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Superconducting differential accelerometers have been used to test Newton’s inverse square law and have been proposed for other sensitive experiments. These include searches for spin-mass coupling, detecting Earth’s gravitomagnetic field, and testing the Equivalence Principle. This article discusses the principle and performance of a sensitive three-axis gravity gradiometer. This device utilizes quantized flux and the Meissner effect to provide stable test mass levitation and signal coupling, and superconducting quantum interference devices to provide very low-noise amplification of the signals. The instrument comprises a total of nine superconducting accelerometers, six linear and three angular. This configuration permits simultaneous measurement of the diagonal components of the gravity gradient tensor as well as platform acceleration in all six degrees of freedom. An analysis of this instrument is presented along with experimental results. Methods to correct for various motion-induced errors are demonstrated. Other error sources are also discussed. The resulting performance of the superconducting gravity gradiometer is 2×10<sup>-11</sup> s <sup>-2</sup> Hz <sup>-1/2</sup>. © 2002 American Institute of Physics.

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... In this section, we describe the relevant mechanisms by which a gravity sensor can couple to gravity perturbations, and give an overview of the most widely used measurement schemes: the (relative) gravimeter (Crossley et al. 2013;Zhou et al. 2011), the gravity gradiometer (Moody et al. 2002;Ando et al. 2010;McManus et al. 2017;Canuel et al. 2018), and the gravity strainmeter, i.e., the large-scale GW detectors Virgo (Acernese et al. (Virgo Collaboration) . Strictly speaking, none of the sensors only responds to a single field quantity (such as changes in gravity acceleration or gravity strain), but there is always a dominant response mechanism in each case, which justifies to give the sensor a specific name. ...

... In the following, we only refer to the stationary design. A gravity gradiometer measures the relative acceleration between two test masses each responding to fluctuations of the gravity field (Jekeli 2014;Moody et al. 2002). The test masses have to be located close to each other so that the approximation in Eq. ...

... If the reference is perfectly stiff, and if we assume as before that there are no cross-couplings between degrees of freedom and the response is linear, then the subtraction of the two gravity channels cancels all of the seismic noise, leaving only the instrumental noise and the differential gravity signal given by the second line of Eq. (3). Even in real setups, the reduction of seismic noise can be many orders of magnitude since the two spheres are close to each other, and the two readouts pick up (almost) the same seismic noise (Moody et al. 2002). This does not mean though that gradiometers are necessarily more sensitive instruments to monitor gravity fields. ...

Terrestrial gravity fluctuations are a target of scientific studies in a variety of fields within geophysics and fundamental-physics experiments involving gravity such as the observation of gravitational waves. In geophysics, these fluctuations are typically considered as signal that carries information about processes such as fault ruptures and atmospheric density perturbations. In fundamental-physics experiments, it appears as environmental noise, which needs to be avoided or mitigated. This article reviews the current state-of-the-art of modeling high-frequency terrestrial gravity fluctuations and of gravity-noise mitigation strategies. It hereby focuses on frequencies above about 50 mHz, which allows us to simplify models of atmospheric gravity perturbations (beyond Brunt–Väisälä regime) and it guarantees as well that gravitational forces on elastic media can be treated as perturbation. Extensive studies have been carried out over the past two decades to model contributions from seismic and atmospheric fields especially by the gravitational-wave community. While terrestrial gravity fluctuations above 50 mHz have not been observed conclusively yet, sensitivity of instruments for geophysical observations and of gravitational-wave detectors is improving, and we can expect first observations in the coming years. The next challenges include the design of gravity-noise mitigation systems to be implemented in current gravitational-wave detectors, and further improvement of models for future gravitational-wave detectors where terrestrial gravity noise will play a more important role. Also, many aspects of the recent proposition to use a new generation of gravity sensors to improve real-time earthquake early-warning systems still require detailed analyses.

... In full-maglev vertical superconducting gravity instruments (VSGI) [1,3], non-mechanical connection exists between the superconducting test mass and the base. In order to keep the test mass levitated stably, side-wall superconducting-current-carrying-coils [2] must be used to provide the needed stiffness to suppress the motion of the test mass in the non-sensitive axis, thus reducing the cross-coupling noise [5,6]. Due to the limit of the critical magnetic field of superconductor, the current in the side-wall coils cannot be too large, so the stiffness provided by these coils is also limited. ...

... The working principle of the VSGI has been described in detail in Ref. [3][4][5][6][7]. In the sensitive axis, planar coils are used to levitate the test mass to construct the magnetic spring-oscillator structure, the displacement of the test mass is detected by the superconducting circuits and superconducting quantum interference device (SQUID) [16][17][18]. ...

... And the control model of the one-degree-of-freedom system is shown in FIG. 3 ωn and ξ are the natural frequency and the damping ratio of the spring oscillator. ωn is determined by the superconducting circuits and can be calculated according to the method in Ref. [5,20]. For translation and rotation mode, the natural frequencies are ...

Non-sensitive axis feedback control is crucial for cross-coupling noise suppression in the application of full-maglev vertical superconducting gravity instruments. This paper introduces the non-sensitive axis feedback control of the test mass in a home-made full-maglev vertical superconducting accelerometer. In the feedback system, special superconducting circuits are designed to decouple and detect the multi-degrees-of-freedom motions of the test mass. Then the decoupled motion signals are dealt with by the PID controller and fed back to the side-wall coils to control the test mass. In our test, the test mass is controlled successfully and the displacement is reduced by about one order of magnitude in the laboratory. Accordingly, the noise level of the vertical superconducting accelerometer in the sensitive axis is also reduced.

... The rotational motion has been utilized for various gravity gradiometers [16][17][18][19]. These instruments have direct analogy with GW detectors. ...

... For this current to be detected, the signal/noise ratio should be greater than one. Thus (19) should be larger than the noise ( J noise ≈ 10 −49 A at currently chosen parameters). That means that the minimally detectable value of h GW for these parameters is ...

... As follows from the previous section, this could be achieved at moderate cooling down to T = 1.2 K. We will again choose the parameters: the loop size a = 10 m, the wire diameter d = 1 mm, and B = 10 T. Then from (19) we have for the current: ...

Following the initial success of LIGO, new advances in gravitational wave (GW) detector systems are planned to reach fruition during the next decades with sensitivity up to 10 −25 Hz −1/2 at 100 Hz frequency range both in Europe (Einstein Telescope) and in the USA (Cosmic Explorer). These systems are interferometric and large (up to 40 km). Here we explore grounds of designing more compact orbital detectors of GW radiation with similar and even higher sensitivity at similar frequency range. These detectors are not interferometric and use superconducting Cooper pairs in magnetic field as transducers of GW-induced mechanical motion into electric current. They can be oriented relative to the source of GW, maximizing signal output and determining the direction of the source. The main idea of this design exploits the fact that an incident GW shifts infinitesimally the orientation of a 3D-superconducting system relative to a static magnetic field. As a result, a superconducting current is generated by the GW action. To utilize the total energy transferable from the GW to the detector one should involve multiple loops. Our modeling based on time-dependent Ginzburg-Landau equations and finite element approach revealed that when the stack of single loops is arranged vertically at a loop separation comparable to the thickness of loop's wire, each loop output is independent: the mutual inductance effects are negligible. These outputs can be added up for converting the total available GW-induced mechanical energy into an electric signal. The suggested arrangement of superconducting signal sources facilitates simple and very effective rejection of common mode. We tested this arrangement in laboratory with the normal multiturn coils and reached the theoretical Johnson-Nyquist noise limit. The same limit, if achieved with superconducting elements of the proposed size (loop area 100 m 2) will deliver phenomenal GW strain sensitivity. To avoid friction and related Coulomb forces counteracting the gravity, the design requires orbital placement. Although we focus our consideration on 100 Hz range of GW frequency, the detector is non-resonant and can be optimized for both higher and lower frequency ranges. For the detector operation, only superconducting elements which constitute much smaller volume than the detector itself should be cryogenically cooled. We consider in detail a design which is cooled down to 1.2 K and delivers 10 −25 Hz −1/2 strain sensitivity.

... To access the subhertz region, which is very rich in GW sources (Harms et al., 2013), three concepts for 0.1-Hz gravity gradiometers are currently under development: superconducting gradiometers (SGG) (Moody et al., 2002;Paik et al., 2016), torsion-bar antennas (e.g., TOBA; Ando et al., 2010, or TorPeDO;McManus et al., 2017), and atom-interferometric gradiometers (Geiger, 2017;Hohensee et al., 2011). In the following, we will refer to these detectors as GG10, that is, gravity gradiometers with high-sensitivity for signals with periods around 10 s. ...

... The most sensitive instrument so far is the superconducting gradiometer with a strain sensitivity of about 10 −10 Hz −1∕2 at 0.1 Hz (Moody et al., 2002). However, extensive gain in experience with these technologies has led to defining more ambitious strain-sensitivity targets: 10 −15 Hz −1∕2 at 0.1 Hz (Ando et al., 2010;Hogan et al., 2011;Hohensee et al., 2011). ...

