High sensitivity accelerometry with a feedback-cooled magnetically levitated microsphere
Charles W. Lewandowski, Tyler D. Knowles, Zachariah B. Etienne, Brian D'Urso
HHigh sensitivity accelerometry with a feedback-cooled magnetically levitatedmicrosphere
Charles W. Lewandowski, Tyler D. Knowles, Zachariah B. Etienne,
3, 4 and Brian D’Urso ∗ Department of Physics, Montana State University, Bozeman, Montana 59717, USA Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506, USA Department of Physics and Astronomy, West Virginia University, Morgantown, West Virginia 26506, USA Center for Gravitational Waves and Cosmology, West Virginia University,Chestnut Ridge Research Building, Morgantown, West Virginia 26505, USA (Dated: March 10, 2020)We show that a magnetically levitated microsphere in high vacuum can be used as an accelerometerby comparing its response to that of a commercially available geophone. This system shows greatpromise for ultrahigh acceleration sensitivities without the need for large masses or cryogenics. Withfeedback cooling, the transient decay time is reduced and the center-of-mass motion is cooled to 9 Kor less. Remarkably, the levitated particle accelerometer has a sensitivity down to 3 . × − g / √ Hzand gives measurements similar to those of the commercial geophone at frequencies up to 14 Hzdespite a test mass that is four billion times smaller. With no free parameters in the calibration, theresponses of the accelerometers match within 3% at 5 Hz. The system reaches this sensitivity due toa relatively large particle mass of 0 . µ g, a low center of mass oscillation frequency of 1 .
75 Hz, anda novel image analysis method that can measure the displacement with an uncertainty of 1 . I. INTRODUCTION
High sensitivity accelerometry has myriad applicationsin fundamental and practical fields of physics and engi-neering. The ability to measure extremely small acceler-ations and forces has uses in absolute gravimeters [1–3],inertial navigation [4], tests of quantum gravity [5, 6],gravitational wave detection [7], precision measurementsof the Newtonian constant of gravitation [8] and othertests of fundamental physics [9].Typical accelerometers are based on clamped resonatorsystems [10–12]. With cryogenic temperatures, forcesensitivities as low as S / F ∼ − N / √ Hz are pre-dicted [13]. Using a Si N membrane [14], quality factorsof 10 can be achieved at room temperature with oscilla-tion frequencies of ∼
150 kHz, and thermal noise limitedforce sensitivities of S / F ∼ − N / √ Hz are possible.Mechanical devices have the advantage of typically be-ing extremely compact [15, 16]. Systems with very testlarge masses, such as LISA Pathfinder, can have acceler-ation sensitivities of S / a ∼ − g / √ Hz [17] where g isstandard gravity, g = 9 . / s . Cold atom interferome-try systems have also been proposed for measuring smallchanges in gravity [18–20] with acceleration sensitivitiesas low as S / a ∼ − g / √ Hz [21, 22].Levitated systems avoid dissipation associated withthe mechanical contact of the resonator with its envi-ronment. Force sensitivities of S / F ∼ − N / √ Hz and S / F ∼ − N / √ Hz have been measured with parti-cles in optical traps [23, 24]. Acceleration sensitivitiesof S / a ∼ − g / √ Hz [25] have been reported using a ∗ [email protected] permanent magnet levitated above a superconductor atcryogenic temperatures.Levitated optomechanical systems in vacuum provideextreme isolation from the environment, making thempowerful candidates for high sensitivity accelerometry.The field has been dominated by optical trapping since itsdevelopment by Ashkin and Dziedzic [26], in which feed-back cooling is typically required for the levitated particleto remain trapped at pressures less than approximately0 .
