Opto-mechanical resonator-enhanced atom interferometry

We combine an optical-mechanical resonator with an atom interferometer. A classical cantilever and matter waves sense their acceleration with respect to a joint reference. Apart from research on macroscopic opto-mechanical quantum objects, applications are in the realm of quantum sensing. We demonstrate its robustness by operating an atom-interferometric gravimeter beyond its reciprocal response in a highly dynamic environment, exploiting the common mode signal.

We combine our atom interferometer with an opticalmechanical resonator. This arrangement allows us to operate the atom interferometer under strong perturbations without loss of the phase information. Joint measurements of matter wave interferometers with commercial accelerometers have already been pioneered to extend the dynamic range or suppress vibration noise [26][27][28][29][30][31][32]. In our approach, the acceleration of a cantilever and a freely falling cloud of atoms is measured relative to a joint reference and both signals are combined to restore the interference fringes of the atomic interferometer. The device merges the complementary benefits of both sensors. While standard cold atom interferometers are employed for accurate and long-term stable measurement of accelerations, but are challenged by their dynamic range and their cyclic measurements, opto-mechanical resonators can measure continuously and have a larger dynamic range, but are in general subject to drifts.
Our hybrid device results in a high common mode noise rejection as both sensors are measuring accelera-tions with respect to the same reference. A common reference is established by directly contacting the resonator to the retroreflector of the atom interferometer, both being made of glass.
With our method, we realized an atom-interferometric gravimeter under influence of artificially induced strong vibrations which would otherwise obscure any phase information. Applications are not restricted to gravimetry, but can also exploit the excellent intrinsic long term stability and accuracy of atom interferometers in navigation [33]. Our method can be employed to enhance any atom interferometric measurement in environments with large inertial noise.
Indeed, replacing bulky vibration isolation and motion sensors by opto-mechanical resonators with a volume of much less than a cubic centimeter shows a great potential for miniaturization, especially for integration on atom-chip fusions such as [34]. We also anticipate a multitude of exciting applications of the combination of optical-mechanical resonators and atom interferometers in fundamental research such as sensing the quantum state of opto-mechanical resonators [35,36].
Our opto-mechanical atom-interferometric hybrid device ( Fig. 1) comprises a rubidium Raman-type interferometer, which was employed as a differential gravimeter in Ref. [6], and a small resonator with a volume on the order of a few hundred mm 3 [24]. The latter is attached utilizing adhesive bonding to a two-inch square mirror . A prototype opto-mechanical resonator is implemented on a 2 square retroreflection mirror resting on a piezoelectric transducer tripod and rests on a commercial vibration isolation platform under atmosphere. Post-correction of the atom interferometer's raw data (c) is achieved by digitally convolving the resonator's signal (d) with the atom interferometer's transfer function to compute phase corrections. The signal from the resonator is read out on a photo diode and digitally processed (e).
retroreflecting the light pulses driving a sequence of three Raman processes. These coherently split, redirect, and recombine matter waves of 87 Rb to form an interferometer. Phase is determined by measuring the number of atoms in each of its output ports, e.g. by means of stateselective fluorescence detection. A constant acceleration a of the atoms along the vector normal to the retroreflection mirror induces a phase shift equal to k eff · a·T 2 , where k eff is the effective wave vector of the beamsplitting light, and T denotes the time between the three light pulses. Typically, the interferometer's response is adjusted by varying T such that ambient noise induces phase shifts well within one fringe.
The opto-mechanical resonator is formed by the opposed ends of two fibers attached to a stiff u-shaped flexible mount made of fused silica, the cantilever, and a rigid counterpart, following the design of Ref. [37]. Our prototype features a finesse F of about two, a resonance frequency 7.7 kHz, and a quality factor Q of a thousand. Due to its stiffness the opto-mechanical resonator can be described as an ideal harmonic oscillator and displacements of the test mass show a linear spectral response below the resonance frequency. The resonator's acceleration-sensitive axis was aligned collinearly with the mirror's normal vector by orienting the outer edges of both devices parallel. The cantilever is read out with a fiber-based optical setup based on telecom components comprising a tunable laser operating at a wavelength near 1560 nm. The beam reflected off the resonator is separated with the help of an optical circulator. The intensity variations of the retroreflected signal depend on the transmission of the opto-mechanical resonator and, hence, the distance of the two fiber ends. The entire setup is operated under normal atmospheric conditions. We place the mirror with the resonator attached onto a solid aluminum plate resting on a piezoelectric transducer (PZT) tripod. The latter is mounted upon a commercial vibration isolation platform and enables a controlled actuation of the assembly to simulate an environment subject to vibrational noise.