... The high-frequency noise spectra differ more strongly. While it is expected that instrumental noise of the superconducting gradiometer keeps falling above 0.1 Hz (in units of gravity strain; Moody et al., 2002), torsion-bar antennas have a flat noise spectrum above 0.1 Hz (Shoda et al., 2014), and atom-interferometric gradiometers reach their best sensitivity only within small frequency bands (Cheinet et al., 2008). ...

Since gravity propagates at the speed of light, gravity perturbations induced by earthquake deformation have the potential to enable faster alerts than the current earthquake early warning systems based on seismic waves. Additionally, for large earthquakes (Mw > 8), gravity signals may allow for a more reliable magnitude estimation than seismic-based methods. Prompt elastogravity signals induced by earthquakes of magnitude larger than 7.9 have been previously detected with seismic arrays and superconducting gravimeters. For smaller earthquakes, down to Mw ≃ 7, it has been proposed that detection should be based on measurements of the gradient of the gravitational field, in order to mitigate seismic vibration noise and to avoid the canceling effect of the ground motions induced by gravity signals. Here we simulate the five independent components of the gravity gradient signals induced by earthquakes of different focal mechanisms. We study their spatial amplitude distribution to determine what kind of detectors is preferred (which components of the gravity gradient are more informative), how detectors should be arranged, and how earthquake source parameters can be estimated. The results show that early earthquake detections, within 10 seconds of the rupture onset, using only the horizontal gravity strain components are achievable up to about 140 km distance from the epicenter. Depending on the earthquake focal mechanism and on the detector location, additional measurement of the vertical gravity strain components can enhance the detectable range by 10–20 km. These results are essential for the design of gravity-based earthquake early warning systems.

... To access the subhertz region, which is very rich in GW sources ( Harms et al., 2013), three concepts for 0.1-Hz gravity gradiometers are currently under development: superconducting gradiometers (SGG) ( Moody et al., 2002;Paik et al., 2016), torsion-bar antennas (e.g., TOBA; Ando et al., 2010, or TorPeDO;McManus et al., 2017), and atom-interferometric gradiometers (Geiger, 2017;Hohensee et al., 2011). In the following, we will refer to these detectors as GG10, that is, gravity gradiometers with high-sensitivity for signals with periods around 10 s. ...

... The most sensitive instrument so far is the superconducting gradiometer with a strain sensitivity of about 10 −10 Hz −1∕2 at 0.1 Hz ( Moody et al., 2002). However, extensive gain in experience with these technologies has led to defining more ambitious strain-sensitivity targets: 10 −15 Hz −1∕2 at 0.1 Hz ( Ando et al., 2010;Hogan et al., 2011;Hohensee et al., 2011). ...

... The high-frequency noise spectra dif- fer more strongly. While it is expected that instrumental noise of the superconducting gradiometer keeps falling above 0.1 Hz (in units of gravity strain; Moody et al., 2002), torsion-bar antennas have a flat noise spec- trum above 0.1 Hz ( Shoda et al., 2014), and atom-interferometric gradiometers reach their best sensitivity only within small frequency bands (Cheinet et al., 2008). ...

Recent studies reported the observation of prompt elastogravity signals during the 2011 M9.1 Tohoku earthquake, recorded with broadband seismometers and gravimeter between the rupture onset and the arrival of the seismic waves. Here we show that to extend the range of magnitudes over which the gravity perturbations can be observed and reduce the time needed for their detection, high-precision gravity strainmeters under development could be used, such as torsion bars, superconducting gradiometers or strainmeters based on atom interferometers. These instruments measure the differential gravitational acceleration between two seismically isolated test masses, and are initially designed to observe gravitational waves around 0.1 Hz. Our analysis involves simulations of the expected gravity strain signals generated by fault rupture, based on an analytical model of gravity perturbations in a homogeneous half-space. We show that future gravity strainmeters should be able to detect prompt gravity perturbations induced by earthquakes larger than M7, up to 1000 km from the earthquake centroid within P-waves travel time and up to 120 km within the first 10 seconds of rupture onset, provided a sensitivity in gravity strain of 10^{-15}/rtHz at 0.1 Hz can be achieved. Our results further suggest that, in comparison to conventional P-wave-based earthquake-early warning systems (EEWS), a gravity-based EEWS could perform faster detections of large off-shore subduction earthquakes (at least larger than M7.3). Gravity strainmeters could also perform earlier magnitude estimates, within the duration of the fault rupture, and therefore complement current tsunami warning systems.

... It should be noted that we only considered the airborne SGG in the following. A number of pioneering works on SGG have been carried out in Paik's group in Maryland [8][9][10][11][12][13]. These works have distinctly shown its advantages as a next-generation gradiometer. ...

... Second, the SGG uses superconducting interference devices to detect the displacements of oscillators, resulting in extremely low instrumental noise. The above two features jointly enable the SGG to be the most unique instrument with a fundamental noise floor lower than 1E/√Hz (1E = 10 −9 s −2 ) in the laboratory today [8]. ...

... Unfortunately, it is quite difficult to apply the rotation strategy to the SGG due to its incompatibility with the cryogenic environment. Instead, the SGG adopts an off-line method to tune common-mode balance [8]. This is to say, the instrument should be finely tuned to the optimum state before measurement. ...

Tuning the stiffness balance is crucial to full-band common-mode rejection for a superconducting gravity gradiometer (SGG). A reliable method to do so has been proposed and experimentally tested. In the tuning scheme, the frequency response functions of the displacement of individual test mass upon common-mode accelerations were measured and thus determined a characteristic frequency for each test mass. A reduced difference in characteristic frequencies between the two test masses was utilized as the criterion for an effective tuning. Since the measurement of the characteristic frequencies does not depend on the scale factors of displacement detection, stiffness tuning can be done independently. We have tested this new method on a single-component SGG and obtained a reduction of two orders of magnitude in stiffness mismatch.

... Another powerful instrument for measuring weak gravity forces is the superconducting gravity gradiometer (SGG). By the early 1990s, the SGG developed at the University of Maryland (UM) achieved a differential acceleration sensitivity of 4 × 10 −12 m s −2 Hz −1/2 [10], which, to this day surpasses the sensitivity of the other Earth-bound gravity gradiometers by three orders of magnitude. By replacing the mechanically suspended test masses (TMs) with magnetically levitated ones, the new SGG under development at UM will improve the sensitivity by another two orders of magnitude and will be free from the irregularities of the mechanical suspension [11]. ...

... The two levitated TMs are coupled by a superconducting circuit to form an in-line-component SGG along the x axis. The principle and design of various models of the SGG have been published in detail [10,11]. Figure 6 is a schematic circuit diagram of SGG. ...

... Due to the temperature sensitivity of the magnetic field penetration depth of a superconductor, the temper ature fluctuations of the detector produce an error [10,11]. Our sensing circuit includes a feature that compensates the temperature sensitivity of the SGG. ...

Newton's gravitational constant G is the least known fundamental constant of nature, partly because the gravity signal produced in the laboratory is extremely weak and difficult to measure accurately. There also seem to be unknown systematics in many G measurements. Since Cavendish made the first measurement of G with a torsion balance over two hundred years ago, torsion balances have been used in many G experiments, but uncorrected anelasticity of torsion fibers make the results questionable. We present a new method of G measurement by using a superconducting gravity gradiometer constructed with levitated test masses, which is free from the irregularities of mechanical suspension. The detector is rotated to null the gravity field from the source mass by centrifugal acceleration, forming an artificial planetary system. We show that this experiment can potentially measure G with accuracy better than 10 ppm.

... So, SOGRO uses low-temperature superconducting magnetic levitation to suspend the test masses [8,9]. As a property of superconductor, there is no resistance when niobium is below its critical temperature of 9.25 K [10], so the current in a superconducting loop will last forever [11], and so does the magnetic field it generates, which makes the suspension of test masses very stable and quiet [12]. Low-temperature superconducting magnetic levitation has an additional advantage that it generates very low stiffness along the directions perpendicular to the levitation direction [13]. ...

... We find a way to make negative stiffness in the superconducting magnetic levitation system which can reduce the frequency of the levitated test mass to the needed level without increasing any hardware and complicity to the system. In addition, because the entire platform of SOGRO is suspended with the suspension cable [1,2] and SOGRO works at low temperatures, the tilt is predicted to be very stable [12]. So we have not considered here the possible impact of slow drifts in the tilt. ...

A new terrestrial gravitational wave detector, Superconducting Omni-directional Gravitational Radiation Observatory (SOGRO), has been proposed in 2016 and seen as a competitive candidate of middle-frequency gravitational wave detector. In this detector, there are three pairs of 5-ton low-temperature superconducting test masses separated by 30–50 m and magnetically levitated by superconducting coils carrying persistent currents. To get a sensitivity of 10− 19 − 10− 20Hz− 1/2, the levitation frequency of the test masses need to be as low as 0.01 Hz. Because of the machining and assembling errors, the levitation coil will tilt with respect to gravity and deviate from the center of the levitation coil. Alignment coils can cancel the tilt but bring extra stiffness to the test mass that enlarges the frequency. We numerically have studied a downscaled prototype of the superconducting magnetic levitation of SOGRO and find a possible design of the superconducting levitation system, which can reduce the levitation frequency to less than 0.01 Hz without increasing any hardware and complexity of the system. This study will benefit the development of SOGRO and other superconducting levitation devices.