08 Torr [27, 28]. A magnetic trap that does not rely ongravity for confinement has been demonstrated down to apressure of ∼ . ∼ − Torr with a feedbackcooled center-of-mass motion from room temperature to140 µ K [32]. Recent cooling experiments in an opticaltrap have demonstrated a center-of-mass motion temper-ature of 50 µ K for large particles ( ≈ µ m) [33]. Coolingto the quantum ground state of a sub-micrometer parti-cle has also been shown, reaching a temperature of 12 µ Kfrom room temperature [34].In this paper, we demonstrate levitation of a diamag-netic borosilicate microsphere in a magneto-gravitationaltrap down to a pressure of ∼ − Torr at room tempera-ture. The relatively large mass of the 60 µ m microsphereand low oscillation frequencies compared to optical trap-ping systems [35] make this a promising optomechanicalsystem for high sensitivity room temperature accelerom-etry. The center-of-mass motion is cooled with feedbackto damp transients on a reasonable timescale. To checkthe calibration, accelerations are directly applied to thesystem via a surface transducer.A critical component of the system is a new offline im-age analysis technique we have developed to determinethe displacement of the trapped particle from photosrecording its motion over time. In particular, we mitigate a r X i v : . [ phy s i c s . a pp - ph ] M a r FIG. 1. (a) The linear magneto-gravitational trap as viewed from the transverse ( x ) direction. The length of the bottom polepieces is approximately 26 mm. The top pole pieces are cut shorter to a length of approximately 21 mm. The asymmetrycombined with the force of gravity constrains the particle in the axial ( z ) direction. (b) The view of the trap as viewed fromthe axial direction showing the quadrupole symmetry in the transverse and vertical ( y ) directions. (c) A rotated view of thetrap showing the quadrupole symmetry and the broken symmetry in the vertical direction. The tapped holes pictured on thepole pieces are used to attach the trap to its mount. image background noise and avoid issues with fractionalpixel translations by constructing a pixel-independent“eigenframe”, against which we compute the cross corre-lation. II. EXPERIMENTAL SETUPA. Loading and Trapping of Microspheres
The magneto-gravitational trap, designed with twosamarium-cobalt (SmCo) permanent magnets and fouriron-cobalt alloy (Hiperco-50A) pole pieces (see Fig. 1),creates a three-dimensional potential well to stably trapdiamagnetic particles. The total potential energy of anobject with volume V of diamagnetic material with mag-netic susceptibility χ and mass m in an external magneticfield subject to standard gravity g is U = − χB V µ + mgy, (1)where B = | (cid:126)B | is the magnitude of the magnetic field, µ is the vacuum permeability, and y is the vertical dis-placement of the material [36]. For diamagnetic materi-als ( χ < x − y )plane. Symmetry is broken in the vertical-axial ( y − z )plane by cutting the top pole pieces shorter along theaxial direction. This asymmetry along with gravity formsthe trapping potential in the axial ( z ) direction [31, 37].To reduce the effect of thermal noise while maintainingsensitivity to acceleration, larger trapped particles arepreferred. A loading method has been developed to allowreliable trapping of large microspheres [32]. In these ex-periments, we chose borosilicate microspheres (Cospheric BSGMS-2.2 53-63um-10g) with greater than 90% of par-ticles in the diameter range of 53 µ m-63 µ m. Insulatingpolyimide tape is attached to the tip of an ultrasonichorn [38] to electrostatically hold large microspheres tothe tip. The ultrasonic horn shakes the particles off andinto the trapping region at atmospheric pressure. An ACvoltage is applied to two pole pieces while the other twoare kept isolated from the AC voltage to form a linearquadrupole ion (Paul) trap [39, 40] for the particles thathave non-zero net charge.Note that Eq. 1 requires a large gradient in B tobalance gravity. The large dimensions of the magneto-gravitational trap form an extremely weak potential. ThePaul trap is much stronger, allowing particles to be suc-cessfully levitated near the center of the trap. A DCvoltage, typically between 20 V and 40 V, is applied fromthe top to the bottom pole pieces to help counter gravityand center large particles in the trap.The DC voltage across the top and bottom pole piecesis supplied from a 1-ppm digital-to-analog converter(DAC, Analog Devices AD5791). The DAC is floatedto a voltage between −
300 V and 0 V using a modifiedstacking of Texas Instruments REF5010 high-voltage ref-erences [41] in steps of 5 V. The voltage reference circuitcan be modified to allow for positive voltages as well.The DAC allows for fine tuning of the voltage, and theresulting potential is estimated to be stable to < − Torr in the vacuumchamber. To eliminate vibrations from these pumps, they
FIG. 2. A simple schematic of the circuit used to apply the ACvoltage for the Paul trap and the DC bias to help counteractgravity. A 50 s filter prevents the addition of high frequencynoise. The red dashed lines indicate where jumpers are addedto prevent image charge currents from going through highresistance paths when the Paul trap is not in use. are closed off from the chamber and turned off whilepumping continues with an ion-sputter pump. A pres-sure of ∼ − Torr was maintained for all measurementsreported.
