Johnson-Nyquist Noise Effects in Neutron Electric-Dipole-Moment Experiments
N. J. Ayres, G. Bison, K. Bodek, V. Bondar, P.-J. Chiu, B. Clement, C. B. Crawford, M. Daum, S. Emmenegger, M. Fertl, A. Fratangelo, W. C. Griffith, Z. D. Gruji?, P. G. Harris, K. Kirch, P. A. Koss, B. Lauss, T. Lefort, P. Mohanmurthy, O. Naviliat-Cuncic, D. Pais, F. M. Piegsa, G. Pignol, D. Rebreyend, I. Rienäcker, D. Ries, S. Roccia, K. U. Ross, D. Rozpedzik, P. Schmidt-Wellenburg, A. Schnabel, N. Severijns, B. Shen, R. Tavakoli Dinani, J. A. Thorne, R. Virot, N. Yazdandoost, J. Zejma, G. Zsigmond
JJohnson-Nyquist Noise Effects in Neutron Electric-Dipole-Moment Experiments
N. J. Ayres, G. Bison, K. Bodek, V. Bondar, P.-J. Chiu,
1, 2, ∗ B. Clement, C. B. Crawford, M. Daum, S. Emmenegger, M. Fertl, A. Fratangelo, W. C. Griffith, Z. D. Gruji´c, P. G. Harris, K. Kirch,
1, 2
P. A. Koss, † B. Lauss, T. Lefort, P. Mohanmurthy,
1, 2, ‡ O. Naviliat-Cuncic, D. Pais,
1, 2
F. M. Piegsa, G. Pignol, D. Rebreyend, I. Rien¨acker,
1, 2
D. Ries, S. Roccia,
4, 13
K. U. Ross, D. Rozpedzik, P. Schmidt-Wellenburg, § A. Schnabel, N. Severijns, B. Shen, R. Tavakoli Dinani, J. A. Thorne, R. Virot, N. Yazdandoost, J. Zejma, and G. Zsigmond ETH Z¨urich, Institute for Particle Physics and Astrophysics, CH-8093 Z¨urich, Switzerland Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland Marian Smoluchowski Institute of Physics, Jagiellonian University, 30-348 Cracow, Poland Universit´e Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 38026 Grenoble, France Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky, 40506, USA Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany Laboratory for High Energy Physics and Albert Einstein Center forFundamental Physics, University of Bern, CH-3012 Bern, Switzerland Department of Physics and Astronomy, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom Institute of Physics Belgrade, University of Belgrade, 11080 Belgrade, Serbia Instituut voor Kern- en Stralingsfysica, University of Leuven, B-3001 Leuven, Belgium Normandie Univ, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France Department of Chemistry - TRIGA site, Johannes Gutenberg University Mainz, 55128 Mainz, Germany Institut Laue-Langevin, CS 20156 F-38042 Grenoble Cedex 9, France Physikalisch-Technische Bundesanstalt, D-10587 Berlin, Germany
Magnetic Johnson-Nyquist noise (JNN) originating from metal electrodes, used to create a staticelectric field in neutron electric-dipole-moment (nEDM) experiments, may limit the sensitivity ofmeasurements. In this study, we derive surface and volume-averaged root-mean-square normal noiseamplitudes at a certain frequency bandwidth for a cylindrical geometry. In addition, we model thesource of noise as a finite number of current dipoles and demonstrate a method to simulate temporaland three-dimensional spatial dependencies of JNN. The calculations are applied to estimate theimpact of JNN on measurements with the new apparatus, n2EDM, at the Paul Scherrer Institute.We demonstrate that the performances of the optically pumped
Cs magnetometers and
Hgco-magnetometers, which will be used in the apparatus, are not limited by JNN. Further, we findthat in measurements deploying a co-magnetometer system, the impact of JNN is negligible fornEDM searches down to a sensitivity of 4 × − e · cm in a single measurement; therefore, theuse of economically and mechanically favored solid aluminum electrodes is possible. I. INTRODUCTION
The search for a permanent electric dipole moment ofthe neutron (nEDM) has been an important topic in fun-damental physics research since the 1950s [1]. These ex-periments have been carried out by comparing the Lar-mor precession frequencies of neutron spins ( f n ) understatic uniform parallel and anti-parallel electric and mag-netic fields, using the Ramsey technique of separated os-cillating fields [2, 3]. For an accurate and precise mea-surement of f n , controlling the stability and uniformityas well as reducing the noise of the magnetic field inthe apparatus are of paramount importance. A potentialsource of magnetic-field disturbance is Johnson-Nyquistnoise (JNN) [4, 5] originating from the thermal motion ofelectrons in metal components within the