... Each of the current leads that enters the Dewar was connected to a highquality Pi feed-through a filter, providing attenuation above 40 kHz. In addition to the Pi filters mounted at room temperature, the leads were also filtered by small capacitors just before they enter the vacuum, and finally filtered by the second set of Pi filters before they were connected to the circuits [9,10]. By the three-step filtering, high-frequency electromagnetic interference was maintained at a very low level, and the stability of the SQUID was improved effectively. ...

... From equation (10) one can find that the larger LB is, the smaller the irms is. Besides insuring the properly working of the SGI, LB can still suppress the noise introduced by the metal layer. ...

... It has been proposed for studying various fundamental physics such as test of the wave function collapse models [2][3][4][5][6][7] and investigation of new physics [8][9][10][11][12][13][14]. For metrology study, it has also been applied in the force detection [15][16][17], the inertia sensing [18][19][20][21][22][23], the thermometry [24,25], and the magnetometry [26,27]. According to different applications, the sizes of the levitated mechanical oscillators range from 26 nm using room temperature optical trap [28] to centimeter level using a superconducting levitation with mass greater than 1 kg [19]. ...

... For metrology study, it has also been applied in the force detection [15][16][17], the inertia sensing [18][19][20][21][22][23], the thermometry [24,25], and the magnetometry [26,27]. According to different applications, the sizes of the levitated mechanical oscillators range from 26 nm using room temperature optical trap [28] to centimeter level using a superconducting levitation with mass greater than 1 kg [19]. ...

Levitated oscillators with millimeter or sub-millimeter size are particularly attractive due to their potential role in studying various fundamental problems and practical applications. One of the crucial issues towards these goals is to achieve efficient measurements of oscillator motion, while this remains a challenge. Here we theoretically propose a lens-free optical detection scheme, which can be used to detect the motion of a millimeter or sub-millimeter levitated oscillator with a measurement efficiency close to the standard quantum limit with a modest optical power. We demonstrate experimentally this scheme on a 0.5 mm diameter micro-sphere that is diamagnetically levitated under high vacuum and room temperature, and the thermal motion is detected with high precision. Based on this system, an estimated acceleration sensitivity of $9.7 \times 10^{-10}\rm g/\sqrt{Hz}$ is achieved, which is more than one order improvement over the best value reported by the levitated mechanical system. Due to the stability of the system, the minimum resolved acceleration of $3.5\times 10^{-12}\rm g$ is reached with measurement times of $10^5$ s. This result is expected to have potential applications in the study of exotic interactions in the millimeter or sub-millimeter range and the realization of compact gravimeter and accelerometer.

... The gravitational gradient excitation produced by a static test mass is a DC component and cannot satisfy the requirements of testing. In 2002, O. J. Paik and M. Vol Moody tested the performance of a three-axis superconducting gravity gradiometer with a 95 kg sphere mass rotating around it 17 . In this study, we have established an analytical model of the RAGG's measurements when a test mass is rotating about the RAGG; and we propose the simultaneous use of multiple test masses rotating about a RAGG at different angular velocities to produce the multifrequency gravitational gradient excitations that are expected in practical applications. ...

... From Eq.(17), the bandwidths of x 1 (t) and x 2 (t) are located in the ranges [0, pω 01 ] Hz and [0, pω 02 ] Hz, respectively. From Eq.(19), we can see that after QAM, the frequency components of the modulated carrier wave in [ω c − pω 01 , ω c + pω 01 ] and [ω c − pω 02 , ω c + pω 02 ] come from x ...

A moving-base rotating accelerometer gravity gradiometer (RAGG) is an instrument for measuring gravitational gradient signals produced by geological bodies with a certain signal bandwidth. Development and improvement of RAGG requires that they be subjected to testing and calibration; however, the zero-frequency gravitational gradient signals produced by static test masses are not suitable for this purpose. We propose a method in which multiple test masses simultaneously rotating about a RAGG at different angular velocities and in different circular orbits produce the multifrequency gravitational gradient excitation required for testing or calibrating the RAGG. We also present a gravitational gradient extraction method that combines a fore-end circuit design, a multirate filter technique, and a quadrature amplitude modulation demodulation technique. We describe in detail the procedures for gravitational gradient extraction. Multifrequency gravitational gradient excitations are applied to evaluate this extraction method. A RAGG physical simulation system substitutes for an actual RAGG in a multifrequency gravitational gradient extraction experiment. The extracted multifrequency gravitational gradient signal is consistent with theoretical predictions. The gravitational gradient extraction error approximates the noise of the RAGG physical simulation system. These experimental results suggest that the proposed gravitational gradient extraction method is feasible. The research presented in this paper is of great significance for engineering applications.

... 7 Furthermore, many new gravity gradiometers with performances better than 1 Eo/ √ Hz, such as ARkex exploration gravity gradiometer (ARkex EGG), Stanford University atomic interferometer gradiometer (Stanford AI), Gedex high density airborne gravity gradiometer (HD-AGG), and University of Western Australia's orthogonal quadrupole responder (UWA OQR), are under development. [8][9][10][11][12][13] It is known that a rotating accelerometer gravity gradiometer (RAGG) is extremely sensitive to its operating environment, and so environmental parameters such as temperature, pressure, and humidity must be strictly controlled. 14,15 Uncontrolled shutdowns will cause loss of temperature control and deactivate the electromagnetic fields constraining RAGG accelerometers, which will change the scale factors of a RAGG; therefore, at each restart, a RAGG requires recalibration. ...

... According to the relation between the gravitational gradient vector and the gravitational gradient tensor, by substituting Eqs. (11) and (12) into Eq. (13), we obtain the following relationship between the self-gradients and Γ P0 S 4 : ...

The purpose of this study is to calibrate scale factors and equivalent zero biases of a rotating accelerometer gravity gradiometer (RAGG). We calibrate scale factors by determining the relationship between the centrifugal gradient excitation and RAGG response. Compared with calibration by changing the gravitational gradient excitation, this method does not need test masses and is easier to implement. The equivalent zero biases are superpositions of self-gradients and the intrinsic zero biases of the RAGG. A self-gradient is the gravitational gradient produced by surrounding masses, and it correlates well with the RAGG attitude angle. We propose a self-gradient model that includes self-gradients and the intrinsic zero biases of the RAGG. The self-gradient model is a function of the RAGG attitude, and it includes parameters related to surrounding masses. The calibration of equivalent zero biases determines the parameters of the self-gradient model. We provide detailed procedures and mathematical formulations for calibrating scale factors and parameters in the self-gradient model. A RAGG physical simulation system substitutes for the actual RAGG in the calibration and validation experiments. Four point masses simulate four types of surrounding masses producing self-gradients. Validation experiments show that the self-gradients predicted by the self-gradient model are consistent with those from the outputs of the RAGG physical simulation system, suggesting that the presented calibration method is valid.

... Researchers have successfully suppressed cross-coupling noise in this manner [13,14]. However, in FM-VSGIs, which are limited by the critical current of the superconducting coils, because the stiffness cannot be as high as that of the mechanically connected one, the displacements of the test mass in the non-sensitive axes are much larger, especially in a dynamic environment [10]. In this situation, the attitude of the test mass changes continually with a large margin. ...

... In the superconducting gravity instruments developed by the University of Maryland and the University of Western Australia [10][11][12], the test mass is mechanically connected to the base. Thus, significant stiffness is provided in the non-sensitive axes, so the displacements of the test mass in the non-sensitive axes are small. ...

For full-maglev vertical superconducting gravity instruments, displacement control in the non-sensitive axis is a key technique to suppress cross-coupling noise in a dynamic environment. Motion decoupling of the test mass is crucial for the control design. In practice, when levitated, the test mass is always in tilt, and unknown parameters will be introduced to the scale factors of displacement detection, which makes motion decoupling work extremely difficult. This paper proposes a method for decoupling the translation and rotation of the test mass in the non-sensitive axis for full-maglev vertical superconducting gravity instruments. In the method, superconducting circuits at low temperature and adjustable gain amplifiers at room temperature are combined to measure the difference between the scale factors caused by the tilt of the test mass. With the measured difference of the scale factors, the translation and rotation are decoupled according to the theoretical model. This method was verified with a test of a home-made full-maglev vertical superconducting accelerometer in which the translation and rotation were decoupled.

... Takes for example the vertical gyro drift: this has the characteristic of increasing over time (see Fig.3 -red line) with slight oscillations. The application of the Kalman's filter allows selective filtering of the error on the output quantity of the system [73]- [78]. From the figure above, he sees that after a few minutes of measurement, the filter operates a damping action that tends asymptotically to decrease the error over time. ...

... At present, the superconducting gravity meters can reach a sensitivity of 1 nGal [22] and superconducting gravity gradiometers can have a sensitivity of 0.02E in the lab [23]- [24]. Considering these developments, we set a noise level of 0.02 Gal for gravity field and 0.02E for gravity gradient tensor signals. ...