B. Table Stabilization
Changes in the tilt of the optical table cause the equi-librium position of the levitated particle to shift. In theweak direction of the trap, very small changes in tilt canhave a significant effect on the equilibrium position. Forsmall tilts, the shift in equilibrium position is describedby ∆ z eq ≈ gω ∆ θ. (2)To avoid any large shifts in the equilibrium position, amethod has been developed to feedback stabilize the rel-ative tilt of the optical table in real time. The tilt of thetable is measured with an ultra-high sensitivity tiltmeter(Jewell Instruments A603-C) and read on a computer.Using two mass flow controllers, air is added or removedfrom one side of the floating table to keep it level.Without stabilization, the relative tilt of the table canchange by ± µ rad or more. For a levitated particlewith an axial oscillation frequency ω / (2 π ) = 2 Hz, thiscorresponds to a ± . µ m shift in equilibrium, which ismuch larger than the typical oscillation amplitudes due toenvironmental vibrations. With feedback stabilization,this value can be 200 times smaller, resulting in onlynegligible shifts in equilibrium position. FIG. 3. Light from a pulsed 660 nm LED illuminates aslit which is imaged onto the particle, as indicated by the redpath. The control laser, indicated by the green path, utilizes a520 nm diode laser. Radiation pressure from the laser appliesa force that heats or cools the center-of-mass motion of theparticle, depending on the phase of the drive relative to themotion of the particle. The scattered green light is blockedby a long-pass filter, while the scattered illumination lightis collected and imaged onto a CMOS camera. The imagesare analyzed in real time to apply the feedback drive to theparticle.
C. Real-Time Image Analysis and FeedbackCooling
The particle is stroboscopically illuminated using a660 nm LED with a repetition rate of 100 Hz and a pulseduration of 1 ms. As shown in Fig. 3, light from the LEDis collimated using an aspheric lens and passed througha 100 µ m slit. The slit is imaged onto the particle andmagnified to illuminate the entire region of interest. Theparticle is imaged onto the CMOS camera with a 0.09 NAtelecentric objective (Mitutoyo 375-037-1). All recordedimages are 256 by 128 pixels, corresponding to a field ofview of 300 µ m by 150 µ m.As shown in Fig. 4(a), the illuminated microsphere ap-pears as a dark disk in each image (or frame). The mi-crosphere diameter of approximately 60 µ m correspondsto a diameter of approximately 60 pixels in each frame.The microsphere never leaves the frame in the data weanalyze.The images from the CMOS camera are analyzed inreal time to track the motion of the particle. Each imageis thresholded to isolate the particle, and the apparentcenter-of-mass is calculated. The movement from frameto frame is used to calculate the velocity of the parti-cle, which is then passed through a second order infi-nite impulse response (IIR) peak bandpass filter with abandwidth of 1 . . FIG. 4. (a) Raw data: dark microsphere on light background with approximate scale. (b) Fifth eigenframe. Note the smoothingof background features in comparison to the raw data. (c) Difference in pixel values between the zeroth eigenframe and thefifth eigenframe. Note that grey pixels denote small differences; dark and light pixels represent the fifth eigenframe having adarker or brighter pixel than the zeroth eigenframe, respectively. Differences in pixel values are scaled by a factor of 9. for imaging.
