# Quantum sensing for gravity cartography

### Experimental overview

A lightweight-pulse atom interferometer is conceptually just like an optical interferometer, with the roles of sunshine and matter interchanged. Atoms, appearing as matter waves, are subjected to a sequence of sunshine pulses that impart momentum to them, appearing analogously to mirrors and beamsplitters. Making use of a light-weight pulse for an acceptable size of time will trigger a transition between the bottom and excited states of an atom, accompanied by the absorption and stimulated emission of a photon. Such a pulse, generally known as a π pulse, acts as an atom optic mirror owing to the momentum that’s transferred. Equally, tuning the sunshine pulse such that it has solely a 50% transition chance, generally known as a π/2 pulse, acts as a beamsplitter via offering a momentum kick to solely half of the atomic chance distribution. A matter-wave equal of the optical Mach–Zehnder interferometer can then be created via making use of a π/2–π–π/2 pulse sequence with an evolution time, T, between the pulses. The ensuing space-time space enclosed by the atomic trajectories (Prolonged Knowledge Fig. 1b) is proportional to the native gravitational acceleration, which may then be measured from the relative inhabitants of the 2 atomic states after the ultimate pulse.

A gravity gradiometer makes use of two such interferometers offset vertically and probed concurrently with the identical pulse sequence. This suppresses common-mode results, resembling noise from vibration or part modifications attributable to variations in tilt with respect to the gravity of the Earth, that are indistinguishable from the gravity anomalies of curiosity in line with Einstein’s equivalence precept. Our machine consists of two subunits (Prolonged Knowledge Fig. 1a), a sensor head and a management system, with mild and electrical alerts transferred via a 5-m umbilical. The gradiometer is proven in Prolonged Knowledge Fig. 2, with an summary of its measurement, weight and energy traits.

The sensor head includes a vacuum system with twin MOT preparation and interrogation areas in an hourglass configuration, with all mild delivered to the atoms via on-axis counter-oriented telescopes. The sunshine is delivered in every course, with parts of the beam being redirected in direction of the atom-trapping area utilizing in-vacuum mirrors, to kind the radial cooling beams in every MOT. The central portion passes via, such that every enter supplies the vertical laser cooling beam in a given course for each MOTs. This makes all fluctuations in depth widespread for the radial cooling beams (stopping lateral offsets), and, via use of a Gaussian beam form, supplies a better depth for the vertical beams to higher saturate the radiation strain pressure on this course. This ends in a vastly improved stability and robustness of the laser cooling course of, decreasing fluctuations in temperature or atom cloud place (Fig. 1b) with out the necessity for extreme laser powers that will inhibit area operation. In a comparable take a look at system, this offered a discount in common cloud centre-of-mass movement to (0.14 ± 0.09) mm as in comparison with (1.19 ± 0.86) mm over an hour in related situations with a six-beam MOT. Each MOT areas have two coils, every shaped of 92 turns of 1-mm-Kapton-coated copper wire wound round an aluminium former (fastened utilizing epoxy), with a slit to stop eddy currents. The coils have a radius of 43 mm and separation of 56 mm, to provide a linear area gradient of 12.5 G cm−1 at a driving present of two.5 A. These are positioned across the vacuum system, such that the sturdy magnetic area axis of their quadrupole area is alongside the course of journey of the cooling beam axis. As well as, two units of rectangular coil pairs, every having 20 turns, are positioned across the MOT areas. These have a separation of 100 mm, and dimensions of 320 mm within the vertical and 90 mm within the horizontal, and can be utilized to compensate residual magnetic fields, or apply offsets. In apply, no compensation fields are used for the molasses part. Within the decrease chamber, one coil pair is used to use a 0.63 G area to regulate the atom cloud horizontal place by roughly 0.5 mm within the MOT part, enhancing the interferometer distinction. A bias coil42 is positioned across the system to outline a quantization axis and take away degeneracy between magnetic sublevels, with different coils being switched off after the magneto-optical trapping part. This has a variable pitch form to account for edge results and enhance area uniformity over the atom interferometry area. The system is enclosed in a magnetic protect that gives 25 dB attenuation of the exterior area. The in situ magnetic area profile is measured (via spectroscopy of the Raman transition) as being homogeneous to beneath 5% throughout the atom interferometry area, restricted by inside magnetic area sources from vacuum pumps.