We present a novel algorithm to locate multiple underwater objects in real time using gravity field vector and gravity gradient tensor signals. This algorithm formulates the task of localization of multiple underwater objects into a regularized non-linear problem, which is solved with the standard Levenberg-Marquardt algorithm. The regularization parameters are estimated by cross-validation. The initial coordinates and masses of these underwater objects are automatically determined by solving a single-object localization problem. A synthetic navigation model with two underwater objects was adopted to validate the proposed algorithm. The results show that it has good stability and anti-noise ability for multiple underwater objects localizations.

... In the 1960s, airborne gravimetry was developed. There have been many kinds of gravity gradiometer instruments (GGIs) or gravity gradiometers since the 1970s, such as the rotating gravity gradiometer, rotating accelerometer gravity gradiometer, Cold-Atom interferometric gravity gradiometer, superconducting gravity gradiometer and so on [7][8][9][10]. Among all of them, the rotating accelerometer gravity gradiometer is the only one that has been used on an airborne and shipborne platform and has been put into commercial operation successfully. ...

In the process of airborne gravity gradiometry for the full-tensor airborne gravity gradiometer (FTAGG), the attitude of the carrier and the fuel mass will seriously affect the accuracy of gravity gradiometry. A self-gradient is the gravity gradient produced by the surrounding masses, and the surrounding masses include distribution mass for the carrier mass and fuel mass. In this paper, in order to improve the accuracy of airborne gravity gradiometry, a self-gradient compensation model is proposed for FTAGG. The self-gradient compensation model is a fuction of attitude for carrier and time, and it includes parameters ralated to the distribution mass for the carrier. The influence of carrier attitude and fuel mass on the self-gradient are simulated and analyzed. Simulation shows that the self-gradient tensor element Γ x x , Γ x y , Γ x z , Γ y z and Γ z z are greatly affected by the middle part of the carrier, and the self-gradient tensor element Γ y z is affected by the carrier’s fuel mass in three attitudes. Further simulation experiments show that the presented self-gradient compensation method is valid, and the error of the self-gradient compensation is within 0.1 Eu. Furthermore, this method can provide an important reference for improving the accuracy of aviation gravity gradiometry.

... Many gravity gradient measurement methods based on different principles have been proposed. For example, Moody, M. V. measured the diagonal components of the gravity gradient tensor using superconducting technology [24]. McGuirk, J. M. improved an absolute-gravity gradiometer by light-pulse atom interference techniques [25]. ...

Gravity gradient plays an important role in many fields of science, and many methods are used to achieve the measurement of it. To improve measurement accuracy, various error analyses have been conducted in previous studies about positioning and orientation errors and system noise, among others. However, knowledge on the influence of omission errors from the theoretical models of gravity gradient measurement is limited. In this study, we investigated omission errors in gravity gradient measurement, which was accomplished with the principle of differential acceleration. First, we determined the source of the omission errors to be the omission of high-order terms. Second, we calculated these terms on the basis of the Earth Gravitational Model 2008. Specifically, the expression of the partial derivative of the high order for the gravity potential in the spherical coordinates and the recursive equations for the high-order partial derivatives of the Legendre function were derived. Moreover, we transformed these high-order terms from the spherical coordinate system to the local north-oriented frame. The analysis led to three findings. First, a positive correlation was found between the omission errors and the distance between two measuring points. Second, the influences of the omission errors varied across different regions. Third, Г zz was the least affected by the omission errors among the components Г xz, Г yz and Г zz. In conclusion, our study demonstrates that omission errors affect gravity gradient measurement.

... This can be done in principle by measuring the gradient of the gravity perturbation between two or more seismically isolated test masses. Relevant technologies are being developed in the context of low-frequency gravitational-wave detectors, with concepts such as torsion bars antennas (26,27), superconducting gravity gradiometers (28,29), and atom interferometers (30,31). In the first two concepts, the test masses are linked to the ground by a common frame; the displacements driven by the seismic noise and affecting the gravity measurement can be made very similar for the two masses, and they are hence rejected by the differential measurement. ...

Lors d'un tremblement de terre, la rupture sismique et la propagation d'ondes qui en résulte perturbent le champ de densité terrestre. Cette redistribution transitoire de masses conduit à un ré-équilibrage global et instantané du champ de gravité terrestre, mesurable avant l'arrivée du front d'ondes sismiques. Un instrument déployé au sol enregistre cette perturbation du champ gravitationnel, ainsi qu'une accélération inertielle induite par gravité : nous appelons le signal total résultant, perturbation élasto-gravitationnelle. Nous modélisons la réponse élasto-gravitationnelle complète d'un instrument à une rupture sismique à travers le formalisme des modes propres. Ce formalisme permet la simulation de perturbations dans des modèles sphériques auto-gravitants, homogènes ou radialement stratifiés. Le tremblement de terre de Tohoku-oki (magnitude 9.1, mars 2011, Japon) a développé l'une des ruptures les plus importantes de ces dernières décennies, à proximité de réseaux intégrant une instrumentation de pointe. Nous présentons la détection et la modélisation précise de perturbations élasto-gravitationnelles induites par cette rupture. Un système de détection précoce de tremblements de terre permet de détecter l'occurence d'un séisme avant l'arrivée des ondes sismiques destructrices générées par la rupture, afin d'en limiter les dégâts humains et matériels. Les systèmes d'alerte actuels reposent sur la différence de temps de propagation entre ondes de compression P (peu destructrices) et ondes de cisaillement S (plus lentes, mais dangereuses). Associé au développement de capteurs du gradient du champ de gravité à haute précision, nous introduisons un système de détection précoce reposant sur la gravité.

... The rotational motion has been used in various gravity gradiometers [26][27][28][29] and suggested for GW detectors in the past by the Braginsky group [30,31], and by Sakharov [32]. Recently, the rotational design for GW detectors was revived by the TOBA group [33] and simultaneously and independently by our group [34]. ...

Following the initial success of LIGO, new advances in gravitational wave (GW) detector systems are planned to reach fruition during the next decades. These systems are interferometric and large. Here we suggest different, more compact detectors of GW radiation with competitive sensitivity. These nonresonant detectors are not interferometric. They use superconducting Cooper pairs in a magnetic field to transform mechanical motion induced by GW into detectable magnetic flux. The detectors can be oriented relative to the source of GW, so as to maximize the signal output and help determine the direction of nontransient sources. In this design an incident GW rotates infinitesimally a system of massive barbells and superconducting frames attached to them. This last rotation relative to a strong magnetic field generates a signal of superconducting currents. The suggested arrangement of superconducting signal sources facilitates rejection of noise due to stray electromagnetic fields. In addition to signal analysis, we provide estimates of mechanical noise of the detector, taking into account temperature and elastic properties of the loops and barbells. We analyze at which parameters of the system a competitive strain sensitivity could be achieved. We have tested the basic idea of the detector in the laboratory and reached the theoretical Johnson-Nyquist noise limit with multiturn coils of normal metal. Realization of full-blown superconducting detectors can serve as viable alternatives to interferometric devices.

... Vibration isolation at such low frequencies is extremely difficult. The CM rejection techniques demonstrated with SGGs [15], which produced a stable CM rejection level of three parts in 10 8 , will be applied to SOGRO and improved to one part in 10 9 . The highly polished, high-purity Nb surfaces will allow for exceptional stability of the balance and error rejection. ...

Detection of gravitational waves (GWs) from merging binary black holes (BHs) by Advanced LIGO has ushered in the new era of GW astronomy. Many conceivable sources such as intermediate-mass BH binaries and white dwarf binaries, as well as stellar-mass BH inspirals, would emit GWs below 10 Hz. It is highly desirable to open a new window for GW astronomy in the infrasound frequency band. A low-frequency tensor detector could be constructed by combining six magnetically levitated superconducting test masses. Such a detector would be equally sensitive to GWs coming from anywhere in the sky, and would be capable of resolving the source direction and wave polarization. I will present the design concept of a new terrestrial GW detector, named SOGRO, which could reach a strain sensitivity of 10−19-10−21 Hz−1/2 at 0.1-10 Hz. Seismic and Newtonian gravity noises are serious obstacles in constructing terrestrial GW detectors at frequencies below 10 Hz. I will explain how these noises are rejected in SOGRO. I will also report the progress made in designing the platform and modelling its thermal noise.

... Table 5 shows that the disturbing gravity gradient caused by the sea level anomaly significantly decreases with increasing depth. The vertical disturbing gravity gradient can still reach 3 E at 50 m below the mean sea level, and the maximum difference of the disturbing gravity gradient at different locations can reach 4 E. Therefore, based on the measurement accuracy of the gravity gradiometer 1 E [20][21][22][23][24][25], the local sea level anomaly will considerably affect the underwater gravity gradient measurements at the depth of 50 m below the mean sea level. ...