III. OFFLINE IMAGE ANALYSIS
If limited to a resolution of one pixel, we could onlytrack the microsphere’s position to about 1 µ m. Sophisti-cated image analysis techniques exist, however, that mea-sure displacement versus some reference frame to a smallfraction of a pixel by incorporating all pixel data fromeach frame. While the image analysis for feedback mustbe completed in real time, a more accurate but more com-putationally intensive algorithm can be used for offlineanalysis of the data. As our first approach, we adoptedthe cross-correlation function register translation() available in the scikit-image Python package [43, 44]to determine the displacement of the particle relative tothe first recorded frame of data (to which we refer asthe “zeroth eigenframe”). While this approach largelyseemed to work well, we noticed jump discontinuities inthe microsphere displacement versus time as can be seenin Fig. 5(a).We attributed these discontinuities to noise in the ze-roth eigenframe. As this frame was chosen arbitrar-ily, we anticipated that any other choice of referenceframe would result in similar displacement discontinu-ities. To minimize the effects of this noise we devised anew “eigenframe” approach, which proceeds as follows:we first compute the translation in z and y of each frameagainst the zeroth eigenframe in the spatial domain us-ing register translation() . Using these translations,we line up all frames to their inferred displacement withrespect to the zeroth eigenframe and construct a globallyaveraged frame. We refer to the resulting averaged frameas the “first eigenframe”. Specifically, the translationsand averaging are performed using the two-dimensionaldiscrete Fourier transforms of the images so that thechoice of pixel alignment in the spatial domain does notresult in loss of information. The averaging smears outthe noise present in the zeroth eigenframe and smoothsthe displacement data (as illustrated in Fig. 5(a)). Wethen refine the translation values by correlating eachframe against a translation of the first eigenframe (againin the Fourier domain) to the inferred particle location.The resulting translations may be used to build a second eigenframe in a manner analogous to building the first,and this process can be iterated as many times as we like.To further refine our position resolution, we modified register translation() to fit a slice of the correlationsurface through the peak in the z -direction to a quadraticfunction using SciPy ’s optimize.curve fit() function.Locating the peak of this quadratic gives another esti-mate of the particle translation between each frame andthe eigenframe.To demonstrate that the translation values convergewith eigenframe number, denote by d n ( t i ) the axial dis-placement of the microsphere at time t i when correlatedagainst eigenframe n ( n = 1 , , . . . , d n ( t i ) − d n − ( t i ) over all t i (seeFig. 5(b) and 4(c)). Incredibly, the position differencesquickly reach a standard deviation of less than 1 nm, thusfalling well below the physical resolution limit. After re-peating this eigenframe procedure five times, the stan-dard deviation of the change in displacements drops tobelow 1 pm. As this is far below other sources of dis-placement error in our experiment, the fifth eigenframeis the final one we compute. IV. ACCELERATION MEASUREMENT
We measure the acceleration sensitivity of the trappedparticle by examining the effect of movement of the pneu-matically isolated optical table (on which the trap andoptics are mounted) on the particle. In the frame ofthe laboratory, consider the displacement of the parti-cle in the axial direction, z , and the displacement of thecamera, z . The camera directly measures z (cid:48) = z − z .The equation of motion for the particle in the laboratoryframe is then ¨ z + Γ ˙ z + ω z = Γ ˙ z + ω z (3)where Γ is the damping rate and ω is the resonant an-gular frequency of the particle.The displacement of the optical table, for example,from vibrations, can be written as an integral over allfrequencies, z ( t ) = (cid:90) ∞−∞ dω (cid:48) A ( ω (cid:48) ) sin( ω (cid:48) t + φ ) (4) FIG. 5. (a) Displacement comparison between correlationagainst the zeroth and fifth eigenframes. Note in particu-lar the discontinuities appearing throughout the zeroth eigen-frame displacement time series. (b) Standard deviation ofdisplacement differences between eigenframes n and n − z ) oscillation frequency of ω / (2 π ) = 1 .
75 Hz and thedecay rate is Γ = 6 . × − s − (black dashed lines). Theamplitude range plotted and analyzed is chosen so that vibra-tional noise is negligible. where A is the strength of the drive as a function offrequency.After substituting Eq. 4 into Eq. 3, we can take theFourier transform of Eq. 3. Simplifying the resultingexpression, we find that the magnitude of the transfer function is (cid:12)(cid:12)(cid:12)(cid:12) Z (cid:48) ( ω ) A ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) = ω (( ω − ω ) + Γ ω ) / (5)where Z (cid:48) ( ω ) is the Fourier transform of the particle’smotion with respect to the camera.The minimum acceleration that can be detected for anoscillator in thermal equilibrium at temperature T is [45] S / a = (cid:114) k B Γ Tm (6)where k B is Boltzmann’s constant, m is the mass, andΓ is the damping rate of the oscillator. Feedback cool-ing at best keeps Γ T constant, damping out potentiallylong-lived transients without a significant impact on sen-sitivity [46]. A. Results
A borosilicate microsphere was levitated with a DCbias across the vertical gap of the magneto-gravitationaltrap of − . − Torr was maintained and the tiltof the optical table was stabilized to within ± . µ rad.With the measured resonant frequency of the micro-sphere, Eq. 2 gives that the equilibrium position of theparticle was stabilized to within ±
60 nm.Before acquiring acceleration data, the system mag-nification, a critical calibration parameter, is measured.By analyzing the recorded image of a USAF1951 cal-ibration target (Edmund Optics S c = 1 . µ m / pixel wasdetermined. For frequency calibration, the digital delaygenerator used to control all of the timing in the experi-ment is tied to a rubidium frequency standard (StanfordResearch Systems, Inc. FS725).In order to eliminate any free parameters of the system,the transient response of the microsphere was measuredafter a small excitation in the axial direction, shown inFig. 6. The resonant frequency of the particle was mea-sured to be ω / (2 π ) = 1 .