experimentalapparatus.Johnson-Nyquist noise was originally observed as a“random variation of potential between the ends of a con-ductor” [4]. The same underlying effect, random ther-mal motion of charge carriers, also results in fluctua- ∗ Corresponding author: [email protected] † Present address: Fraunhofer-Institut f¨ur Physikalische Messtech-nik IPM, 79110 Freiburg i. Breisgau, Germany ‡ Present address: University of Chicago, 5801 S Ellis Ave,Chicago, IL 60637, USA. § Corresponding author: [email protected] tion of the electromagnetic field near a conductor. Mag-netic JNN is relevant in various research domains, allrelated to measurements with highest precision. Thefirst published numerical analysis of JNN came fromthe bio-magnetic measurements [6] using superconduct-ing quantum interference devices (SQUIDs). Johnson-Nyquist noise often exceeds the intrinsic noise of modernhigh-sensitivity detectors such as SQUIDs [6–9] and high-density alkali atomic magnetometers [10]. More recently,it was observed that these magnetic-near-field fluctua-tions induce spin-flip processes, and in turn are a crucialelement of decoherence in magnetic traps which limitsthe trapping lifetime of atoms [11–16]. In addition, relax-ations of spin states in the presence of magnetic fluctu-ations were also studied in the context of magnetic res-onance force microscopy and quantum computation [17].In EDM measurements, following the requirement of sen-sitivity enhancement, potential constraints from JNN re-ceived extensive attentions in the last few decades [18–21].Johnson-Nyquist noise from metal parts in high precisionexperiments impose a limit on the measurement sensitiv-ity. Quantifying the impact of JNN in the design of then2EDM experiment [22, 23] to search for an nEDM atthe Paul Scherrer Institute (PSI) in Villigen, Switzerland,with the PSI ultracold-neutron (UCN) source [24, 25] mo-tivates the presented research.As in the past, the challenges in measuring nEDM, d n ,are the increase of the statistical sensitivity and the cor-responding control of systematic effects. New sources of a r X i v : . [ nu c l - e x ] F e b HV electrodeground electrodeground electrodeCsM 12 cm40 cm 12 cm
FIG. 1: Cross-sectional drawing of the central storagechambers of the n2EDM apparatus located inside thevacuum tank. Included are only components relevant tothis study. For better visibility, the support structure ofthe Cs magnetometers (CsM) and the top UCN shutteris omitted. The electrodes are separated by polystyreneinsulating rings.UCN world wide [26, 27] improve the statistical sensitiv-ity, σ d n ∝ N − / , by increasing the number of neutronsavailable after storage, N . Nevertheless, owing to system-atic effects, e.g. random frequency shifts due to possiblemagnetic-field noise or drifts, the improvement in purecounting statistics might be compromised. This studyinvestigates the impact of JNN on nEDM experiments,focusing especially on the n2EDM spectrometer [22, 23]currently under construction at the PSI.The n2EDM apparatus features two cylindrical storagechambers, 12 cm in height and 40 cm in radius, stackedvertically. The two chambers share the central plane, anelectrode to which a high voltage up to 200 kV can beapplied. The top and bottom of the cylinder pair areclosed with grounded plates. A detailed description andschematics of the experimental apparatus can be foundin Figs. 1 and 2 of Ref. [22], whereas Fig. 1 shows a sim-plified drawing of the central storage chambers with thecomponents relevant to this paper. The three electrodesare made of aluminum plates, whose surfaces pointing to-wards the inside of the UCN storage chambers are coatedwith diamond-like carbon (DLC) [28, 29] in order to op-timize the UCN reflection properties.Thermal motion of charge carriers in the bulk alu-minum results in magnetic JNN, which might affectthe sensitivity of the magnetometers in its vicinity. Inthe following, we investigate the effects of JNN onthe cesium magnetometers (CsM) [30, 31], glass bulbsfilled with saturated Cs vapor positioned around theprecession chambers, the UCN and the mercury co-magnetometers (HgM), polarized
Hg atoms occupyingthe same volumes as the UCN in the chambers and readout by resonant light beams [32–34].