The laser system consists of telecom lasers which can be frequency doubled to 780 nm, to be close to the D2 line of rubidium-87 (refs. 43,44). The sunshine for laser cooling is generated by passing the laser output via an electro-optic modulator (EOM) and producing a sideband at a frequency of roughly 1.2 GHz output from the service. That is used to offer a locking sign utilizing the (F=3 > to |{F}^{{prime} }=4 > ) transition in rubidium-85, inserting the service frequency such that it’s tuneable round resonance with the (F=2 > to |{F}^{{prime} }=3 > ) transition of rubidium-87 to offer the cooling mild. A separate EOM is used to offer repumping mild resonant with the (F=1 > to |{F}^{{prime} }=2 > ) transition. Atom interferometry is realized via two-photon stimulated Raman transitions. The Raman laser used to drive these has a linewidth of 73 kHz and is locked with an offset of 1.9 GHz to the (F=2 > to |{F}^{{prime} }=3 > ) transition. The second Raman frequency is generated utilizing a pair of EOMs working at 6.835 GHz. Performing the differential measurement suppresses part noise which will come up owing to optical path-length modifications between the 2 Raman beams (resembling these attributable to vibration and thermally induced modifications within the refractive index of fibres). This permits the 2 beams to be delivered independently with out the necessity for a part lock between them, facilitating an implementation through which the modulated spectrum is utilized to solely one of many enter beams. This avoids parasitic Raman transitions that give rise to systematic offsets and dephasing when utilizing typical modulation-based schemes, resembling these together with a retro-reflected beam31. To understand a sensible implementation of space-time space reversal30, also referred to as wavevector reversal, the system has an EOM in every enter course of the Raman beams, and the modulation sign is utilized to at least one arm in every measurement. This permits the course of the momentum kick imparted to the atoms to be modified between measurements, by altering which arm the modulation sign is utilized to utilizing a radiofrequency change (see Prolonged Knowledge Fig. 1). The contributions to the interferometer phases attributable to acceleration below gravity are delicate to the course of the recoil imparted by the sunshine, whereas these arising from many different results, resembling these attributable to magnetic fields, will not be. This permits these results to be eliminated when interleaved measurements are carried out within the two recoil instructions.

The sunshine is delivered to the sensor head utilizing polarization-maintaining optical fibres, with separate fibres for the cooling and Raman beams. These fibres ship the sunshine to optical telescopes that collimate the sunshine on the desired beam measurement. The cooling beams have a waist of 24 mm, and include a typical most energy of 130 mW. These impinge on the in-vacuum mirrors, that are 15-mm right-angle prisms (Thorlabs, MRA15-E03), to ship the horizontal cooling beams. The mirrors are mounted to a titanium construction (connected utilizing Epo-Tek H21D adhesive) in a cross configuration such that there’s a 15-mm aperture of their centre. The central portion of the cooling beams passes via these apertures to offer the sixth beam required for the other MOT. The Raman beams are overlapped with the cooling beams utilizing a polarizing beamsplitter dice, such that they’re then delivered alongside the identical beam axis because the cooling mild. The Raman beams, every containing a typical most energy of 300 mW, have their waist set to six.2 mm to restrict aperturing and diffraction on the central aperture of the in-vacuum mirrors, permitting the Raman beams to move via the system with out being redirected by the prisms. Though aperturing is proscribed on the mirrors within the present instrument, it could be fascinating to make use of a bigger Raman beam than the aperture in additional compact techniques or these aiming to additional cut back dephasing induced by laser beam inhomogeneity. Diffraction from the aperture would have to be given due consideration if pursuing this, as would the potential for additional mild shifts attributable to, on this case, one interferometer seeing additional mild fields from mirror reflections. The polarization of the sunshine is about to the suitable configuration for cooling or driving Raman transitions via use of voltage-controlled variable retarder plates within the higher and decrease telescopes used to ship the sunshine. The depth of the Raman beams is actively stabilized utilizing suggestions from a photodiode to regulate acousto-optic modulators, that are additionally used to provide the laser pulses.