75 Hz. While feedback coolingthe center-of-mass motion of the microsphere, the damp-ing rate was measured to be Γ = 6 . × − s − .For comparison, we also place an L-4C geophone (Ser-cel, Inc. [47]) on the optical table. The sensitivity of thisinstrument and other critical parameters are given by themanufacturer. We added an additional amplification cir-cuit with a gain of approximately 180 to boost the signalbefore digitization (modeled after that in [48]).The response of the particle to movement of the opti-cal table is tested by applying 5 Hz sinusoidal drive witha surface transducer, oriented to push the table in theaxial direction. While applying this external drive, a setof five 60 s measurements were recorded. Each measure-ment consists of 6000 images from the CMOS camera,which are analyzed with the algorithm described above. FIG. 7. (a) The raw data of the particle response in units of acceleration compared to the raw data of the L-4C geophoneresponse in acceleration units with a 5 Hz drive to verify the calibration. The large peak at 1 .
75 Hz is the axial motion and thepeaks at 10 . . The averaged spectra of the resulting particle accelera-tion over five data sets is shown in Fig. 7(a). For com-parison, the measured acceleration of the test mass ofthe geophone is shown on the same plot. The vibrationbetween 1 Hz and 2 Hz is believed to be a resonance ofthe optical table and overlaps with the resonance of theparticle, causing an on-resonance excitation illustratedby the large peak at 1 .
75 Hz. The transverse and ver-tical motion of the particle are at 10 . . B. Noise Analysis
The noise contributions for both the geophone and par-ticle are plotted in Fig. 8 along with the (undriven) accel-eration of the table as determined by the geophone and levitated particle.The noise of the L-4C geophone and its accompany-ing amplification circuit can be broken down into fourterms [48]. As displacement equivalent noise sources,they are: n therm ( ω ) = (cid:115) k B T ω mQ ω (7)n Johnson ( ω ) = (cid:112) k B T R ( ω ) G ( ω ) (8)n voltage ( ω ) = N V ( ω ) G ( ω ) (9)n current ( ω ) = N A ( ω ) R ( ω ) G ( ω ) (10)The thermal noise of the damped harmonic oscillator isgiven by Eq. 7 where T is the temperature of the oscilla-tor, m is the mass, ω is the resonant angular frequency,and Q is the quality factor. The thermal fluctuations areapproximately 2 . × − g / √ Hz for the geophone withthe parameters listed in Table I. The Johnson noise ofthe geophone coil is given by Eq. 8, where R is the real Parameter Description Geophone Value Particle Value T Temperature 300 K 9 K ω / π Resonant frequency of oscillator 0 .