II. SPECTRUM OF MAGNETICJOHNSON-NYQUIST NOISE
The relevant magnetic JNN spectrum was first analyti-cally derived for research in biophysical applications [6, 8].There it is shown that for an infinite-slab conductor of thickness a and conductivity σ at a temperature T ,the normal component of the amplitude spectral den-sity (ASD) found in a distance d to the conductor surfacewithin a finite frequency interval ∆ f , with the z -axis de-fined to be perpendicular to the conductor surface, canbe written as [6] B z ( d, f ) = (cid:113) B z ( d, f ) / ∆ f = µ (cid:114) σk B Tπ (cid:20)(cid:90) ∞ R ( ρ, a, σ, f ) e − ρd ρ d ρ (cid:21) / , (1)where k B is the Boltzmann constant, ρ is the radialcomponent of the infinite conductor, and B z ( d, f ) is themagnetic-field amplitude normal to the surface at a givenfrequency f [35]. R ( ρ, a, σ, f ) in Eq. (1) is a function ofconductor properties, a and σ , at a given frequency f ,with the original expression defined in Eq. (39) of Ref. [6].For the horizontal components, due to symmetry andMaxwell’s equations, one can infer that B x = B y , and [6] B x ( d, f ) = B y ( d, f ) = 1 √ B z ( d, f ) . (2)All three components of the noise spectrum depend onthe normal distance between the point of interest and thesurface of the infinite slab. Figure 2 shows the normalASD of a 2.5-cm-thick aluminum infinite-slab conductor, σ = 3 . × S/m, under 20 ° C at various distances,using Eq. (1). The spectral line flattens at low frequenciestowards a constant value equal to the root-mean-square(RMS) limit for f →
0. With increasing frequency, thenoise amplitude decreases due to self-damping of high-frequency noise within the conductor slab. d = d = d = d =
10 cm f ( Hz ) ℬ z ( d , f )( f T / H z ) FIG. 2: Normal amplitude spectral density of a2.5-cm-thick aluminum ( σ = 3 . × S/m)infinite-slab conductor at various distances under 20 ° C.To verify these noise spectra, measurements of themagnetic-field noise created by an aluminum sheet werecarried out in the magnetically shielded room BMSR-2 at the Physikalisch-Technische Bundesanstalt (PTB),Berlin, Germany [36], using the 304-channel SQUID-vector-magnetometer system [37]. The system is based onnineteen modules placed on a hexagonal grid with eachcomprising of sixteen SQUIDs placed at various verticalplanes. A total of 304 SQUIDs each with a 7 mm ef-fective pickup-coil diameter permit to calculate all threevector components of the magnetic field on three mea-surement planes. The noise of a sheet made of 99.5%aluminum with dimensions of 1.3 m × × d = d =
35. mm d = d =
65. mm d = d = d = d = - f ( Hz ) ℬ z ( d , f )( f T / H z ) (a) d = d = d = d = d = d = - f ( Hz ) ℬ x ( d , f )( f T / H z ) (b) FIG. 3: (a) Vertical and (b) horizontal magnetic-fieldnoise component of a 0.5-mm-thick aluminum sheetmeasured by the SQUID system at PTB w.r.t. thetheoretical ASD at various distances. The spectra wereaveraged over 5 s samples from 300 s measurements.measured. In the first measurement, the sheet was placedto touch the flat bottom of the dewar cryostat, result-ing in a minimal distance of 27.5 mm between sampleand SQUIDs, due to the cold-warm distance of the de-war. Another measurement was carried out by adding awood plate in between aluminum foil and dewar to pro-vide an additional 7.5 mm distance to the dewar bottom.In each measurement, seven SQUIDs at different heightsfrom two central modules were used, with four measur-ing the vertical field component w.r.t. the probe and threemeasuring the horizontal field component.Figures 3a and 3b are the combined results from thetwo measurements, with or without the wood plate, withthe former showing the vertical field component and thelatter displaying the horizontal component. The spectraare results from 300 s measurements averaged over 5 ssamples. The background noise measured without alu-minum sheet was subtracted. The increase of noise below2 fT / √ Hz is due to the SQUID white noise. Mechanicalvibrations influence the measurement between 5-25 Hz atthe level of 10 fT / √ Hz. The spectra agree fairly to thetheoretical ASD calculated with Eqs. (1) and (2), with σ = 3 . × S/m and T = 22 ° C.For JNN studies in EDM experiments, a principle sim-plification can be made assuming that the frequency ofthe fluctuating field is low enough such that the eddycurrents generated in the bulk material can be neglected.This is stated as the static approximation by Lamore-aux [18], which is valid when the thickness of the conduc- tor is smaller than the skin depth λ = (cid:114) πµσf , (3)where f is the fluctuation frequency. In the context ofEDM experiments, this corresponds to approximately thereciprocal of the spin-coherence time, T . In the n2EDMexperiment, the free-spin-precession period for a singlemeasurement will be ∆ t ∼
200 s, approximately twotimes the spin-coherence time of mercury and a fractionof the spin coherence time of neutrons. Hence, we assume f − = ∆ t ∼
200 s ≈ T , Hg will be the free-spin-precessionperiod for a single measurement. At 5 mHz, λ ∼
116 cm,so the low-frequency assumption can be applied safely forconductors with a thickness of less than 10 cm.
III. MAGNETIC-FIELD FLUCTUATIONOBSERVED BY FIELD-SENSING PARTICLES
During an nEDM-measurement cycle,
Hg atomsoccupying the same volumes as UCN, are used as co-habiting magnetometers (HgM) [32–34]. As the HgM andthe UCN measure the magnetic field simultaneously, theratio of the two precession frequencies, f n /f Hg , is robustagainst magnetic-field changes. Nonetheless, the two spinspecies sample the magnetic field differently. The UCNsample the field adiabatically and have a negative center-of-mass offset whereas the mercury atoms sample the fieldnon-adiabatically [38]. For a nominal field of B = 1 µ T,we investigate the degree to which the effects of JNN arecanceled when taking the frequency ratio of two ensem-bles within one precession chamber.