To demonstrate the capability of our opto-mechanical resonator enhanced interferometer, we operate the device at a time T = 10 ms. We set the interferometer's inertial reference mirror in motion by driving the PZT tripod with a sinusoidal signal at 350 Hz corresponding to a weighted RMS acceleration noise of 3.25 × 10 −3 m/s 2 per cycle such that the phase excursion exceeds a single fringe and the readout appears to be random due to the underlying 2π phase ambiguity and shows a bimodal distribution (Fig. 2, left). In general, the histogram's shape depends on the PZT amplitude, reflecting the spectral type of vibration noise. Exploiting the recordings of the optical-mechanical resonator, we post-correct the interferometry data by using the respective spectral response function and reconstruct the atomic interference pattern (Fig. 2, right).
Post-correcting the interferometer with our resonator improves the short-term stability of the hybrid device by a factor 16 (Fig. 3). Moreover, we show the capability to perform long-term measurements of gravitational acceleration g in a 19.5 h drop-out free measurement. For this purpose, the interferometer is operated in two settings, in which the atoms are either scattered upward or downward during the first beam splitting process in order to suppress systematic effects [38,39]. After extracting the interferometer response's amplitude and offset through histogram fits (Fig. 2, left) [40] on bins of 545 shots or about 25 min, we subsequently determine g by fitting data sets of 50 shots for each direction of momentum transfer, solely leaving the interferometer phase as a free parameter from which an acceleration value for upward and downward operation is determined and g is derived as the mean value.
With all other noise sources being two to three orders of magnitude lower, the acceleration noise of our resonator at 5 × 10 −4 m/s 2 / √ Hz exceeds by far the intrinsic noise of the atom interferometer. It is caused by both residual intensity noise of the source laser and optical fiber to the resonator. Indeed, millimeter-sized opto-mechanical resonators have demonstrated sensitivities of 1 × 10 −6 m/s 2 / √ Hz over bandwidths up to 12 kHz.  Moreover, spectroscopy techniques developed for ultrastable resonators [41,42] can be exploited to improve the performance of the readout. As the resonator's sensitivity to accelerations depends quadratically on the resonance frequency and linearly on the finesse, there is large room for improvements by trading sensitivity against smaller dynamic range [43]. For an opto-mechanical resonator with a resonance frequency of 1500 Hz and an order of magnitude improved readout compared to Ref. [24], we project a compact gravimeter [34] with a pulse separation time of T = 35 ms and repetition rate of 1 Hz (Fig. 4) to be vibration limited to σ a = 6 × 10 −8 m/s 2 at 1 s without seismic isolation and assuming ideal correlation with the opto-mechanical accelerometer. Outside of its bandwidth we assume background noise according to the Peterson new high noise model (NHNM) [44]. The achievable performance would be comparable with the lowest noise obtained in a quiet environment with an active vibration isolation [15] and outperform transportable, commercial devices [19]. Noting that many atomic gravimeters employ rubidium and generate the light for manipulating the atoms by second harmonic generation with telecom fiber lasers, the inclusion of this sensor requires only minor hardware changes in this case and can be performed with an all-fibered setup.
Moreover, the resonator is fully vacuum compatible, does not emit notable heat, and is non-magnetic and consequently does not induce related errors [27,45,46], and can be easily merged with the retroreflection mirror of the atom interferometer. Ideally, the latter is placed in the vacuum chamber of the atom interferometer, which is also beneficial for the performance of the opto-mechanical resonator. Here, the small volume device offers great prospects for being integrated on atom chip sensors, and, hence, a large potential for miniaturization of the sensor head.
In conclusion, we have demonstrated an atom interferometer enhanced by an opto-mechanical resonator. We show operation of the atom interferometer under circumstances otherwise impeding phase measurements. Inertial forces on the atoms and on the cantilever carrying one 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 navigation [32]. We also anticipate the continuation of opto-mechanical resonators and atom interferometry employing the cantilever as link between the atom interferometer and the resonator, e.g. to probe the resonator with the atom interferometer [47][48][49][50] or even to couple the atom interferometer to non-classical states of the mechanical oscillator.
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) and 50WM1137 (QUANTUS-IV-Fallturm), by "Niedersächsisches Vorab" through the "Quantum-and Nano-Metrology (QUANOMET)" initiative within the project QT3, and by "Wege in die Forschung (II)" of the Leibniz University Hannover.