The experimental sequence begins by gathering roughly 108 rubidium-87 atoms in every MOT from a background vapour over 1–1.5 s. Molasses cooling is then used to cut back the upper- and lower-cloud temperatures to (2.86 ± 0.09) μK and (3.70 ± 0.20) μK, respectively (see Fig. 1b). The variations in temperature come up from variations in native residual magnetic fields, arising primarily from the magnetic protect geometry, and small variations in optical alignment. Optical state and velocity choice is carried out to pick solely atoms within the (F=1,{m}_{{rm{F}}}=0 > ) magnetic sublevel and desired velocity class. That is achieved via utility of π pulses and a collection of blow-away pulses to take away atoms in undesired states and velocity lessons. Atom interferometry is then carried out with a pulse separation of T = 85 ms and π-pulse size of 4 µs. The interferometers are learn out utilizing bistate fluorescence detection to find out the atomic state inhabitants ratios of the |F = 2> and |F = 1> floor states, for which (2.7 ± 0.1) × 105 and (1.7 ± 0.1) × 105 atoms take part within the higher and decrease interferometers, respectively, with a typical measurement price of 0.7 Hz. The differential part, from which the gravity gradient is derived, is extracted by plotting the higher interferometer outputs in opposition to the decrease interferometer outputs, to kind a Lissajous plot as proven within the inset of Fig. 2. Along with random noise arising from vibration, we add a deliberate random part worth, from between 0 and 2π, to the ultimate pulse of the interferometer. At ellipse phases that don’t correspond to a circle, a clustering of factors across the extremal factors of the ellipse is seen even for uniform noise.

The quantum projection noise of the system based mostly on the taking part atom quantity is roughly 44 E/√Hz. The whole noise finances consists of contributions from additional phrases, and is proven in Prolonged Knowledge Desk 1, alongside related systematics noticed in the course of the survey. The noise finances was investigated via laptop simulation of noise processes, in comparison with experimental knowledge, and ellipse becoming.

### Survey apply and processing of the measurement knowledge

For every measurement on the survey, 600 runs of the atom interferometer had been usually taken with the sensor head in a single location (with the horizontal place being measured utilizing a complete station, Leica TS15, and the vertical place from the highway floor being roughly 0.5 m for the decrease sensor and 1.5 m for the higher sensor), giving twelve 25-point ellipses in every of the interferometer instructions and due to this fact 12 separate estimates of the gravity gradient. Repeat measurements had been taken on every measurement place, with usually three factors on every place. A measurement was taken at a base station between every measurement level, with the ultimate base-station measurement for one location used as the primary for the subsequent. The standard of becoming to every ellipse was recognized utilizing the error metric, (varepsilon ), outlined as

$$varepsilon =frac{left(frac{1}{a}+frac{1}{c}proper)}{2}{left(frac{1}{N}mathop{sum }limits_{i=1}^{N}{L}_{i}^{2}proper)}^{frac{1}{2}},$$

through which N is the variety of knowledge factors, L is the minimal distance between every knowledge level and a degree on the ellipse, and a and c correspond to an ellipse outlined parametrically by equations (x=a{rm{sin }}theta +b) and (y=c{rm{sin }}left(theta +varphi proper)+d), respectively. Errors within the ellipse becoming are delicate to modifications within the ellipse opening angle47. On the premise of numerical simulations, we estimate this impact to be lower than just a few elements in a single thousand; due to this fact, a 100 E change can be topic to an error of lower than 0.5 E. Such errors are due to this fact small in comparison with different errors. Such a 100 E change in gradient would correspond to an 11.6 mrad change within the ellipse form. This part shift will be in comparison with a 2π measurement vary, which means that measurement vary of the instrument on this configuration is related to nearly all of sensible options of curiosity (these being usually beneath 400 E).

