A scalable high-performance magnetic shield for Very Long Baseline AtomInterferometry

E. Wodey, D. Tell, E. M. Rasel, and D. Schlipperta)

Leibniz Universitat Hannover, Institut fur Quantenoptik, Welfengarten 1, 30167 Hannover,Germany

R. Baur, U. Kissling, B. Kolliker, M. Lorenz, M. Marrer, U. Schlapfer, and M. WidmerIMEDCO AG, Industriestrasse West 14, 4614 Hagendorf, Switzerland

C. UfrechtUniversitat Ulm, Institut fur Quantenphysik and Center for Integrated Quantum Science and Technology (IQST),Albert-Einstein-Allee 11, 89069 Ulm, Germany

S. Stuiber and P. FierlingerTechnische Universitat Munchen, Physikdepartment, 85748 Garching, Germany

(Dated: 29 December 2021)

We report on the design, construction, and characterization of a 10 m-long high-performance mag-netic shield for Very Long Baseline Atom Interferometry (VLBAI). We achieve residual fields below4 nT and longitudinal inhomogeneities below 2.5 nT/m over 8 m along the longitudinal direction.Our modular design can be extended to longer baselines without compromising the shielding per-formance. Such a setup constrains biases associated with magnetic field gradients to the sub-pm/s2

level in atomic matterwave accelerometry with rubidium atoms and paves the way towards tests ofthe universality of free fall with atomic test masses beyond the 10−13 level.

I. INTRODUCTION

Light pulse atom interferometers are powerful toolsfor modern precision metrology.1–3 They exploit thefine and well understood control of matter waves bylight to achieve record instabilities and inaccuraciesin the precision measurement of forces and other in-ertial quantities.4–6 In the conventional Kasevich-Chugeometry,7 the phase sensitivity of these interferome-ters scales linearly with the enclosed space-time area.By extending the free fall distance from tens of cen-timeters to the order of ten meters, large scale atominterferometers8,9 target more than a factor fifty increaseof their scale factor. When applied to the measurementof the local gravitational acceleration, the combinationof such long baselines with high-performance inertial ref-erence platforms brings short-term instabilities compet-ing with state of the art superconducting gravimetersin reach, while in addition providing absolute measure-ments.

However, spurious field gradients along the free evo-lution path of the atoms can mimic the signal of inter-est and therefore limit the measurement accuracy andinstrumental stability. In particular, magnetic field in-homogeneities generate bias accelerations on the atomsdue to the Zeeman effect. Magnetic gradients at thefew nT/m level or better along baselines of several me-ters are therefore required to satisfy the accuracy bud-get of these large scale atom interferometers at the sub-nm/s2 level, compatible with their target instability.10

Magnetic shielding is typically achieved by channelingexternal magnetic flux inside a high permeability shellaround the volume to isolate.11 Here, the homogeneityof the shielding material’s permeability is key to ensurea uniform magnetization of the shield. This is however

a)Electronic mail: schlippert@iqo.uni-hannover.de

challenging on the lengths required by large scale atominterferometers since the production of homogeneous ex-tended curved sheet metal is problematic, as well as thegap-free and reproducible junction of several pieces. Acommonly used solution is to produce a fully welded as-sembly which is subsequently hydrogen annealed to en-sure homogeneous properties of the shielding material.12

This is however impractical due to the limited availabilityof suitable furnaces and lacks scalability for future, pos-sibly larger applications. Here, inspired by the layout ofmagnetically shielded rooms,13 we report on the design,construction, and characterization of a 10 m-long mag-netic shield for the Hannover Very Long Baseline AtomInterferometry (VLBAI) facility.10 We first describe thekey design points in achieving a fully length-scalable,large length-to-diameter ratio magnetic shield which doesnot require overall annealing. We then assess the shield’sperformance with residual field and dynamical shieldingfactor measurements and finally discuss its application inprecision atom interferometry.

II. SHIELD DESIGN AND CONSTRUCTION

In our atom interferometer, atoms fall freely inside avertically oriented 10.5 m long aluminum ultra-high vac-uum pipe14 with inner diameter 18 cm and outer diam-eter 20 cm. The magnetic shielding enclosure for thiscylindrical vacuum chamber consists of two concentricoctagonal prism shells of high permeability Ni-Fe alloy15

(“permalloy”) as shown in figure 1. The inner and outerpermalloy shells have circumscribed diameters of 450 mmand 750 mm respectively. Both shells are closed individu-ally by end-caps resulting in a total length of 9.7 m for theinner shell and 10 m for the outer one. On the main axisof symmetry, a 22 cm diameter circular opening is left inthe end-caps as clearance for the vacuum chamber.

