Optomechanical resonator-enhanced atom interferometry

TL;DR

This paper presents a hybrid quantum sensor combining atom interferometry and optomechanical spectroscopy, enabling stable measurements in noisy environments and enhancing the capabilities of quantum sensors.

Contribution

It introduces a novel method that integrates atom interferometry with optomechanical resonator spectroscopy, overcoming limitations of each technique individually.

Findings

Successful measurement of mirror and matter wave motion in noisy environments

Enhanced stability and dynamic range of quantum sensors

Potential for significant improvements through quantum engineering

Abstract

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.

Equations

$$\Delta \varphi ={\vec{k}}_{\text{eff}}\cdot \vec{a}\cdot T^2,$$

$$\frac{X(\omega )}{A(\omega )}=-\frac{1}{{\omega }_0^2-{\omega }^2+i\frac{{\omega }_0}{Q}\omega },$$

Full text

Matter-wave interferometers are employed in fundamental physics Fixler et al. (2007); Rosi et al. (2014); Spagnolli et al. (2017); Fray et al. (2004); Bonnin et al. (2013); Schlippert et al. (2014); Tarallo et al. (2014); Zhou et al. (2015); Kovachy et al. (2015); Pandey et al. (2019); Hamilton et al. (2015); Jaffe et al. (2017); Bouchendira et al. (2011); Décamps et al. (2016); Parker et al. (2018); Shayeghi et al. (2020), metrology, and inertial sensing Peters et al. (1999); Hu et al. (2013); Fang et al. (2016); Freier et al. (2016); Hardman et al. (2016); Ménoret et al. (2018); Bidel et al. (2018); Stockton et al. (2011); Berg et al. (2015); Dutta et al. (2016); Hinton et al. (2017); Chen et al. (2020). 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 Cervantes et al. (2014); Pisco et al. (2018). In addition, cold atoms were coupled to micromechanical cantilevers Hunger et al. (2010) and nanomembranes Camerer et al. (2011).

So far, commercial sensors have been exploited to track the motion of atom interferometers’ inertial references, thus extending the measurement dynamic range, suppressing vibration noise Kohel et al. (2008); Le Gouët et al. (2008); Merlet et al. (2009); Lautier et al. (2014); Geiger et al. (2011); Barrett et al. (2016); Cheiney et al. (2018), and allowing for comparisons with other atom interferometers Peters et al. (2001); Merlet et al. (2010); Freier et al. (2016).

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 quantum engineering usually exploited in optical spectroscopy and photonics to a highly customisable device. To this end, extensions of our method will make use of the tunability of the mechanical resonance of optomechanical resonators. In addition, devices with different resonance frequencies or different orientations will grant access to larger bandwidth and multiple sensitive axes. In combination with the use of more advanced spectroscopy and feedback methods this opens up atom interferometry to rough environments. Our method is not restricted to gravimetry or navigation Jekeli (2005) but can help to enhance any interferometric measurement with matter as well as with light Niebauer et al. (1995) in environments with large inertial noise, thus replacing bulky vibration isolation and motion sensors.

Finally, optomechanical resonators with a volume of much less than a cubic centimetre show a great potential for miniaturisation of future quantum sensors.

I Results

We operate a Kasevich-Chu interferometer Kasevich and Chu (1991) combined with an optomechanical resonator attached to the mirror providing the phase reference for the interferometer as a novel gravimeter (Fig. 1 (a) & (b) and details in the Methods section). Ambient vibrational noise couples to the retroreflector at a weighted acceleration level of $3\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{s}^{2}$ 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 $2\pi$ phase ambiguity (Fig. 1 (c)). Accordingly, the atom interferometer signal, i.e. the relative population of the two ports, features a bimodal distribution visible in Fig. 2 (a). 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. 1 (d)). The records of the optomechanical resonator make it possible to reconstruct the atomic interference pattern (Fig. 2 (b)). The signal from the optomechanical resonator is constrained to the band of interest. We apply high-pass filters at $0.8\text{\,}\mathrm{H}\mathrm{z}$ to suppress low-frequency drifts, as well as a digital low-pass filter at $50\text{\,}\mathrm{H}\mathrm{z}$, the atom interferometer’s corner frequency (see Methods section). We subsequently sample it digitally over $60\text{\,}\mathrm{m}\mathrm{s}$ 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 Cheinet et al. (2008); Bonnin et al. (2015). Residual systematic biases can be experimentally analysed as shown for commercial sensors in Ref. Merlet et al. (2009).