97 Hz 1 .
75 Hz m Mass of oscillator 0 .
957 kg 2 . × − kg Q Quality factor of oscillator 1.845 175 S g , S c Sensitivity (Note different units) 281 . / m 1 . µ m / pixel R c Resistance of geophone coil 5546 Ω S g Sensitivity of geophone oscillator 281 . / m G a Gain of amplification circuit 180.2 N V Input-referred voltage noise 8 . / √ Hz N A Input-referred current noise Negligible I Energy density of scattered light 2 . × − J / m ∆ z Readout noise 160 pm / √ HzTABLE I. Critical parameters for the amplified L-4C geophone and levitated particle accelerometers. The geophone values arefrom the datasheets of the L-4C geophone and OPA188 operational amplifier used in the geophone amplifier. part of the complex impedance of the coil given by R ( ω ) = R c + iS g ωω − ω + i Γ ω (11)and where R c is the resistance of the coil, S g is the sensi-tivity of the oscillator, and Γ is the damping rate of theoscillator. The harmonic oscillator response G is givenby G ( ω ) = ω S g (cid:112) ( ω − ω ) + Γ ω . (12)The input voltage and current noise of the amplificationcircuit is given by Eq. 9 and Eq. 10, respectively. N V is the input-referred voltage noise of the OPA188 oper-ational amplifier [49] used in the amplification circuit,assumed to be constant over the frequency range of in-terest. The current noise of the amplification circuit isnegligible compared to all other noise sources for the geo-phone. The noise sources add in quadrature to give thetotal noise of the geophone system n total ,g asn ,g = n + n + n + n . (13)The noise of the levitated particle accelerometer hastwo contributions. First, the thermal noise of the particleis given by Eq. 7, where the parameters are now thatof the particle (given in Table I). With feedback coolingapplied, we measure the damping rate to determine the Q of 175, but the effective temperature may be significantlyreduced relative to the ambient temperature. Inspectionof the minimum signal recorded (around 0 . . × − g / √ Hz and a limiton the effective temperature associated with the dampingof the particle of 9 K.The second noise source is readout noise from the cam-era and image analysis which is expected to be dominatedby shot noise of the light and camera noise. To place alower bound on the readout noise, we consider the preci-sion to which a diffraction limited spot can be determinedin the presence of shot noise. This is described by (cid:68) (∆ z ) (cid:69) = σ N (14) where σ PSD is the standard deviation of the point spreadfunction (PSF) of the imaging optics and N is the num-ber of photons collected, or in our case, blocked, by theparticle.For the lower bound on the noise, the PSF is calculatedfor a diffraction limited spot. The standard deviation is σ PSD = 0 . λ/ NA where λ is the wavelength of thescattered light and NA is the numerical aperture of thecollection objective. For our system, σ PSD = 1 . µ m.The number of photons is estimated from the brightnessof the illumination in the CMOS camera and the num-ber of pixels blocked by the 30 µ m radius particle, re-sulting in an uncertainty in the location of the point of (cid:10) (∆ z ) (cid:11) ≈ . z = 1 . / √ Hz, reasonablyabove the point source diffraction limit. The two noisesources add in quadrature, so that the total noise of theparticle response is n ,p = n + (∆ z ) . (15) V. DISCUSSION
We have experimentally demonstrated levitation of a2 . × − kg borosilicate microsphere in high vacuum.This system shows great promise for ultrahigh acceler-ation sensitivities without the need for large masses orcryogenics. Feedback cooling reduces the transient decaytime of the system, while also cooling the center-of-massmotion. With no free parameters in the calibration, theacceleration determined from the apparent motion of the FIG. 8. The particle and geophone responses divided by theharmonic oscillator responses with no drive. Contributions tothe noise for the geophone and particle are also shown. particle both follows that of a commercial geophone be-low 14 Hz and matches the response to an external drivewithin 3% at 5 Hz, despite the particle having a massthat is 4 × times smaller than the test mass in thegeophone.The sensitivity limit in the levitated particle ac-celerometer is estimated to be below 3 . × − g / √ Hz at low frequencies, limited by either by the vibrationsbeing measured or thermal noise associated with damp-ing at 9 K; a quieter environment would be needed tounambiguously determine the limiting factor and the ef-fective temperature. Much lower center-of-mass temper-atures have been reached with trapped particles in othersystems, so there is room for significant improvement.For example, feedback cooling to 140 µ K in a magneto-gravitational trap [32] and 50 µ K in an optical trap [33]have been demonstrated. Lower center-of-mass temper-atures in the current system could result in a sensitiv-ity improvement of at least an order of magnitude, andmight be reached by using a more precise real-time im-age analysis system for feedback cooling. Further im-provements are possible using an even lower center-of-mass oscillation frequency or a higher camera frame rate.This high-sensitivity, self-calibrating system with negli-gible test mass may be particularly valuable for space-based accelerometry at low frequencies.