A. Analytical derivation of spatial properties
In the first step to calculate the RMS magnetic-fieldnoise sensed by the particles, it is useful to derive thespatial correlation of JNN at different locations withinthe volume. For this purpose we calculated the magneticnoise originating from thermal noise currents by dividinga volume conductor into infinitesimal cuboidal elements,∆ V = ∆ x ∆ y ∆ z , similar to the seminal calculation inRefs. [6, 8]. There an equivalent current dipole for thevolume element is defined, whose component P α = I α ∆ α ( α = x, y, z ) in the direction α is the product of thisshort-circuit current and the finite size of the element.Following this concept, the source of thermal magneticnoise is represented by a great number of randomly ori-ented current dipoles on the surface of the conductor.We consider an infinite conductor and assume its sur-face is an x − y plane on the reference of the verticalcoordinate z = 0. A current dipole element on an in-finitesimal surface area d s located at ( x , x nowdenotes a two-dimensional vector on the x − y plane, iswritten as I ( x ) d s , where the z = 0 component is omittedfor simplicity. The magnetic field created by this dipoleat a point ( r , z ) can be calculated, according to the Biot-Savart law, as d B = µ π I ( x ) d s × k d , (4)where k = r − x d + zd ˆ e z (5)is the unit directional vector pointing from ( x ,
0) to ( r , z ),and d = (cid:113) | r − x | + z (6)is the distance between the dipole and the observationpoint. Now, we obtain the normal component ˆ e z of thefield d B z ( r , z ) = d B · ˆ e z = µ π (cid:18) I ( x ) d s × r − x d (cid:19) · ˆ e z = µ π (ˆ e z × I ( x ) d s ) · r − x d = µ π F ( r − x , z ) · I ( x ) d s, (7)with F ( x , z ) ≡ x (cid:16) | x | + z (cid:17) / (8)and I ( x ) ≡ ˆ e z × I ( x ) (9)being the rotated current component transformed fromthe triple product.
1. Variance of a disk-averaged field
Consider a disk parallel to the conductor, which hasa radius R and is located at a normal distance z abovethe conductor. The average normal magnetic field overthis disk generated by thermal noise in a finite elementd s from the conductor, can be written asd ¯ B z ( R, z ) ≡ (cid:90) S R d rπR d B z ( r , z )= µ π (cid:90) S R d rπR F ( r − x , z ) · I ( x ) d s = µ π I ( x ) d s · ¯ M ( x , R, z ) , (10)where ¯ M ( x , R, z ) ≡ (cid:90) S R d rπR F ( r − x , z ) (11)is the average over the disk. For an infinite conductor,we integrate over all dipoles¯ B z ( R, z ) = (cid:90) d ¯ B z ( R, z ) . (12)The variance of this surface average is then calculated as (cid:10) ¯ B z ( R, z ) (cid:11) = (cid:16) µ π (cid:17) (cid:90) d s (cid:90) d s (cid:48) (cid:10)(cid:0) I ( x ) · ¯ M ( x , R, z ) (cid:1) (cid:0) I ( x (cid:48) ) · ¯ M ( x (cid:48) , R, z ) (cid:1)(cid:11) . (13)As shown in Eq. (1) in Ref. [6], based on Nyquist’s the-orem, (cid:104) I ( x ) I ( x (cid:48) ) (cid:105) = 4 σk B T ∆ f aδ ( x − x (cid:48) ) , (14) where the conductor properties are identical to those indi-cated in Eq. (1). With the change of variables and furtherderivations, the variance of the surface-averaged field canbe expressed as (cid:10) ¯ B z ( R, z ) (cid:11) = C π z I (cid:18) Rz (cid:19) = (cid:10) B z ( r , z ) (cid:11) I (cid:18) Rz (cid:19) = (cid:10) B z (0 , z ) (cid:11) I (cid:18) Rz (cid:19) , (15)normalized to the variance of the single-point field at arandom location of distance z , (cid:10) B ( r , z ) (cid:11) = (cid:10) B (0 , z ) (cid:11) ,with C ≡ (cid:16) µ π (cid:17) σk B T ∆ f a. (16) I ( R/z ) is an integration over three two-dimensional vec-tors calculated as I ( ξ ) ≡ π ξ (cid:90) S d u (cid:90) S d v (cid:90) d X u − X (cid:16) ξ | u − X | + 1 (cid:17) / · v − X (cid:16) ξ | v − X | + 1 (cid:17) / . (17) u and v are two observation points on the disk, where anintegration over a unit circle S is performed, and X is adipole on the conductor integrated from zero to infinity.