Considering the theoretical research needs of gravity gradient detection and navigation, this study uses the right rectangular prism method to calculate the disturbing gravity gradient from sea level anomalies in the range of 5° × 5° in the Kuroshio extension area of the western Pacific with large sea level anomalies. The disturbing gravity gradient is obtained in different directions within a depth of 50 m below the mean sea level based on the principle of the disturbing gravity gradient. The calculation results show that the sea level anomalies at local positions significantly impact the underwater gravity gradient measurements, with the maximum contribution exceeding 10 E and the maximum difference between different locations exceeding 20 E. The change of the sea level anomaly over time also significantly impacts the measurement of the underwater gravity gradient, with the maximum change value exceeding 20 E. The impact will have a corresponding change with the seasonal change of the sea level anomaly. Therefore, the underwater carrier needs to consider the disturbing gravity gradient caused by sea level anomalies when using the gravity gradient for underwater detection and navigation.

... The readout is done using a SQUID. Such systems already form part of the most sensitive gravitygradient sensors (Moody et al. 2002;Griggs et al. 2017), and similar but simpler systems can, for example, be found in superconducting gravimeters of the Global Geodynamics Project (Goodkind 1999;Crossley & Hinderer 2010). It is expected that relatively high Q-values of order 10 6 or potentially even higher can be achieved even with low resonance frequencies, which gives the cryomagnetic concept an advantage concerning thermal noise. ...

Monitoring of vibrational eigenmodes of an elastic body excited by gravitational waves was one of the first concepts proposed for the detection of gravitational waves. At laboratory scale, these experiments became known as resonant bar detectors first developed by Joseph Weber in the 1960s. Due to the dimensions of these bars, the targeted signal frequencies were in the kHz range. Weber also pointed out that monitoring of vibrations of Earth or the Moon could reveal gravitational waves in the mHz band. His Lunar Surface Gravimeter experiment deployed on the Moon by the Apollo 17 crew had a technical failure, which greatly reduced the science scope of the experiment. In this article, we revisit the idea and propose a Lunar Gravitational-Wave Antenna (LGWA). We find that LGWA could become an important partner observatory for joint observations with the space-borne, laser-interferometric detector LISA and at the same time contribute an independent science case due to LGWA's unique features. Technical challenges need to be overcome for the deployment of the experiment, and development of inertial vibration sensor technology lays out a future path for this exciting detector concept. © 2021. The Author(s). Published by the American Astronomical Society.

... In parallel, efforts are ongoing to study the feasibility of sub-Hz Earth-based gravitational-wave detectors [23]. Three main concept are explored: torsion bar antennas, superconducting gravity gradiometers [24][25][26] and laser atom interferometers [27][28][29]. ...

Density changes in the atmosphere produce a fluctuating gravity field that affect gravity strainmeters or gravity gradiometers used for the detection of gravitational-waves and for geophysical applications. This work addresses the impact of the atmospheric local gravity noise on such detectors, extending previous analyses. In particular we present the effect introduced by the building housing the detectors, and we analyze local gravity-noise suppression by constructing the detector underground. We present also new sound spectra and correlations measurements. The results obtained are important for the design of future gravitational-wave detectors and gravity gradiometers used to detect prompt gravity perturbations from earthquakes.

... The readout is done using a SQUID. Such systems already form part of the most sensitive gravity-gradient sensors [97,98], and similar, but simpler systems can for example be found in superconducting gravimeters of the Global Geodynamics Project [99,100]. It is expected that relatively high Q-values of order 10 6 or potentially even higher can be achieved even with low resonance frequencies, which gives the cryomagnetic concept an advantage concerning thermal noise. ...

Monitoring of vibrational eigenmodes of an elastic body excited by gravitational waves was one of the first concepts proposed for the detection of gravitational waves. At laboratory scale, these experiments became known as resonant-bar detectors first developed by Joseph Weber in the 1960s. Due to the dimensions of these bars, the targeted signal frequencies were in the kHz range. Weber also pointed out that monitoring of vibrations of Earth or Moon could reveal gravitational waves in the mHz band. His Lunar Surface Gravimeter experiment deployed on the Moon by the Apollo 17 crew had a technical failure rendering the data useless. In this article, we revisit the idea and propose a Lunar Gravitational-Wave Antenna (LGWA). We find that LGWA could become an important partner observatory for joint observations with the space-borne, laser-interferometric detector LISA, and at the same time contribute an independent science case due to LGWA's unique features. Technical challenges need to be overcome for the deployment of the experiment, and development of inertial vibration sensor technology lays out a future path for this exciting detector concept.

Laser interferometer gravitational-wave (GW) detectors are observing signals from merging black hole and neutron star binaries with a frequency window from 10Hz to several kHz. Future space-based laser interferometers will open a new window of 0.1mHz to 0.1Hz. In this paper, we discuss the possibility of constructing a terrestrial GW detector named Superconducting Omni-directional Gravitational Radiation Observatory (SOGRO), which can fill the missing frequency window, 0.1 to 10Hz, with astronomically interesting sensitivity. SOGRO measures all five tensor components of the spacetime metric, which results in uniform sensitivity for all-sky directions and enables identification of the source direction and wave polarization with a single detector. Seismic and Newtonian gravity noise pose the greatest challenges for constructing ground-based detectors below 10Hz. SOGRO utilizes enhanced mechanical and electrical stabilities of materials at cryogenic temperatures to reject common-mode seismic noise to a very high degree. Further, its full-tensor characteristic gives an advantage in the rejection of the Newtonian noise over conventional detectors.

In this paper, a newly designed accelerometer based on a SQUID detection technology and the experimental results are presented. The levitated proof mass was manufactured in the shape that combines a disk and a cylinder on the basis of an earlier patent by the authors. The advantage of this shape is, given that the bottom part is cylindrical, even if the proof mass is mounted mechanically tilted, it can be moved to the center depending on the magnetic flux. The inside of the accelerometer is unobservable after the sealing of the superconducting housing; therefore, the initial set current values on both the solenoid and flat spiral coil are crucial. It was easily able to determine the levitation status at equilibrium position of the proof mass by investigating the inductance change according the persistent current for sensing and solenoid coils. At the levitation status, the movement of the proof mass caused by an external acceleration could be detected with a SQUID optimization. Accordingly, levitation experimental results and the flux noise spectra of the levitated proof mass are described.

Cette thèse porte sur la conception et la réalisation d’une nouvelle expérience d’interféromètre atomique au SYRTE. Elle permettra de réaliser des mesures ultrasensibles du gradient vertical de gravité. Cette expérience fonctionnera à terme en utilisant comme source des atomes ultra-froids, préparés sur une puce à atomes. Elle utilisera comme séparatrices des transitions multiphotoniques, obtenues par diffraction de Bragg d’ordre élevé. Le transport des atomes sera assuré par des réseaux optiques en mouvement. Une première partie du dispositif expérimental a été assemblée et son fonctionnement a été validé en réalisant un interféromètre dual. Cet interféromètre est réalisé sur deux ensembles d’atomes produits successivement à partir de la même source d’atomes froids, et interrogés par une même paire de faisceaux Raman. Une nouvelle méthode d’extraction de la phase différentielle a été démontrée expérimentalement. Elle repose sur l’exploitation des corrélations entre les mesures de phase des interféromètres et une estimation de la phase sismique fournie par la mesure annexe d’un capteur classique.

Newton's gravitational constant G is the least known fundamental constant of nature. Since Cavendish made the first measurement of G with a torsion balance over two hundred years ago, the best results of G have been obtained by using torsion balances. However, the uncorrected anelasticity of torsion fibers makes the results questionable. We present a new method of G measurement by using a superconducting gravity gradiometer constructed with levitated test masses, which is free from the irregularities of mechanical suspension. The superconducting gravity gradiometer is rotated to generate a centrifugal acceleration that nulls the gravity field of the source mass, forming an artificial planetary system. This experiment has a potential accuracy of G better than 10 ppm.

Due to absence of friction, a superconducting rotor can achieve a very high speed to obtain a large angular momentum for high-precision measurement of angular displacement and angular velocity. However, in practice, the superconducting rotor will do vibration and collide with the inner wall of the rotor cavity at a certain speed during startup, which will result in failure of the rotor' acceleration. In this paper, an improved starting strategy was proposed. This new method can avoid the peak of vibration amplitude by changing the bearing stiffness of the rotor quickly. Startup and shutdown process of the rotor was redesigned. Several experiments were carried out to verify its correctness. The results show that the superconducting rotor can be accelerated to the rated speed of 200 Hz (12000 r/min) successfully by the proposed method. Furthermore, start time of the rotor was greatly reduced to 20 minutes from more than one hour for the original method.

We demonstrate basic operations of a two-component superconducting gravity gradiometer (SGG) that is constructed with a pair of magnetically levitated test masses coupled to superconducting quantum-interference devices. A design that gives a potential sensitivity of 1.4×10−4 E Hz−1/2 (1 E≡10−9 s−2) in the frequency band of 1 to 50 mHz and better than 2×10−5 E Hz−1/2 between 0.1 and 1 mHz for a compact tensor SGG that fits within a 22-cm-diameter sphere. The SGG has the capability of rejecting the platform acceleration and jitter in all 6 degrees of freedom to one part in 109. Such an instrument has applications in precision tests of fundamental laws of physics, earthquake early warning, and gravity mapping of Earth and the planets.