Despite the cylindrical symmetry of the volume to iso-

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(a)

70 cm

Inner shell

with end-cap

Outer shell(b)

FIG. 1. Photographs of the assembled magnetic shielding en-closure. (a) Side view of the final assembly. (b) End viewwith the end-cap for the outer shell removed. Two octagonalprism permalloy shells are supported by a plywood structure.Between the shells, PVC pipes guide the magnetic equilibra-tion coils. Colored lines show the routing of the two coils usedfor the equilibration of the inner layer. In the center, an alu-minum pipe holds the space for the interferometer’s vacuumchamber.

late, an octagonal geometry turned out to be more prac-tical. It allows using mainly planar sheet material withwell defined and homogeneous permeability after anneal-ing which enables more reliable finite element simulationof the assembly. Figure 2 shows the construction prin-ciple of the shielding enclosure. The octagonal prismshape is maintained by a structure made of 21 mm thickplywood. The plywood panels are assembled inside anextruded aluminum frame using aluminum fixtures andtitanium and stainless steel bolts and nuts. The 1 mmthick annealed permalloy sheets are stacked and pressedagainst the supporting plywood structure by means of ti-tanium wood screws. High pressure laminate strips helpdistributing the load applied by the screws. Clearanceholes for the screws in the permalloy sheets are laser cutafter the annealing procedure. The gaps on the edgesof the octagonal prism shape are closed using angledpermalloy strips. The strips are bent after the anneal-ing procedure. Angled and planar sheets are alternatedas shown on figure 2. In the longitudinal direction, thepermalloy sheets are close to 3 m long. The few mm gapbetween consecutive planar sheets is bridged by offset-ting the next sheet on the stack with 50 % overlap, hencethe pair of planar sheets shown on figure 2. In order toavoid saturation effects and help the magnetic equilibra-tion field to penetrate effectively, the nominal thicknessof the shells varies from 3 mm at the ends up to 8 mm in

Plywoodstructure

Planar permalloy sheet

Angledpermalloy

sheet

Titanium wood screw

High pressurelaminate strip

FIG. 2. Principle for the assembly of the permalloy sheets fora single shell with a nominal thickness of 2 mm (transversecut view, not to scale). The 1 mm thick permalloy sheets arestacked and pressed against a supporting plywood structure.Planar and angled strips are alternated. High pressure lami-nate strips help distributing the load applied by the titaniumwood screws. We use one screw every 250 mm to ensure uni-form pressure on the permalloy sheets.

the central region.

This construction allows precise positioning of eachsheet while keeping the material stress-free and hencereducing build-up of inhomogeneities in its magnetic per-meability. It is in particular crucial that the mechanicalrobustness is given by the plywood structure and not bythe shielding material itself. Also, the plywood structureand the air gap between the layers provide an electricalisolation over 10 MΩ. This design is also fully scalable inlength. The permalloy sheet material is annealed beforeassembly on the plywood structure, making the largestpieces to anneal only 3 m long. This removes the need fora large hydrogen furnace12 irrespective of the final lengthof the shield. Finally, the possibility to open individualfaces of the shells provides great access and flexibility forthe installation of the vacuum chamber.

In order to minimize the free energy of the magneticdomains in the shielding material, we use a magneticequilibration (“degaussing”) procedure similar to the onedescribed by Altarev et al.13 We apply the equilibrationfield using coils routed around the faces of the octagonalprism shells using PVC pipes as guides (see figure 1). Forthe inner shell, we use two sets of five turns of 6 mm2 cop-per wire connected in parallel, building up a coil enclos-ing each face of the permalloy octagonal prism twice (redand green lines on figure 1). The resistive impedance ofthe equilibration coils for the inner layer is around 2.5 Ωper coil. We connect these coils in parallel to relax thevoltage requirements on the coil driver. For the outerlayer, we use only a single pass per face with five turns ofthe same wire which amounts to 2.7 Ω of resistive load.

3

Finally, we set up another pair of coils for the end-capswith five turns of 2.5 mm2 cross-section wire per end-cap.These coils are wired in series and sum up to a resistanceof 0.3 Ω. We drive the equilibration coils sequentially,first for the inner shell, then the end-caps, the outer shell,and finally the inner shell again. Each step starts byfeeding a 6 Hz, 10 A RMS sinusoidal current in the coil,then decreasing the peak current linearly over 100 s. Thefull equilibration sequence therefore lasts around 7 min.The target current waveform is calculated on a computerand fed into a 16-bit voltage output DAC that drives anoffset-trimmed current-mode amplifier. Residual offsetsat the output of the current amplifier are further reducedby an external low distortion transformer.

III. MEASUREMENTS

We determine the dynamical shielding factors in themiddle of the shield (section III A) and the residual vec-tor field along the full longitudinal axis (section III B)using the setup shown in figure 3. The end-caps are fullyclosed on both sides. We effectively reproduce the con-ditions of the final experimental apparatus by using areplicate of the vacuum chamber’s aluminum pipe as aguide for the magnetic field sensors. For practicality rea-sons however, the shield is in horizontal position whereasit will be implemented vertically in the final configura-tion.