2. Variance of a cylinder-averaged field
In our case, we are interested in the average field ob-served within a cylinder of radius R and height H on thesurface of the conductor,d ¯¯ B z ( R, H ) ≡ (cid:90) H d zH (cid:90) S R d rπR d B z ( r , z ) . (18)Physically, a direct contact between a dipole and an ob-servation point will result in a divergent magnetic field;hence, we regularize the integration by starting from asmall distance h ( h (cid:28) H, R ) to the conductor surfaced ¯¯ B z ( R, H ) ≈ d ¯¯ B z ( R, H, h )= (cid:90) h + Hh d zH (cid:90) S R d rπR d B z ( r , z )= µ π (cid:90) h + Hh d zH (cid:90) S R d rπR F ( r − x , z ) · I ( x ) d s = µ π I ( x ) d s · ¯¯ M ( x , R, H, h ) , (19)with ¯¯ M ( x , R, H, h ) ≡ (cid:90) S R d rπR ¯ F ( r − x , H, h ) (20)and ¯ F ( x , H, h ) ≡ (cid:90) h + Hh d zH F ( x , z ) . (21)Similarly, the contributions of all dipoles are integratedover ¯¯ B z ( R, H, h ) = (cid:82) d ¯¯ B z ( R, H, h ) and the variance ofthe volume-averaged field can be carried out as (cid:68) ¯¯ B z ( R, H, h ) (cid:69) = C π R J (cid:18) HR , hR (cid:19) , (22)where J ( η, ζ ) ≡ η (cid:90) S d u (cid:90) S d v (cid:90) d X ( η + ζ ) ( u − X ) | u − X | (cid:104) | u − X | + ( η + ζ ) (cid:105) / − ζ ( u − X ) | u − X | (cid:16) | u − X | + ζ (cid:17) / · ( η + ζ ) ( v − X ) | v − X | (cid:104) | v − X | + ( η + ζ ) (cid:105) / − ζ ( v − X ) | v − X | (cid:16) | v − X | + ζ (cid:17) / . (23)At the limit of H →
0, the variance of the volume averagereduces to the variance of the disk average at a distance h , which gives (cid:68) ¯¯ B z ( R, H → , h ) (cid:69) ≈ (cid:10) ¯ B z ( R, h ) (cid:11) . (24) B. Analytical derivation computed with MonteCarlo integration
The variances of surface and volume averages are im-portant for practical purposes. To calculate the corre-sponding results, integrals in Eqs. (17) and (23) were com-puted using the method of Monte Carlo integration [39].As described in Ref. [8], SQUID detectors used to mea-sure magnetic fields have pickup coils with finite sur-face areas within which the correlation of JNN shouldbe taken into account. Nenonen et al. [8] calculated themagnetic noise observed by a single circular coil of di-ameter d parallel to a conducting slab at a distance z , B coil n,z , and plotted the ratio to the single-point spectraldensity, B coil n,z /B n,z , as a function of d/z , shown in Fig. 4of Ref. [8]. B coil n,z and B n,z are the notations used in theoriginal paper of Nenonen et al. , where n in the subscriptstands for JNN. They are equivalent to ¯ B z ( R, z ) and B z ( r , z ) in our study, respectively. The ratio B coil n,z /B n,z is the same as I ( R/z ) / in Eq. (15). We computed thisintegral I ( R/z ), and compared it to the calculation madeby Nenonen et al. shown in Fig. 4. The black solid linein the graph is the result from Ref. [8]. The red dashedline is our result using the Monte Carlo integration, aver-aged over thirty random numerical solutions. The othercolored points were calculated with a numerical finite-element method which will be explained in Secs. III C andIII D. All methods agree with one another, and the re-maining small deviations are inconsequential for our prag-matic intent. R / z < B z ( R , z ) > < B z ( r , z ) > FIG. 4: Root-mean-square normal noise averaged over afinite area w.r.t. a random single point ( r , z ).Comparison among results obtained with Monte Carlointegration based on analytical derivation (red dashedline), calculation shown in Ref. [8] (black solid line) andnumerical finite-element method computed at variousdistances (colored points).As described in Eq. (24), with reduction of cylinderheight, the volume variance converges to the surface vari-ance. Figure 5 displays both (cid:68) ¯¯ B z ( R, H → , h ) (cid:69) / (red,square) and (cid:10) ¯ B z ( R, h ) (cid:11) / (blue, circular) for variouschamber radii. Integrations were performed using tento fifty random solutions in the Monte Carlo methodfor various radii, where the mean values are plottedwith their standard errors shown as error bars. Filleddata points were computed with a regularization distance h = 2 . h = 10 µ m. Both methods agree with each other whichconfirms the validity of the convergence of the volumecalculation to the surface solution in the limit of zerochamber height ( H → ∘ ∘ ∘∘ ∘ R ( cm ) < B z ( R , h ) > ( f T ) < B z ( R , H → , h ) > ( f T ) FIG. 5: Comparison of RMS normal noise amplitude ata frequency bandwidth ∆ f = 1 / (2 ×