ACKNOWLEDGMENTS
We acknowledge Lin Yi from JPL for discussions on po-tential mission requirements. This work was supportedby the National Science Foundation under awards PHY-1707789, PHY-1757005, PHY-1707678, PHY-1806596,and OIA-1458952; the National Aeronautics and SpaceAdministration under awards ISFM-80NSSC18K0538and TCAN-80NSSC18K1488; and a block gift fromthe II-VI Foundation. Offline data analysis was com-pleted on the Spruce Knob Super Computing Systemat West Virginia University (WVU), which is funded inpart by the National Science Foundation EPSCoR Re-search Infrastructure Improvement Cooperative Agree-ment [1] T. Niebauer, G. Sasagawa, J. Faller, R. Hilt, and F. Klop-ping, A new generation of absolute gravimeters, Metrolo-gia , 159 (1995).[2] Y. Bidel, O. Carraz, R. Charriere, M. Cadoret, N. Za-hzam, and A. Bresson, Compact cold atom gravimeter forfield applications, Appl. Phys. Lett. , 144107 (2013).[3] J. Liu and K.-D. Zhu, Nanogravity gradiometer based ona sharp optical nonlinearity in a levitated particle op-tomechanical system, Phys. Rev. D , 044014 (2017).[4] B. Battelier, B. Barrett, L. Fouch´e, L. Chichet,L. Antoni-Micollier, H. Porte, F. Napolitano, J. Lautier,A. Landragin, and P. Bouyer, Development of compactcold-atom sensors for inertial navigation, in QuantumOptics , Vol. 9900 (International Society for Optics andPhotonics, 2016) p. 990004.[5] D. Kafri, J. Taylor, and G. Milburn, A classical channelmodel for gravitational decoherence, New J. Phys. ,065020 (2014).[6] A. Albrecht, A. Retzker, and M. B. Plenio, Testingquantum gravity by nanodiamond interferometry with nitrogen-vacancy centers, Phys. Rev. A , 033834(2014).[7] B. P. Abbott, R. Abbott, T. Abbott, F. Acernese,K. Ackley, C. Adams, T. Adams, P. Addesso, R. Ad-hikari, V. Adya, et al. , Gw170817: observation of gravi-tational waves from a binary neutron star inspiral, Phys.Rev. Lett. , 161101 (2017).[8] H. Cavendish, Xxi. experiments to determine the densityof the earth, Philos. Trans. Royal Soc. , 469 (1798).[9] D. C. Moore, Tests of fundamental physics with opti-cally levitated microspheres in high vacuum, in OpticalTrapping and Optical Micromanipulation XV , Vol. 10723(International Society for Optics and Photonics, 2018) p.107230H.[10] O. Gerberding, F. G. Cervantes, J. Melcher, J. R. Pratt,and J. M. Taylor, Optomechanical reference accelerome-ter, Metrologia , 654 (2015).[11] Y. Bao, F. G. Cervantes, A. Balijepalli, J. R. Lawall,J. M. Taylor, T. W. LeBrun, and J. J. Gorman, Anoptomechanical accelerometer with a high-finesse hemi- spherical optical cavity, in (IEEE, 2016) pp.105–108.[12] F. Guzm´an Cervantes, L. Kumanchik, J. Pratt, andJ. M. Taylor, High sensitivity optomechanical referenceaccelerometer over 10 khz, Appl. Phys. Lett. , 221111(2014).[13] J. Moser, A. Eichler, J. G¨uttinger, M. I. Dykman, andA. Bachtold, Nanotube mechanical resonators with qual-ity factors of up to 5 million, Nat. Nanotechnol. , 1007(2014).[14] R. A. Norte, J. P. Moura, and S. Gr¨oblacher, Mechanicalresonators for quantum optomechanics experiments atroom temperature, Phys. Rev. Lett. , 147202 (2016).[15] A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, andO. Painter, A high-resolution microchip optomechanicalaccelerometer, Nat. Photonics , 768 (2012).[16] Y. L. Li and P. Barker, Characterization and testing ofa micro-g whispering gallery mode optomechanical ac-celerometer, J. Light. Technol. , 3919 (2018).