As soon as an earthquake starts, the rupture and the propagation of seismic waves redistribute masses within the Earth. This mass redistribution generates in turn a long-range perturbation of the Earth gravitational field, which can be recorded before the arrival of the direct seismic waves. The recent first observations of such early signals motivate the use of the normal mode theory to model the elastogravity perturbations recorded by a ground-coupled seismometer or gravimeter. Complete modeling by normal mode summation is challenging due to the very large difference in amplitude between the prompt elastogravity signals and the direct P-wave signal. We overcome this problem by introducing a two-step simulation approach. The normal mode approach enables a fast computation of elastogravity signals in layered self-gravitating Earth models. The fast and accurate computation of gravity perturbations indicates instrument locations where signal detection may be achieved, and may prove useful in the implementation of a gravity-based earthquake early warning system.

Planetary gravity fields are measured by various means. The accuracy of the recovered gravitational potential model depends, of course, on the number and accuracy of the measurements, and in non-trivial ways upon the measurement configuration and orbit geometry. We derive and present simple analytic expressions which yield estimates of the resulting error spectra, with reasonable accuracy. Actual recovery of gravity models from the relevant data requires much more rigorous analysis, but these models are expected to be helpful in early mission planning activities. We present results of an application to Vesta, used as a validation exercise, and show possible future results for Mars and Venus.

With continuous advances in related technologies, precision tests of modern gravitational theories with orbiting gradiometers becomes feasible, which may naturally be incorporated into future satellite gravity missions. In this work, we derive, at the post-Newtonian level, the new secular gravity gradient signals from the non-dynamical Chern–Simons modified gravity for satellite gradiometry measurements, which may be exploited to improve the constraints on the mass scale \(M_{CS}\) or the corresponding length scale \({\dot{\theta }}\) of the theory with future missions. For orbiting superconducting gradiometers, a bound \(M_{CS}\ge 10^{-7}\ \mathrm{eV}\) and \({\dot{\theta }} \le 1\ \mathrm{m}\) could in principle be obtained, and for gradiometers with optical readout based on the similar technologies established in the LISA PathFinder mission, an even stronger bound \(M_{CS}\ge 10^{-6}\)–\(10^{-5}\ \mathrm{eV}\) and \({\dot{\theta }} \le 10^{-1}\)–\(10^{-2} \ \mathrm{m}\) might be expected.

This paper presents a specialized horizontal linear vibrator with small accompanying motions for precisely measuring the cross-coupling coefficient of superconducting gravity gradiometers(SGGs). The vibrator is suspended by leaf springs and actuated by superconducting coils(SCs) and permanent magnets(PMs). Innovative distribution of SCs and PMs can generate frictionless, small nonlinear and controllable driving forces, and it also can provide large force moments to cancel the accompanying angular motions. Besides, the vibrator can be incorporated into the liquid helium Dewar and work at a low temperature. The experimental results show that the accompanying motions related to the amplitude ratio of the accompanying vertical acceleration(az) and angular accelerations(αx and αy) to the actuated horizontal linear acceleration(ax) at 1 Hz are as small as 1.4 × 10⁻⁴, 4.7 × 10⁻⁴ rad/m and 2.8 × 10⁻⁴ rad/m, respectively. With this vibrator, the measurement resolution for cross-coupling coefficients can be estimated to be 1.8 × 10⁻⁸ rad.

Korea Research Institute of Standards and Science (KRISS) has started to design a Superconducting Gravimeter system based on a SQUID sensor. In this paper, we describe the design and fabrication of fundamental components of the superconducting circuit. Heat switches (HSWs) were fabricated using surface-mounted device (SMD) resistors and niobium (Nb) wires in a variety of ways to provide DC current to superconducting coils. The HSWs were tested regarding response time, wave form, and aging features at a cryogenic temperature (4.2 K) for 40 days. These results allowed us to optimize the HSW designs, superconducting housing, and superconducting circuits of the system. The system was designed as a drive controller to inject constant current into the superconducting coils and to control the operating times of the HSWs and the constant current. In addition, we also describe a suitable method for calculating the inductance of the closed superconducting loops.

State-of-the-art detectors are necessary to measure very tiny variations of gravity produced by spiraling neutron stars, merging black holes, moving tectonic plates. We are developing a superconducting gravity gradiometer and aim to achieve 0.1 mE Hz^{-1/2} in the frequency band of 0.1 mHz to 0.1 Hz. The superconducting test masses are levitated by a superconducting current-carrying monolayer pancake coil, which is one of the key components of the instrument. However, the nonlinear aspect of the magnetic field trapped between the test mass and the pancake coil imposes one of the main constraints to achieve that such low frequencies. In this paper, we investigated the causes of that nonlinearity by finite element method using COMSOL Multiphysics® simulation software. First, inductances were measured with an experimental setup where a gap spacing, created by a pancake coil and a niobium plate, could be adjustable. The inductances computed with a 2D axis-symmetric model satisfactorily agreed to the experimental data. Finally, we extensively studied several mechanisms for cancelling the nonlinearity of the inductance. A solenoid next to the pancake coil is the most effective and practical way to mitigate it. Furthermore, our approach can also be useful for those seeking a simple and effective model to study magnetostatic problems in a superconductor

Non-sensitive axis feedback control is crucial for cross-coupling noise suppression in the application of full-maglev vertical superconducting gravity instruments. This study proposes the non-sensitive axis feedback control of a test mass in a custom-made full-maglev vertical superconducting accelerometer. In the feedback system, special superconducting circuits are designed to decouple and detect multiple degree-of-freedom motions of the test mass. The decoupled motion signals are then passed through a PID controller and fed back to the side-wall coils to control the test mass. The displacement of the test mass is controlled within ±2 nm and is reduced by approximately one order of magnitude in a laboratory environment. Accordingly, the noise level of the vertical superconducting accelerometer in the sensitive axis is reduced by a factor of 4–6 @ 0.03–0.1 Hz.

Levitated oscillators of millimeter or submillimeter size are particularly attractive due to their potential role in studying various fundamental problems and practical applications. One of the crucial issues towards these goals is to achieve efficient measurements of oscillator motion, although this remains a challenge. Here we theoretically propose a lens-free optical detection scheme, which can be used to detect the motion of a millimeter or submillimeter levitated oscillator with a measurement efficiency close to the standard quantum limit with a modest optical power. We demonstrate experimentally this scheme on a 0.5-mm-diameter microsphere that is diamagnetically levitated under high vacuum and room temperature, and the thermal motion is detected with high precision. Based on this system, an estimated acceleration sensitivity of 9.7×10−10g/Hz is achieved, which is an improvement of more than 1 order of magnitude over the best value reported for a levitated mechanical system. Due to the stability of the system, the minimum resolved acceleration of 3.5×10−12g is reached with measurement times of 105 s. This result is expected to have potential applications in the study of exotic interactions in the millimeter or submillimeter range and the realization of compact gravimeters and accelerometers.

Broadband seismometers and gravitational wave detectors make use of mechanical resonators with a high quality factor to reduce Brownian noise. At low frequency, Brownian noise is ultimately dominated by internal friction in the suspension, which has a 1/f noise compared with the white noise arising from viscous dissipation. Internal friction is typically modeled as a frequency-dependent loss and can be challenging to measure reliably through experiment. In this work, we present the physics and experimental implementation of electrostatic frequency reduction (EFR) in a mechanical oscillator—a method to measure dissipation as a function of frequency. By applying a high voltage to two parallel capacitor plates, with the center plate being a suspended mass, an electrostatic force is created that acts as a negative stiffness mechanism to reduce the system’s resonance frequency. Through EFR, the loss angle can be measured as a function of frequency by measuring amplitude decay response curves for a range of applied voltages. We present experimental measurements of the loss angle for three metal helical extension springs in the nominal frequency range 0.7–2.9 Hz at 0.2 Hz intervals, demonstrating the possibility for fine adjustment of the resonance frequency for loss angle measurements. A quality factor proportional to the resonance frequency squared was measured, an indication that internal friction and other non-viscous dissipation elements, such as electrostatic damping, were the prominent loss mechanisms in our experiments. Finally, we consider the implications of Brownian noise arising from internal friction on a low 1/f noise seismometer.

A superconducting gravimeter based on the superconducting quantum interference device system is under development. As the main source of low-frequency noise, temperature fluctuations affect the resolution of superconducting gravimeters. In this study, a set of experimental devices was built to investigate the primary coupling processes of temperature fluctuations in superconducting gravimeters. Under the temperature modulation method, the effects of temperature fluctuations can be expressed as dΦ/dT = 342(2)Φ0/K, which, according to theoretical analysis, corresponds to a displacement change of (1.38 ± 0.04) × 10⁻⁷ m/K. Based on these results, the ambient temperature is controlled to within ±100 µK, and the equivalent effect of temperature fluctuations on our superconducting gravimeter is 0.5 μGal.