200 s) betweensurface average (blue, circular) and volume average withan infinitesimal cylinder height (red, square). Tworegularization distances, h = 2 . µ m, areshown as filled or open data points, respectively.By integrating over a larger cylinder height, H , thevolume-averaged JNN decreases as a result of averag-ing over uncorrelated noise at relative larger distances.Results of (cid:68) ¯¯ B z ( R, H, h ) (cid:69) / with various H and R aredisplayed in Fig. 6. Again, filled and open points werecomputed with h = 2 . µ m, respectively. ∘ ∘ ∘ ∘ ∘∘ ♢ ♢ ♢ ♢ ♢♢ R =
40 cm R = R = ∘ ∘ ∘ ∘ ♢ ♢ ♢ ♢ - H ( cm ) < B z ( R , H , h ) > ( f T ) FIG. 6: Volume-averaged RMS normal noise amplitudeat a frequency bandwidth ∆ f = 1 / (2 ×
200 s) withvarious cylinder dimensions. Two regularizationdistances, h = 2 . µ m, are shown as filled oropen data points, respectively.From Figs. 5 and 6, one can see that the larger thecylinder volume ( R or H ), the smaller the influence ofthe regularization distance, h . C. Finite-element method with discrete dipoles
To estimate the JNN originating from the electrodes,instead of infinite slabs, conductors of finite size need tobe considered. Lee and Romalis [40] calculated magneticnoise from conducting objects of simple geometries. TheJNN calculation for a thin circular planar conductor isshown in Tab. VI of Ref. [40], which accords with the ge-ometry of the electrodes and was used in our study. Toestimate an upper limit of the average-field difference ob-served by UCN and HgM in the presence of JNN, it issufficient to apply the static approximation, where onlya white-noise spectrum at the limit of f → z component of theASD measured at a normal distance d generated by athin film of radius R , thickness a and conductivity σ ata temperature T is [40] [41]1 √ π µ √ k B T σad
11 + d R =: B thin z ( d, f → . (25)However, for a calculation of the magnetic field sam-pled by particles within the chamber, the JNN spectrumshown above, Eq. (25), which depends only on the nor-mal distance between the source of noise and the obser-vation point need to be replaced with a time-dependentthree-dimensional magnetic-noise source. For this reason,we used a supplementary method by considering a finitenumber of random magnetic dipoles on the surface of theconductors as noise sources.Although the n2EDM apparatus consists of a double-chamber, to study the possible cancellation of field fluc-tuation deploying a co-magnetometer system, we consid-ered only the field measurements taken place in one pre-cession chamber. In general, only the contribution from the two electrodes defining top and bottom of the relevantchamber needs to be considered, as the effect from metalplates further away is exponentially suppressed. Ran-dom time series were generated for each discrete dipoleon both electrodes. The superposition of all these timeseries and the applied field B at discrete positions withinthe volume of the chamber permits to calculate the dis-tinct magnetic field B ( r , t ) for any time and locations.Following the idea of equivalent current dipoles intro-duced initially in Refs. [6, 8] described in Sec. III A, wedivided the surface of the electrodes into triangular ar-eas. The motivation of using triangular grids instead ofcommon quadrilateral meshing methods will be explainedlater. For each triangular element, three noise-currentsources located at the center of the triangle and orientedalong the three Cartesian coordinates representing thenormal component of the three directions were created.At a time t , the magnetic field created by a number of n dip dipoles and measured at position r can be calculatedby the discrete Biot-Savart law, B ( r , t ) = µ π n dip (cid:88) i =1 (cid:88) α = x,y,z I α,i ( t ) d l × ( r − r (cid:48) i ) | r − r (cid:48) i | , (26)where r (cid:48) i and I α,i ( t ) are the position and current of anindividual dipole in direction α , and the unit-length vec-tor d l was defined to be the average side length of thetriangles.The white-noise ASD of the thermal current is [6, 8, 40] I = (cid:112) k B T σa, (27)having a unit of A/ √ Hz. By using the power spectraldensity (PSD = ASD ), the variance of the dipole-currenttime series can be calculated as σ ( I ( t )) = 2 I ∆ f BW = 2 k B T σa ∆ f BW , (28)where ∆ f BW is the bandwidth, corresponding to 1 / (2∆ t )with ∆ t being the free-spin-precession period of one mea-surement.The surface of the aluminum electrode was dividedinto approximately 1500 finite surface elements, whoseaverage side length was about 28 mm. The electrode-division layout was optimized to provide a theoreticallycompatible noise spectrum and to be computationally ef-ficient. A dipole with three Gaussian-distributed timeseries created from a defined variance lies at the centerof each surface element. With this set of time series us-ing ∆ t = 200 s for dipole currents on one electrode, thetime-domain magnetic-field distribution along a horizon-tal cut line, x = − . . .