Ellipse suits discovered to have (varepsilon > 0.05) had been robotically discarded. This resulted in 98.4% of all knowledge being usable in regular operation, representing a beneficial knowledge up time in comparison with that of comparable typical geophysical gadgets.

To course of the info, a straight line was fitted to the base-station factors, with this line then being subtracted from all knowledge factors. That is normal apply to take away drift in geophysical surveys. The main supply of drift is believed to be as a result of a.c. Stark shift, with this additionally being related owing to the distinction within the temperature of the 2 clouds. The gravity gradient worth is then taken as the typical of the measurement factors, leading to an estimate of the distinction in gradient between the measurement location and the bottom station. Moreover, the variations within the knowledge factors are used to make an estimate of the error within the distinction worth. When a number of measurements from the identical location had been mixed, a weighted common was used, giving much less weight to measurements with higher errors. The weighting issue is proportional to the reciprocal of the variance of every measurement48. The information, as proven in Fig. 3a, will not be corrected for terrain or results resembling tides. Tidal results will not be corrected, being negligible via the differential measurement of the gravity gradient.

The typical of the gravity gradient error discovered throughout the measurement positions of the survey is 17.9 E. Evaluating this to an approximate sign measurement of 150 E provides an approximate signal-to-noise ratio of 8.

### Inference from gravity gradiometer knowledge

Bayesian inference is a framework inside which prior beliefs will be up to date with info contained in knowledge. For a mannequin parameter vector ((theta )) and an information vector ((d))

$$pleft(theta |dright)=fractheta proper)pleft(theta proper){pleft(dright)},$$

through which (pleft(d|theta proper)) is the chance, (pleft(theta proper)) is the prior, (pleft(dright)) is a normalization fixed and (pleft(theta |dright)) is the posterior distribution.

The chance operate supplies the misfit between the measured knowledge, (d), and the modelled knowledge values calculated from the mannequin parameter vector, (theta ). The mannequin used right here is that of a three-dimensional cuboid35; the free mannequin parameters are proven in Prolonged Knowledge Fig. 3, together with the useful type of the respective prior distributions. The rationale behind the chosen prior distributions is detailed in Prolonged Knowledge Desk 2. The whole uncertainty for every measurement level is calculated utilizing the Pythagorean sum of the usual error and the mannequin uncertainty random variable multiplied by the typical of the usual error throughout the entire measurement positions.

The probabilistic Python bundle pymc3 (ref. 49) is used to implement the cuboid mannequin, outline the mannequin parameter prior distributions and pattern the posterior distribution, utilizing a no U-turn sampler50. Prolonged Knowledge Fig. 4 exhibits the Bayesian posterior distribution for choose mannequin parameters.

The parameter posterior distributions symbolize the up to date beliefs in regards to the mannequin parameters, given the measurement knowledge. To assist interpretation of the posterior distribution, the POE36 is calculated, which represents the spatial chance of the anomaly underground, given the mannequin and prior distributions (as proven in Fig. 3c). The horizontal place of the tunnel centre is set as (0.19 ± 0.19) m alongside the survey line, with the distribution being roughly Gaussian. The depth from the origin, outlined within the vertical utilizing the bottom level on the survey line, to the centre is (1.7 −0.59/+2.3) m. On the horizontal place of the tunnel, the gap to the floor from the origin is roughly 0.19 m, which means that the overall distance from the floor to the tunnel centre is (1.89 −0.59/+2.3) m. From the tunnel geometry, this locations the highest of the tunnel at roughly 0.89 m depth from the floor.

The alerts arising from native options are used to create a definite web site mannequin. That is used to offer an estimate of the anticipated form of the gravity gradient sign over the positioning, for comparability with the inference output. These options embody the tunnel of curiosity, basements from close by buildings, partitions and a drain. They’re proven within the scale drawing of Fig. 3b.