Excitation coilsNulling coil

z x yIII B III A

FIG. 3. Setup for the characterization measurements. In sec-tion III A, we use three orthogonal coil pairs around a fixedprobe in the middle of the shield ( ) to measure the dynam-ical shielding factors in all three spatial directions. For theresidual field measurements in section III B, the probe ( )travels along the longitudinal axis (z direction) of the shieldand we coarsely null the field at the ends of the shield usingtwo constant current coils while the excitation coils are notused.

A. Dynamical shielding factor

We first measure the response of the shield to a sinu-soidal external field perturbation. We define the dynam-ical shielding factor as the ratio of the amplitude of theapplied perturbation to the measured amplitude insidethe shield.13 We apply the external perturbation using aset of three calibrated rectangular coil pairs of dimensions3 m× 2 m around the center of the shield. The resultingfield in the corresponding frequency band is recorded bya three axes fluxgate sensor placed in the middle of theshield. To avoid saturating the magnetometer’s digitizer,we vary the applied field between 700 nT and 2000 nT.Figure 4 shows the measured dynamical shielding factorversus frequency for all three directions. In the trans-verse directions, the shielding ratio reaches values above

10−2 10−1 100 101 102 103

102

103

104

105

106

Frequency (Hz)

Dynam

ical

shieldingfactor

Transverse XTransverse YLongitudinal

FIG. 4. Measurement of the dynamical shielding factor at thecenter of the shield with end-caps installed. The effect of thealuminum pipe used as a guide for the probe is visible throughthe increased shielding factor at frequencies above 1 Hz.

4000 at 0.01 Hz, similar to other two-layer designs.13

The corresponding longitudinal damping is more than100 times lower, as expected from the large length-to-diameter ratio.12 Owing to the cancellation of the field’sdivergence in vacuum, the longitudinal field must be ho-mogeneous if the transverse gradients are nulled and itcan therefore be adjusted by a simple solenoid. Finally,we note that the change of slope observed in all directionsaround 1 Hz is characteristic of the crossover between ef-fective shielding by the permalloy sheets at low frequen-cies and by the aluminum pipe above this threshold.

B. Residual field

We also map the three components of the residual fieldalong the symmetry axis of the shield. We mount a threeaxes magnetometer16 on a wooden mount assembled withnon-metallic connectors to enable it travelling on theshield’s axis guided by the aluminum pipe. We measurethe position of the sensor in the shield using a remotecontrolled laser distance meter. We place two constantcurrent loops around the end-caps of the magnetic shieldto coarsely null the longitudinal field component nearthe entrance of the shield (figure 3). The shield is other-wise fully passive and we in particular did not implementany active external field stabilization. The magnetome-ter’s output signals are conditioned by commercial pre-amplifiers17 and digitalized using auto-zeroed 6.5 digitsdigital multimeters18. We perform six scans of the fieldon the shield’s axis over two consecutive days. Each scanconsists of ca. 90 points separated by ca. 10 cm to mapthe full 10 m of shielded region. Each point is a 20 s av-erage, corresponding to 50 acquisition cycles from thedigital multimeters19. Since the probe offsets are onlycalibrated to ±5 nT in the factory, we perform a customoffset calibration step between all scans. For this, wemeasure both the parallel and antiparallel components ofthe field and take their average to find the probe offset.

4

01234

Bx(nT)

−4−3−2−10

By(nT)

0 1 2 3 4 5 6 7 8 9 10−10123

Position z (m)

Bz(nT)

FIG. 5. Measurements of the residual magnetic field components along the shield’s symmetry axis. The magnitude of theresidual field does not exceed 4 nT over the inner 8 m with a reproducibility better than 1.5 nT between scans. The directionsx and y are transverse while z is longitudinal. The data consists of 570 points from 6 full length scans distributed over twoconsecutive days with magnetic equilibration between the scans.

This is better realized in low field regions to limit errorsdue to imperfect inversion of the sensitive axis direction.We observe that measured offsets are reproducible at theone nanotesla level and therefore attribute an uncertaintyof ±500 pT to the absolute magnetic field measurements.We note however that probe offsets can vary more sig-nificantly when cycling the sensor’s power or stressingreadout connectors.20 Finally, to account for the largevariability of the external field, up to 400 nT peak-to-peak on the 1 h scale, due to industrial activity in thesurrounding area, we also apply the equilibration proce-dure between each run. Figure 5 shows the measuredfield maps with a residual magnetic field below 4 nT overthe inner 8 m of the shielded region. The reproducibilityof the scans over 20 cm windows is better than 1.5 nTmax-to-min and better than 500 pT on the standard de-viation, demonstrating the robustness of the magneticequilibration procedure.