40 cm within the diameter of thechamber is shown in Fig. 7, where (a)-(d) indicate fielddistributions at various distances from the electrode. Theshorter the distance to the source, the larger the ampli-tude of the field. Note that also the fluctuation of thefield is larger in close vicinity to the source. This vali-dates the argument in the beginning of this section, thata normal-distance dependent JNN spectrum, Eq. (25), isinadequate for the purpose of calculating the impact ofJNN on the sensitivity of field-sensing spin- particles,since the spatial correlation between adjacent observationpoints is not considered in the spectral-density formula-tion.This finite-element method was used to calculate thetime-and-volume-averaged magnetic field observed by - - B z ( d , s )( f T ) ( a ) d =
10 cm - - B z ( d , s )( f T ) ( b ) d = - - B z ( d , s )( f T ) ( c ) d = - - - -
10 0 10 20 30 40 x ( cm )- - B z ( d , s )( f T ) ( d ) d = FIG. 7: Normal component of time-domainmagnetic-field distribution at a certain time instantalong a horizontal cut line at various distances.field-sensing particles within the chamber over one mea-surement, in the presence of JNN. Monte Carlo simu-lations matching experimental results [42] show that thewhole volume of the chamber is sampled isotropically dur-ing one 200 s measurement in the case of a large numberof particles; therefore, it is sufficient to divide the cham-ber into equally-sized finite volumes, and calculate themagnetic fields observed at the center of each of theserectangular cuboids. A good balance between numericalaccuracy and computational efficiency was reached witha size of 10 × × for these voxels, which was se-lected as the optimal voxel size and used for the resultspresented below.The reason of using triangular meshes for the conduc-tor is to avoid aligned patterns between the conductorgrids and the voxels within the chamber volume. In apreliminary study, we found that an alignment betweenthe two meshing patterns could result in artifacts whichshould be avoided. D. Comparison between the analytical derivationand the finite-element method
With the finite-element method, the magnetic noise av-eraged over a disk or a cylinder can be easily estimated.With the optimal voxel height of 5 mm, the n2EDM pre-cession chamber was divided into 24 layers each consistsof 5024 pieces. First, we calculated the average fieldover different numbers of adjacent voxels on the samelayer, corresponding to radii ranging from 10 . . .
35 mm,w.r.t. a central piece, where 100 random central pieceswere selected. Results from four different layers, withdistances of 7 . . . . . (cid:104) B z (cid:105) ,was computed and shown in Fig. 8. Each entry in thehistogram is the result of one simulated cycle. For onefinite-element calculation, i.e. one simulated cycle, ap-proximately 1500 dipoles were created on the conduc-tor using three random noise currents at a bandwidth of∆ f BW = 1 / (2 ×
200 s). A total of more than 3000 ran-dom configurations were generated to accumulate statis-tics. The standard deviation of these random solutionsis σ Bz ∼ . R = 40 cm,the standard deviation of the volume average with twentyrandom solutions for the Monte Carlo integration is (cid:68) ¯¯ B z (40 cm , , . (cid:69) / = 2 . ± .
005 fT , (29)where the error is the statistical error of the Monte Carlosample. The results from the two methods agree withina femto tesla. The small deviation is negligible for ourpurpose and confirms the use of voxels for volume-averagecalculation with the finite-element method. - -
10 0 10 200100200300400 < B z > ( fT ) c oun t Entries: 3450Mean: - ( fT ) Std Dev: 3.060 ( fT ) FIG. 8: Volume average of normal JNN component (cid:104) B z (cid:105) over half of the n2EDM chamber calculated with thenumerical finite-element method. IV. EFFECTS ON THE n2EDM EXPERIMENTA. Magnetic fields observed by UCN and HgM
Due to the difference in the velocity spectrum and theLarmor precession frequency, UCN and HgM sample thevolume differently under a nominal 1 µ T B field. Muchfaster thermal Hg atoms fall into the non-adiabaticregime. The spins precess under a vectorial volume av-erage of the field; hence, the average magnetic field ob-served by
Hg atoms is calculated as (cid:104) B Hg (cid:105) = |(cid:104) B (cid:105)| = (cid:113) (cid:104) B x (cid:105) + (cid:104) B y (cid:105) + (cid:104) B + B z (cid:105) . (30)By contrast, due to the much smaller velocity and largerprecession frequency, UCN sample the volume in the adi-abatic regime, such that their spins precess under thevolume average of the modulus of the field. In addition,taking into account the negative center-of-mass offset (cid:104) z (cid:105) of the ensemble of UCN, the average field sampled byUCN is (cid:104) B UCN (cid:105) = (cid:104)| B | ρ UCN ( z ) (cid:105) = (cid:28)(cid:113) B x + B y + ( B + B z ) ρ UCN ( z ) (cid:29) , (31)where ρ UCN ( z ) = 1 H (cid:18) (cid:104) z (cid:105) H z (cid:19) (32)is the normalized vertical UCN density function.To estimate the time-and-volume average of the mag-netic fields over one precession chamber sandwiched be-tween two electrodes, the finite-element method was em-ployed. Figure 9 shows the average magnetic fields ob-served by UCN and HgM over one simulated cycle, cal-culated with (cid:104) z (cid:105) = − . t = 200 s time series simulated with approximately 1500dipoles created on both electrodes. The average magneticfields were computed with Eqs. (30) and (31). Histogramsfor UCN and Hg atoms are shown in Figs. 9a and 9b,respectively. The standard deviation of these distribu-tions are approximately equal, σ ( (cid:104) B UCN (cid:105) ) ≈ σ ( (cid:104) B Hg (cid:105) ) ∼ . ensembles bylooking at the difference of the average magnetic fields(see Fig. 10). The standard deviation of the differencesof average magnetic fields is σ ( (cid:104) B UCN (cid:105) − (cid:104) B Hg (cid:105) ) ∼ . σ JNN ∼ . σ d n = (cid:126) E γ n σ JNN = 4 × − e · cm , (33)assuming an electric field E = 15 kV / cm and γ n =29 .
