/ 2 − π − π/2 pulse sequence, stimulated two-photon Raman transitions coherently split, redirect, and recombine matter waves of 87 Rb. The interferometer phase is determined by measuring the number of atoms in output ports 1 and 2 with stateselective fluorescence detection. To leading order, a constant acceleration a ! of the atoms induces a phase shift where _ k ! eff is the photon recoil transferred to the atoms via a Raman process, and T denotes the time between two subsequent light pulses. A chirp of the relative frequency of the lasers cancels the phase induced by acceleration, and is a measure for the latter. The atom interferometer's response to vibrational noise can be described using the sensitivity formalism 42 . Notably, the response is flat in a band between DC and up to the corner frequency 1/(2T), above which it features a low pass behaviour. Typically, the interferometer's response is adjusted by varying T such that ambient noise induces phase shifts well within one fringe. At quiet conditions, i.e., with an operating vibration isolation system, the interferometer, which we operate at a cycle rate of 0.6 Hz, features a fringe contrast of ≈30% and a Raman phase locked loop-limited phase noise of 30 mrad for a time T = 10 ms.
Data analysis. In order to suppress systematic shifts independent of the direction of momentum transfer we use the k-reversal method 58,59 . The interferometer measures ten times in each direction of momentum transfer over a period of 18 s. For each scattering direction, we create histograms out of the normalised output population of the interferometer. Hereby each individual histogram comprises data accumulated over a 9000 s period. From each histogram, we extract the interferometer response's amplitude and offset 60 as shown in Fig. 2. We subsequently extract g for intervals of 180 s (approximately 50 shots per direction) by fitting the postcorrected data to the expected atom interferometer response using the parameters determined by the histogram fit, solely leaving the interferometer phase as a free parameter. After 18,000 s integration we determine a local gravitational acceleration 9.812675 m s −2 with a standard uncertainty of 4 × 10 −6 m s −2 . We estimate the ambient noise in Fig. 3 from the phase corrections made during postcorrection. Accordingly, the data resembles the underlying ambient acceleration noise after weighting it with the atom interferometer's transfer function. The first uncorrected value of ≈3 mm s −2 per cycle also manifests in the 1/e width of the Gaussian-shaped spread in Fig. 2c.
Optomechanical sensor. Mirrors forming the optomechanical resonator, which has a volume on the order of a few hundred mm 3 , are made from the flat tip of a polarisation maintaining fibre and a side face of the fused silica test mass supported by a stiff u-shaped flexible mount, the cantilever (Fig. 1a), following the design of ref. 61 . Our sensor features an optical finesse of about two, a resonance frequency of ω 0 = 2π 678.5 Hz, and a mechanical quality factor of Q = 630. Due to its stiffness the optomechanical resonator can be described as an ideal harmonic oscillator. Below the resonance frequency, displacement of the test mass X as a function of vibration frequency ω linearly depends on the acting acceleration A, and is therefore flat. By means of more advanced data analysis, the upper limit of the usable bandwidth can be extended beyond the mechanical resonance. Using adhesive bonding, the resonator is attached to a two-inch square mirror retroreflecting the light pulses driving the atom interferometer. The resonator's acceleration-sensitive axis is aligned collinearly with the retroreflector's normal vector (Fig. 1a) by orienting the outer edges of both devices parallel. The motion of the test mass is read out with a fibre-based optical setup based on telecom components comprising a tunable laser operating at a wavelength near 1560 nm protected by an optical isolator (Fig. 5). The sensor has a quarter wave plate incorporated in its lead fibre thereby enabling us to separate the signal reflected off the resonator using a polarising beam splitter. Additionally, a small fraction of the laser light is split off before the resonator using a 90:10 splitter. Making use of differential data acquisition of photo detectors PD 1 and 2 we can therefore cancel common mode laser intensity noise on the resonator signal. Finally, the processed signal depends on the transmission of the optomechanical resonator and hence the distance between the two reflective surfaces. Consequently, it is a direct measure of the acting acceleration. The optomechanical resonator and the retroreflection mirror are operated under normal atmospheric conditions. We place the mirror with the resonator attached onto a solid aluminium plate in strapdown configuration on the laboratory floor. During the initialisation of the optomechanical sensor, a commercial force-balance accelerometer (Nanometrics Titan) was utilised to perform test measurements for comparison as well as post correction for reference purposes.

Data availability
The data used in this manuscript are available from the corresponding author upon reasonable request.
Received: 1 May 2020; Accepted: 14 October 2020;  . Both signals are passed to current-to-voltage converters (I/V) followed by high-pass filters (HPF) at 0.8 Hz and are subsequently enhanced by amplifiers (Amp). Finally, the difference signal is digitally lowpass filtered (LPF) at 50 Hz.