IV. APPLICATION TO PRECISION ATOMICACCELEROMETRY

For our application in precision atom interferometry,the leading systematic error associated with magneticfields is due to the Zeeman effect. For atomic states withzero magnetic quantum number, the linear component ofthe Zeeman effect vanishes and the potential seen by theatoms due to a longitudinal magnetic field profile B(z)is quadratic in the total field strength:

V (z) = −1

2hαB(z)2. (1)

Here, h is the reduced Planck constant and α

the atomic species’ clock transition Zeeman coefficient(57.5 GHz/T2 for the 87Rb atom21). Using perturbationtheory,22,23 we evaluate the phase φ of an atom interfer-ometer in the presence of an arbitrary but small pertur-bation potential V (z). To first order, we get

φ = φ0 −1

h

∮dt V (z0(t)) (2)

where φ0 is the phase of the corresponding unper-turbed interferometer and the integral spans over theoriented loop formed by the unperturbed classical tra-jectories z0(t). In the presence of a local gradient withno curvature and writing the magnetic field profile asB(z) = B0 + εb(z), ε 1, the local bias acceleration onatoms of mass m reads

δa = εhαB0

m

∂b(z)

∂z+O(ε2). (3)

We numerically calculate the magnetic field gradientfrom the data of figure 5. The resolution is limited bythe finite spatial sampling of the data. However, betweenall six scans presented in figure 5, the sensor positionswere not exactly reproduced, making the sampling grideffectively finer than 10 cm. Moreover, due to smooth-ness requirements of the magnetic field, we do not ex-pect spurious features to have been missed in the residualfield measurement and therefore interpolate the data lin-early and take the local slope as the local gradient. Overthe inner 8 m of shielded region, the local longitudinalmagnetic field gradient never exceeds 3 nT/m which cor-responds to a maximum local acceleration of 1.2 pm/s2

for 87Rb atoms when B0 = 1.5 µT. This effect can be

5

constrained further by applying the perturbation theoryresult of equation 2 since the integral over the unper-turbed classical trajectories smoothes local spikes in themagnetic field profile. For a simple drop mode operation,we find the bias for an interferometer spanning the inner8 m of shielded region to be smaller than five parts in1015 of the Earth’s local gravitational acceleration.

V. CONCLUSION

We reported on the design, construction, and char-acterization of a 10 m high-performance magnetic shieldwith application in Very Long Baseline Atom Interfer-ometry, achieving residual fields below 4 nT and longi-tudinal gradients smaller than 2.5 nT/m over the cen-tral 8 m. Owing to the use of pre-annealed permalloysheet material in an octagonal prism geometry and care-ful, stress-free assembly, the design is fully scalable inlength, effectively removing the need for a large scale hy-drogen furnace for annealing while improving the homo-geneity of the shielded region’s magnetic field by an orderof magnitude compared to previous work.12 This namelyopens shielding possibilities for ultra large scale experi-ments proposed to detect gravitational waves or searchfor exotic matter with atomic matter-waves24,25 wheremonolithic designs cannot be considered. For interfer-ometer geometries using the full length of the baseline,our shield leads to a Zeeman effect associated bias for87Rb atoms below five parts in 1015 of the Earth’s lo-cal gravitational acceleration. This enables a new classof absolute gravimeters for long-term gravity monitoringand reference networks.26 Finally, when comparing theacceleration of two different atomic species, this pavesthe way towards Galilean tests of the universality of freefall with atomic test masses beyond the 10−13 level.10

ACKNOWLEDGEMENTS

The Hannover Very Long Baseline Atom Interferome-try facility is a major research equipment funded by theGerman Research Foundation (DFG). This work is sup-ported by the Collaborative Research Centers 1128 “geo-Q” (project A02) and 1227 “DQ-mat” (project B07).D. S. acknowledges funding from the German FederalMinistry of Education and Research (BMBF) throughthe funding program Photonics Research Germany (con-tract number 13N14875). E. W. acknowledges supportfrom “Niedersachsisches Vorab” through the “Quantum-and Nano-Metrology (QUANOMET)” initiative (projectQT3). St. S. and P. F. acknowledge support by the DFGCluster of Excellence 153 “Origin and Structure of theUniverse”. The work of C. U. is supported by the Ger-man Aerospace Center (DLR) with funds provided bythe Federal Ministry for Economic Affairs and Energy(BMWi) due to an enactment of the German Bundestagunder grants no. DLR 50WM1556 and 50WM1956.E. W., D. T., E. M. R., and D. S. thank I. Fan andA. Schnabel for valuable insight at the beginning of theproject. E. W. thanks T. Wendrich, K. M. Knaak,

T. Hensel, and T. Rehmert for logistics help and the loanof measurement equipment.

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