Since gravity propagates at the speed of light, gravity perturbations induced by earthquake deformation have the potential to enable faster alerts than the current earthquake early warning systems based on seismic waves. Additionally, for large earthquakes (M_w > 8), gravity signals may allow for a more reliable magnitude estimation than seismic-based methods. Prompt elastogravity signals induced by earthquakes of magnitude larger than 7.9 have been previously detected with seismic arrays and superconducting gravimeters. For smaller earthquakes, down to M_w ≃ 7, it has been proposed that detection should be based on measurements of the gradient of the gravitational field, in order to mitigate seismic vibration noise and to avoid the canceling effect of the ground motions induced by gravity signals. Here we simulate the five independent components of the gravity gradient signals induced by earthquakes of different focal mechanisms. We study their spatial amplitude distribution to determine what kind of detectors is preferred (which components of the gravity gradient are more informative), how detectors should be arranged, and how earthquake source parameters can be estimated. The results show that early earthquake detections, within 10 seconds of the rupture onset, using only the horizontal gravity strain components are achievable up to about 140 km distance from the epicenter. Depending on the earthquake focal mechanism and on the detector location, additional measurement of the vertical gravity strain components can enhance the detectable range by 10–20 km. These results are essential for the design of gravity-based earthquake early warning systems.

Ambient seismic noise contains valuable information about the structural properties of Earth’s subsurface. The superconducting gravimeter (SG) is the most suitable instrument for temporal gravity, but not for microseisms due to its limited bandwidth. We report an alternative type of vertical superconducting accelerometer with a bandwidth as high as 0.7 Hz. Its sensitivity is comparable with the current SG. The accelerometer consists of a superconducting mass-spring oscillator and a displacement sensor based on a superconducting quantum interference device. The oscillator is constructed by levitating a Meisnner-state proof mass using combined positive-stiffness and negative-stiffness superconducting coils. Its natural frequency can be adjusted by the coil currents over a large range. The enhanced sensitivity of displacement sensing compensates the sensitivity loss imposed by the broadened bandwidth. Experimental tests show that both the gravity variation and the microseism can be extracted from the measurement data. It would be quite meaningful to monitor the temporal gravity and structural evolution using such an accelerometer array in sensitive zones, for example, the crustal fault and volcano zones.

In this paper, a superconducting gravity gradiometer system that is designed to measure full-tensor gravity gradients based on levitation-type component accelerometers without using mechanical springs is presented. All components of the gradiometer system are made of a superconducting material, niobium. Working principles are the flux quantization and the perfect diamagnetism of the superconductor. Each component of the gravity gradient tensor is obtained from the differential gravity measured by using two accelerometers, each of which is composed of a cubic test mass, a flat sensing coil, and a flat counter coil to balance the repulsive force of the sensing coil and ambient gravity. The measured signals are amplified by using superconducting quantum interference devices. The full-tensor gradiometer system consists of three perpendicularly aligned sets of gradiometers, each of which measures one diagonal and two off-diagonal components. The details of the structure of the system, the dynamics of the gradiometer, the transfer function, the common mode rejection, and the optimum conditions are discussed.

A high precision superconducting levitation system for gravity measurement has been developed which used the levitation of a superconducting sphere by the magnetic field of two superconducting coils. The magnetic levitation is designed to provide independent adjustment of the levitating force and the force gradient. A GM cryocooler is employed to cool down the system. This paper reviews the construction and operating characteristics of the system. The test results show that the earth tide signal was detected by the system.

Based on the Meissner effect a superconducting hollow sphere rotor is levitated in the vacuum housing. During the high speed rotation, the dynamic deformations caused by the centrifugal force will generate magnetic disturbing torque on the superconducting rotor. The deformations of the rotor are analyzed and simulated by finite element method, and the deformation laws are obtained. The structure of spherical-like rotor is designed to compensate the centrifugal deformation at the high rotational velocity. It has significant theoretical value for the design and optimization of the rotor structure.

We report here on experimental and theoretical de-velopments of a new method for gravitational wave (GW) detec-tion. It is based on two principal steps: 1) conversion of the GW action into rotational motion and 2) conversion of the rotational motion into an electric current. The ability to detect extremely tiny currents empowers this approach. Preliminary experiments con-firm the theoretical expectations and suggest that gravitational waves beyond the reach of present interferometric devices can be detected.

The angular momentum of the Earth produces gravitomagnetic components of the Riemann curvature tensor, which are of the order of 10/sup /minus/10/ of the Newtonian tidal terms arising from the mass of the Earth. These components could be detected in principle by sensitive superconducting gravity gradiometers currently under development. We lay out the theoretical principles of such an experiment by using the parametrized post-Newtonian formalism to derive the locally measured Riemann tensor in an orbiting proper reference frame, in a class of metric theories of gravity that includes general relativity. A gradiometer assembly consisting of three gradiometers with axes at mutually right angles measures three diagonal components of a 3/times/3 ''tidal tensor,'' related to the Riemann tensor. We find that, by choosing a particular assembly orientation relative to the orbit and taking a sum and difference of two of the three gradiometer outputs, one can isolate the gravitomagnetic relativistic effect from the large Newtonian background.

In January 1990, a test of the feasibility of airborne gravimetry from a small geophysical survey aircraft, a Cessna 404, was conducted over the Long Island Sound using a Bell Aerospace BGM-3 sea gravity meter. Gravity has been measured from large aircraft and specially modified de Havilland Twin Otters but never from small, standard survey aircraft. The gravity field of the Long Island Sound is dominated by an asymmetric positive 30 mGal anomaly which is well constrained by both marine and land gravity measurements. Using a Trimble 4000 GPS receiver to record the aircraft's horizontal position and radar altimeter elevations to recover the vertical accelerations, gravity anomalies along a total of 65 km were successfully measured. The root mean square (rms) difference between the airborne results and marine measurements within 2 km of the flight path was 2.6 mGal for 15 measured values. The anomalies recovered from airborne gravimetry can also be compared with the gridded regional free air gravity field calculated using all available marine and land gravity measurements. The rms difference between 458 airborne gravity measurements and the regional gravity field is 2.7 mGal. This preliminary experiment demonstrates that gravity anomalies, with wavelengths as short as 5 km, can be measured from small aircraft with accuracies of 2.7 mGal or better. The gravity measurements could be improved by higher quality vertical and horizontal positioning and tuning the gravimeter's stabilized platform for aircraft use.

The Naval Research Laboratory (NRL) has developed a prototype airborne gravity measurement system. The core of the system is a LaCoste and Romberg air‐sea gravity meter mounted on a three‐axis stable platform. Corrections to the gravimeter data for altitude and variations in altitude are determined from a combination of highly precise radar and pressure altimeters. The original prototype system was designed for use over oceanic areas. We recently incorporated the pressure measurement to extend use of the airborne system to terrestrial regions where occasional radar altitudes over points of known topographic height can be obtained. The radar heights are used to relate the pressure altitudes to absolute altitudes and to determine the slopes of the isobaric surfaces. Vertical accelerations due to horizontal velocity over a curved, rotating earth (the Eötvös correction) and precise two‐dimensional positions are determined from a Texas Instrument P-code global positioning system. The updated system was tested over eastern North Carolina and the Outer Banks, an area that is difficult to survey by conventional means. Over one‐third of the region consists of low lying swampy terrain and another one‐third is the shallow water of the Pamlico and Albemarle Sounds. Neither the land method nor the shipboard gravity surveying method is well suited for these types of areas. Flying at an altitude of 600 m at 375 km/hr, we were able to cover an area over 10 000 km 2 with a nominal track spacing of 9 km by 9 km in less than 18 hours of flying time. A comparison by the Defense Mapping Agency showed a 2.8 mGal rms and a −0.2 mGal mean difference between ground truth data and the airborne data at grid points when both data sets were interpolated to a common 9 km grid.

The Gravity Gradiometer Survey System (GGSS) was designed to measure the local and regional gravity field from a ground or airborne moving platform. With the first and only airborne field test, the GGSS was able to recover five-arcminute by five-arcminute mean gravity anomalies to an accuracy of a few mGal. These results were obtained by flying the system, with an operational precision of about 10 Eotvos (ten-second average), on a grid of orthogonal tracks spaced 5 km apart at an altitude of about 700 m above the terrain. Despite perpetual navigation problems with the Global Positioning System and several periods of excessive system noise, the results of a performance analysis on 19 out of 128 tracks demonstrated the potential accuracy and efficiency of the GGSS as an airborne gravity mapping system. The ground tests (both road and railway), suffering from undue vehicle vibrations and from a lack of ground truth data, were correspondingly less successful, but they also showed no surprises in the system corrupted by these adverse conditions. Unfortunately, the GGSS program has terminated; and it is appropriate to reflect on its accomplishments. Without going into technical details, this somewhat historical review summarizes the field tests, the data reduction algorithms, and the test results, which together portray the breadth of expertise the program engendered in the area of gravity gradiometry.

An experiment has been performed to demonstrate a new source-independent null test of the inverse square law of gravitation. A single-axis superconducting gravity gradiometer was rotated into three orthogonal orientations to measure the Laplacian of the gravitational potential produced by a 1600-kg lead pendulum at an average distance of 2.3 m. The result is that if one assumes a potential of the form ϕ(r)=-(GM/r)[1+αexp(-μr)], the value of α is + 0.024±0.036 at μ-1=1 m.