Using our method, we measure the local gravitational acceleration $g$ in an approximately $22\text{\,}\mathrm{h}$-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).

II Discussion

Fig. 4 illustrates the present and projected features of the atom interferometer, the optomechanical resonator, and the combination of both. We expect a large potential for improvements for both quantum-optical devices. Our optomechanical resonator features a sensitivity comparable to commercial accelerometers. Its sensor fusion performance is nevertheless limited by low frequency noise (Fig. 4, solid blue trace). It displays a RMS white acceleration noise of $1\text{\times}{10}^{-5}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}^{2}\mathrm{/}\sqrt{Hz}$ between 10 and $50\text{\,}\mathrm{Hz}$. Pink noise ($\propto f^{-1}$) dominates from 1 to $10\text{\,}\mathrm{Hz}$. Below $1\text{\,}\mathrm{Hz}$, Brownian noise ($\propto f^{-2}$) 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 111Increasing the sensitivity through lower natural frequency of the resonator, the dynamic range is necessarily reduced since the device is operated in the non-linear regime.. We foresee an optimisation of the hybrid sensor (Fig. 4, solid orange trace) by tuning the resonance frequency to $1500\text{\,}\mathrm{H}\mathrm{z}$ to increase the bandwidth, improving the optical finesse to 1600 by high-reflectivity coating Cervantes et al. (2014), and the readout by an order of magnitude as compared to Ref. Cervantes et al. (2014) by means of spectroscopy techniques developed for ultrastable resonators, e.g., Pound-Drever-Hall locking Pound (1946); Drever et al. (1983). Millimetre-sized optomechanical resonators have already demonstrated sensitivities of $1\text{\times}{10}^{-6}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}^{2}\mathrm{/}\sqrt{Hz}$ over bandwidths up to $12\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$.

Additionally, pathways exist for future atom interferometers customised for gravimetry, as discussed in Ref. Abend et al. (2016). The sensitivity can be enhanced by operating the device with $T=$35\text{,}\mathrm{m}\mathrm{s}$$, a cycle rate of $1\text{\,}\mathrm{H}\mathrm{z}$, higher-order Bragg processes transferring $4\cdot\vec{k}_{\textnormal{eff}}$ and a reduced phase noise of $3\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$. By improving the atom interferometer and tuning the optomechanical resonator, it is plausible that the intrinsic noise can be lowered to $6\text{\times}{10}^{-8}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}^{2}\mathrm{/}\sqrt{Hz}$. This target performance is comparable to the noise obtained in a quiet environment with an active vibration isolation Hu et al. (2013) and outperforms transportable, commercial devices Ménoret et al. (2018). This is based on the assumption of an environment described by the “Peterson new high noise model” Peterson (1993).

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 Luo et al. (2019); Theron et al. (2017). 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 non-magnetic. Consequently, it does not induce systematic errors due to black body radiation Haslinger et al. (2017) or due to spurious magnetic fields coupling to the matter waves Le Gouët et al. (2008); Hu et al. (2017), 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. Last but not least, the small volume of cubic millimetres offers great prospects for being integrated on atom chip sensors Abend et al. (2016), and, hence, a large potential for miniaturisation of the sensor head.

Our method shares analogies with atomic clocks by hybridisation of long- 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 Grotti et al. (2018).

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. In particular, the achievable large dynamic range opens up great perspectives for the use of atomic sensors for inertial navigation Cheiney et al. (2018) and airborne gravimetry Bidel et al. (2020). 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 light-mediated coupling of matter waves and mechanical systems Camerer et al. (2011); Karg et al. (2019) with pulsed instead of cw-interaction.

III Methods

Atom Interferometer –

Our setup (Fig. 1 (a) & (b)), which was employed as a differential gravimeter in Ref. Schlippert et al. (2014); Albers et al. , comprises a Kasevich-Chu interferometer Kasevich and Chu (1991). In a $\pi/2-\pi-\pi/2$ pulse sequence, stimulated two-photon Raman transitions coherently split, redirect, and recombine matter waves of 87Rb. The interferometer phase is determined by measuring the number of atoms in output ports 1 and 2 with state-selective fluorescence detection. To leading order, a constant acceleration $\vec{a}$ of the atoms induces a phase shift

[TABLE]

where $\hbar\vec{k}_{\textnormal{eff}}$ 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 Cheinet et al. (2008). 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 $0.6\text{\,}\mathrm{H}\mathrm{z}$, features a fringe contrast of $\approx$ $30\text{\,}\mathrm{\char 37\relax}$ and a Raman phase locked loop-limited phase noise of $30\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ for a time $T=$10\text{,}\mathrm{ms}$$.