[17] M. Armano, H. Audley, G. Auger, J. Baird, M. Bas-san, P. Binetruy, M. Born, D. Bortoluzzi, N. Brandt,M. Caleno, et al. , Sub-femto-g free fall for space-basedgravitational wave observatories: Lisa pathfinder results,Phys. Rev. Lett. , 231101 (2016).[18] N. Yu, J. Kohel, J. Kellogg, and L. Maleki, Developmentof an atom-interferometer gravity gradiometer for gravitymeasurement from space, Appl. Phys. B , 647 (2006).[19] G. Stern, B. Battelier, R. Geiger, G. Varoquaux,A. Villing, F. Moron, O. Carraz, N. Zahzam, Y. Bidel,W. Chaibi, et al. , Light-pulse atom interferometry in mi-crogravity, Eur. Phys. J. D , 353 (2009).[20] G. Biedermann, Gravity tests, differential accelerome-try and interleaved clocks with cold atom interferometers (Stanford University, 2008).[21] M.-K. Zhou, Z.-K. Hu, X.-C. Duan, B.-L. Sun, L.-L.Chen, Q.-Z. Zhang, and J. Luo, Performance of a cold-atom gravimeter with an active vibration isolator, Phys.Rev. A , 043630 (2012).[22] G. Biedermann, X. Wu, L. Deslauriers, S. Roy, C. Ma-hadeswaraswamy, and M. Kasevich, Testing gravity withcold-atom interferometers, Phys. Rev. A , 033629(2015).[23] G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham,and A. A. Geraci, Attonewton force detection using mi-crospheres in a dual-beam optical trap in high vacuum,Phys. Rev. A , 051805 (2015).[24] G. Ranjit, M. Cunningham, K. Casey, and A. A. Geraci,Zeptonewton force sensing with nanospheres in an opticallattice, Phys. Rev. A , 053801 (2016).[25] C. Timberlake, G. Gasbarri, A. Vinante, A. Setter, andH. Ulbricht, Acceleration sensing with magnetically lev-itated oscillators above a superconductor, Appl. Phys.Lett. , 224101 (2019).[26] A. Ashkin and J. Dziedzic, Optical levitation by radiationpressure, Appl. Phys. Lett. , 283 (1971).[27] F. Monteiro, S. Ghosh, A. G. Fine, and D. C. Moore, Op-tical levitation of 10-ng spheres with nano-g accelerationsensitivity, Phys. Rev. A , 063841 (2017).[28] A. D. Rider, C. P. Blakemore, G. Gratta, and D. C.Moore, Single-beam dielectric-microsphere trapping withoptical heterodyne detection, Phys. Rev. A , 013842(2018).[29] M. O’Brien, S. Dunn, J. Downes, and J. Twamley, Magneto-mechanical trapping of micro-diamonds at lowpressures, Appl. Phys. Lett. , 053103 (2019).[30] J. Houlton, M. Chen, M. Brubaker, K. Bertness, andC. Rogers, Axisymmetric scalable magneto-gravitationaltrap for diamagnetic particle levitation, Rev. Sci. In-strum. , 125107 (2018).[31] J.-F. Hsu, P. Ji, C. W. Lewandowski, and B. DUrso,Cooling the motion of diamond nanocrystals in amagneto-gravitational trap in high vacuum, Sci. Rep. ,30125 (2016).[32] W. M. Klahold, C. W. Lewandowski, P. Nachman,B. R. Slezak, and B. D’Urso, Precision optomechanicswith a particle in a magneto-gravitational trap, in Opti-cal, Opto-Atomic, and Entanglement-Enhanced PrecisionMetrology , Vol. 10934 (International Society for Opticsand Photonics, 2019) p. 109340P.[33] F. Monteiro, W. Li, G. Afek, C.-l. Li, M. Mossman, andD. C. Moore, Force and acceleration sensing with opti-cally levitated nanogram masses at microkelvin temper-atures, arXiv preprint arXiv:2001.10931 (2020).[34] U. Deli´c, M. Reisenbauer, K. Dare, D. Grass, V. Vuleti´c,N. Kiesel, and M. Aspelmeyer, Cooling of a levitatednanoparticle to the motional quantum ground state, Sci-ence (2020).[35] C. W. Lewandowski, W. R. Babbitt, and B. D’Urso,Comparison of magneto-gravitational and optical trap-ping for levitated optomechanics, in