We report the laboratory operation of an rf SQUID-based superconducting gravity gradiometer designed to measure from an aircraft, off-diagonal components of the earth's gravity gradient tensor. A detailed numerical model of the multi-mode electromechanical system shows excellent agreement with the experimental observations. We have demonstrated a common mode rejection ratio to linear accelerations of 180dB, as well as 60dB of active thermal compensation below 0.01Hz. Short term measurements show the instrument to be SQUID noise limited at 0.5Eo¨/√Hz in a frequency band from 50mHz to 1Hz. After correcting for the remaining effects of thermal fluctuations and flux creep the noise below 50mHz has a 1/f characteristic which is also dominated by the SQUID.

The superconducting gravity gradiometers for the European STEP and GEM missions have common design features. Both gradiometers have their test masses magnetically levitated, stiff against all unwanted degrees of freedom. The sensitive axes of the component accelerometers are aligned to with respect to each other, by adjusting persistent currents in the alignment coils, to improve the common-mode rejection ratio to . The axial displacements of the two test masses in each gradiometer are coupled through two superconducting circuits to two DC SQUIDs. Persistent currents are stored in the circuits such that the acceleration signals are summed and differenced at the respective SQUID inputs. This signal differencing before detection reduces the linearity and dynamic range requirements of the electronics by several orders of magnitude. The STEP gradiometer will be a single-axis device with a baseline of about 60 cm and with its sensitive axis oriented along the orbit normal. Its intrinsic noise is expected to be above . Below this frequency 1/f power noise should appear. A compact three-axis superconducting gravity gradiometer with a baseline of 12 cm is proposed for GEM. This gradiometer will have an intrinsic noise of above . Below this frequency 1/f noise will dominate.

A very sensitive resonant superconducting accelerometer has been developed as a component of a cryogenic gravitational‐radiation detector. The device consists of a superconducting test mass and superconducting coils carrying a persistent current. The displacement of the test mass modulates the inductances of the coils and generates an ac magnetic field which is detected by a Josephson‐junction magnetometer. The restoring force provided by the magnetic field is used to tune the resonant frequency of the transducer. The expected sensitivity of the system is better than 10<sup>-12</sup>g E /Hz<sup>1/2</sup> (g E =9.8 m/s<sup>2</sup>) when used to detect accelerations at frequencies lower than 50 Hz. The system has been thoroughly tested and is being used to detect small accelerations of a gravitational‐wave antenna caused by the Brownian motion and other external disturbances. When used as a resonant displacement sensor in a gravitational‐wave detector cooled to 3 mK, this transducer is capable of converting a displacement of 4×10<sup>-20</sup> m at 1 kHz into an electrical signal detectable with unity signal‐to‐noise ratio for 1‐Hz bandwidth. The gravitational‐radiation‐flux sensitivity implied by this is 0.1 erg/cm<sup>2</sup> Hz. This will make not only the observation of expected galactic events possible, but will allow one to extend the scope of observation beyond the Milky Way. The system can be modified to make a sensitive gravity gradiometer. When two accelerometers are coupled to the same Josephson‐junction magnetometer with their transformer coils wound in the opposite sense, direct subtraction of acceleration signals can be accomplished. The system will be easy to build and mechanically rugged. The device in various applications is discussed and the theory of transducer energy coupling, frequency tuning, and parameter optimization is presented. Som-
e experimental results confirming the theory are reported. Included are data showing the temperature dependence of the Q of a niobium diaphragm and the measurement of the low‐frequency background acceleration of a magnetically levitated gravitational‐wave antenna.

We show that theories in which supersymmetry is broken via Scherk-Schwarz compactification at the weak scale possess at least one scalar particle with Compton wavelength in the millimetre range, which mediates a force with strength of gravity. Such forces are going to be explored in upcoming experiments using micro-electromechanical systems or cantilever technology. We also present a simple way of understanding some decoupling aspects of these theories by analogy with finite-temperature field theory.

Nearly 95 per cent of geophysical expenditures for oil exploration is for reflection seismograph surveys. Most of the other 5 per cent is for gravity and magnetic surveys. To some extent the disproportionately large expenditures for seismograph work probably are from a lack of understanding and appreciation of the usefulness of the other methods. The paper reviews the basic principles of the gravity and magnetic methods, points out their similarities and differences, and gives examples of their application. These examples include samples of basement depth maps from magnetic surveys where independent control on actual depths was available. A comparison with this control indicates that depth determinations from magnetic surveys may have a reliability of approximately ± per cent. Regional gravity effects and their removal by graphical and grid calculation schemes are illustrated by examples from the Gulf Coast and California. Methods of quantitative interpretation of gravity are demonstrated by examples from the salt dome area of the Gulf Coast.

A three-axis superconducting gravity gradiometer with a potential sensitivity better than Eotvos per sq root Hz is currently under development for applications in space. Although such a high sensitivity may be needed for only a limited number of terrestrial applications, superconductivity offers many extraordinary effects which can be used to obtain a gravity gradiometer with other characteristics necessary for operation in a hostile moving-base environment. Utilizing a number of recently devised techniques which rely on certain properties of superconductors, a design for a sensitive yet rugged gravity gradiometer with a high degree of stability and a common-mode rejection ratio greater than 10 to the 9th is produced. With a base line of 0.11 m, a sensitivity of 0.1 Eotvos per sq root Hz is expected in an environment monitored to a level of 0.01 m/sq sec sq root Hz for linear vibration and 7 x 10 to the -6th rad/s sq root Hz for angular vibration. A conventional stabilized platform can be used at this level. The intrinsic noise level, which is two orders of magnitude lower, could be achieved by monitoring the attitude with a superconducting angular accelerometer which is under development. In addition, the new gradiometer design has the versatility of adapting the instrument to different gravity biases by adjusting stored dc currents.

A sensitive superconducting gravity gradiometer has been constructed and
tested. Coupling to gravity signals is obtained by having two
superconducting proof masses modulate magnetic fields produced by
persistent currents. The induced electrical currents are differenced by
a passive superconducting circuit coupled to a superconducting quantum
interference device. The experimental behavior of this device has been
shown to follow the theoretical model closely in both signal transfer
and noise characteristics. While its intrinsic noise level is shown to
be 0.07 E Hz-1/2 (1 E≡10-9
sec-2), the actual performance of the gravity gradiometer on
a passive platform has been limited to 0.3-0.7 E Hz-1/2 due
to its coupling to the environmental noise. The detailed structure of
this excess noise is understood in terms of an analytical error model of
the instrument. The calibration of the gradiometer has been obtained by
two independent methods: by applying a linear acceleration and a gravity
signal in two different operational modes of the instrument. This device
has been successfully operated as a detector in a new null experiment
for the gravitational inverse-square law. In this paper we report the
design, fabrication, and detailed test results of the superconducting
gravity gradiometer. We also present additional theoretical analyses
which predict the specific dynamic behavior of the gradiometer and of
the test.

Because of the equivalence principle, a global measurement is necessary to distinguish gravity from acceleration of the reference frame. A gravity gradiometer is therefore an essential instrument needed for precision tests of gravity laws and for applications in gravity survey and inertial navigation. Superconductivity and SQUID (superconducting quantum interference device) technology can be used to obtain a gravity gradiometer with very high sensitivity and stability. A superconducting gravity gradiometer has been developed for a null test of the gravitational inverse-square law and space-borne geodesy. Here we present a complete theoretical model of this instrument. Starting from dynamical equations for the device, we derive transfer functions, a common mode rejection characteristic, and an error model of the superconducting instrument. Since a gradiometer must detect a very weak differential gravity signal in the midst of large platform accelerations and other environmental disturbances, the scale factor and common mode rejection stability of the instrument are extremely important in addition to its immunity to temperature and electromagnetic fluctuations. We show how flux quantization, the Meissner effect, and properties of liquid helium can be utilized to meet these challenges.

A null test of the gravitational inverse-square law can be performed by testing Gauss's law for the field. We have constructed a three-axis superconducting gravity gradiometer and carried out such a test. A lead pendulum weighing 1500 kg was used to produce a time-varying field. This experiment places a new (2-sigma) limit of alpha = (0.9 + or - 4.6) x 10 exp -4 at lambda of 1.5 m, where alpha and lambda are parameters for the generalized potential phi = -(GM/r)(l + alpha e exp -r/lambda).

The scientific aims, design, and mission profile of the Superconducting Gravity Gradiometer Mission (SGGM), a NASA spacecraft mission proposed for the late 1990s, are discussed and illustrated with drawings and diagrams. SGGM would complement the two other planned gravimetry missions, GRM and Aristoteles, and would provide gravitational-field measurements with accuracy 2-3 mGal in 55 x 55-km blocks. The principal instruments are a (1) three-axis superconducting gravity gradiometer with intrinsic sensitivity 100 microeotvos/sq rt Hz, (2) a six-axis superconducting accelerometer with sensitivity 100 fg(E)/sq rt Hz linear and 10 prad/sec squared sq rt Hz angular, and (3) a six-axis shaker for active control of the platform. Consideration is given to the error budget and platform requirements, the orbit selection criteria, and the spacecraft design.

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