Data Analysis –

In order to suppress systematic shifts independent of the direction of momentum transfer we use the $k$-reversal method McGuirk et al. (2002); Louchet-Chauvet et al. (2011). The interferometer measures 10 times in each direction of momentum transfer over a period of $18\text{\,}\mathrm{s}$. 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 $9000\text{\,}\mathrm{s}$ period. From each histogram, we extract the interferometer response’s amplitude and offset Varoquaux et al. (2009) as shown in Fig. 2. We subsequently extract $g$ for intervals of $180\text{\,}\mathrm{s}$ (approximately 50 shots per direction) by fitting the post-corrected 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 $18\,000\text{\,}\mathrm{s}$ integration we determine a local gravitational acceleration $9.812\,675\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}^{2}$ with a standard uncertainty of $4\text{\times}{10}^{-6}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}^{2}$. We estimate the ambient noise in Fig. 3 from the phase corrections made during post correction. Accordingly, the data resembles the underlying ambient acceleration noise after weighting it with the atom interferometer’s transfer function. The first uncorrected value of $\approx$3\text{,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{s}^{2}$$ per cycle also manifests in the $1/e$ width of the Gaussian-shaped spread in Fig. 2 (c).

Optomechanical Sensor –

Mirrors forming the optomechanical resonator, which has a volume on the order of a few hundred mm3, 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. 1 (a)), following the design of Ref. Gerberding et al. (2015). Our sensor features an optical finesse of about two, a resonance frequency of $\omega_{0}=$678.5\text{,}\mathrm{H}\mathrm{z}$$, 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 $\omega$ linearly depends on the acting acceleration $A$,

[TABLE]

and is therefore flat. By means of more advanced data analysis, 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. 1 (a)) 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 $1560\text{\,}\mathrm{n}\mathrm{m}$ 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.

IV Acknowledgements

We thank H. Ahlers, J. Lautier-Gaud, L. Timmen, J. Müller, S. Herrmann, S. Schön, K. Hammerer, D. Rätzel, P. Haslinger, and M. Aspelmeyer for comments and fruitful discussions and acknowledge financial support from Deutsche Forschungsgemeinschaft (DFG) within CRC 1128 (geo-Q), projects A02, A06, and F01 and CRC 1227 (DQ-mat), project B07, and under Germany’s Excellence Strategy – EXC-2123 QuantumFrontiers – 390837967 (research unit B02). D.S. acknowledges support by the Federal Ministry of Education and Research (BMBF) through the funding program Photonics Research Germany under contract number 13N14875. This project is furthermore supported by the German Space Agency (DLR) with funds provided by the Federal Ministry for Economic Affairs and Energy (BMWi) due to an enactment of the German Bundestag under Grant No. DLR 50WM1641 (PRIMUS-III), 50WM1137 (QUANTUS-IV-Fallturm), and 50RK1957 (QGYRO), and by “Niedersächsisches Vorab” through the “Quantum- and Nano-Metrology (QUANOMET)” initiative within the project QT3, and by “Wege in die Forschung (II)” of Leibniz University Hannover.

V Author contributions

W.E., E.M.R., C.S., and D.S. designed the atom interferometer and its laser system. L.L.R., H.A., D.N., and D.S. contributed to the design of the atom interferometer and its laser system and realised the overall setup. A.R., M.M., L.K., L.C., R.S., C.B., and F.G. designed, built, and tested the optomechanical resonator and designed the readout laser system. A.R., C.M., D.T., and É.W. built and characterised the laser system for readout. L.L.R., F.G., L.K., and A.R. implemented the optomechanical resonator in the atom interferometer setup. L.L.R., H.A., D.N., and A.R. operated the final experimental setup. Sv.A., M.G., and C.S. contributed to the data acquisition system utilised for post correction. A.R., D.N., C.S., and L.L.R. performed the analysis of the data presented in this manuscript. L.L.R., D.S., and F.G. drafted the initial manuscript. A.R., C.M., C.S., D.T., É.W., E.M.R., and L.K. provided major input to the manuscript and all authors critically reviewed and approved of the final version.