16 MHz / T is the gyro-magnetic ratio of the neutron.The experiment will consist of M measurements to im-prove the statistical sensitivity. Note that the uncer-tainty on σ d n scales statistically with M − / . Resultsshown in Figs. 9 and 10 had been crosschecked with an-other 2000 random configurations which showed similarresults, confirming the negligibility of the statistical errorarising from the sampling size. - -
10 0 10 20050100150200 < B UCN >- B ( fT ) c oun t Entries: 3450Mean: 0.015 ( fT ) Std Dev: 3.781 ( fT ) (a) - -
10 0 10 20050100150200 < B Hg >- B ( fT ) c oun t Entries: 3450Mean: 0.013 ( fT ) Std Dev: 3.777 ( fT ) (b) FIG. 9: Deviations of time-and-volume-averaged field tothe nominal constant B magnetic field sampled by(a) UCN and (b) Hg atoms. - - < B UCN >-< B Hg > ( fT ) c oun t Entries: 3450Mean: 0.002 ( fT ) Std Dev: 0.097 ( fT ) FIG. 10: Difference between the average fields sampledby UCN and mercury ensembles, (cid:104) B UCN (cid:105) − (cid:104) B Hg (cid:105) . B. Magnetic field measured by CsM
The design of the n2EDM experiment [23] deploysmore than 100 CsM being installed above and belowthe precession-chamber stack in order to provide essen-tial information about the homogeneity and stabilityof the magnetic field. They are arranged radially ingroups of four on vertical modules. Each CsM containsa glass bulb filled with saturated vapor pressure of
Csatoms. They are operated as Bell-Bloom type [44] magne-tometers. Tensor-polarization (alignment) is created byamplitude-modulated linearly-polarized laser light thattraverses the bulb, at a frequency roughly matched withthe
Cs Larmor precession frequency, similar as inRefs. [45, 46]. Once the atomic vapor is spin aligned,the light intensity is reduced and kept constant. As thespin-polarized atoms precess under the influence of B with a frequency proportional to the magnitude of themagnetic field, the intensity of transmitted light is peri-odically modulated by precessing atoms and detected bya photodiode.Consider a CsM with a radius of 1.5 cm placed abovethe top-most electrode. Polarized Cs atoms at differ-ent locations within the bulb are exposed to magneticnoise from the electrode which decreases with distanceaccording to Eq. (1). The finite-element method intro-duced in Sec. III C was employed to calculate the averagemagnetic field measured by
Cs atoms in the presenceof JNN. For a CsM bulb, the measurement time of themagnetic field is δt = 70 ms, which is roughly two timesthe spin coherence time of Cs atoms. The skin depthat 14 Hz is 2.2 cm which is outside of the thickness rangein which the static approximation is valid. Nonetheless,the static approximation can be used to obtain an upperlimit for the field fluctuation.In this case, only the closest electrode which was rel-evant to a specific CsM was considered. Similarly, thenoise source was represented by a number of dipoles lyingon the surface of the electrode, each with three randomnoise currents at the bandwidth of ∆ f BW = 1 / (2 ×
70 ms).The bulb was divided into about 14000 voxels of size1 mm . The cesium atoms sense the field in the same wayas the mercury atoms in the precession chamber. Hence,the average magnetic field observed by a CsM was cal-culated by averaging over the fields in all voxels usingEq. (30). More than 3000 random dipole sets were sim-ulated. The average magnetic fields from JNN for fourCsM bulbs placed on one module with different distancesto the electrode were simulated. The corresponding stan-dard deviations of the time-and-volume-averaged fields atthese positions are shown as orange points in Fig. 11.In a perfectly spherical CsM bulb, the Cs atomsare uniformly distributed over the volume. Due to thefast movement of cesium atoms, the average magneticfield over the sphere is sampled homogeneously and itsvalue is equal to the field measured at the center basedon the mean-value theorem [47, 48], assuming all sourcesare outside the sphere. The RMS magnetic noise can beestimated by the noise observed at the center of the bulbwithin a time span δt . At a distance d measured to thecenter of the bulb, the RMS magnetic noise is B CsM i ( d, δt ) = (cid:40)(cid:90) δt B i ( d, f ) d f (cid:41) / , (34)with i being x, y or z . In the presence of an applied B (cid:107) B z field of about 1 µ T, the lateral components B x , B y (cid:28) B of JNN are quadratically suppressed, hencenegligible. For this reason, we only take the vertical com-ponent into account. The normal RMS magnetic noise estimated at the center of the CsM, B CsM z ( d, δt ), as afunction of distance, is also displayed in Fig 11.In the figure, both methods deliver similar resultswith small differences which can be understood by thefollowing explanations. Equation (34) is the frequency-bandwidth integrated RMS noise generated by an infiniteslab, whereas the finite-element method took a finite sizeof the electrode and only the low-frequency noise wasconsidered. Therefore, the results calculated from thefinite-element method will in principle be larger due tothe use of static approximation which is true for three ofthe cases. As for the result calculated at d = 16 . R = 55 cm, whichis larger than the electrode radius; hence, the effect ofnoise from the electrode will be smaller compared to thetheoretical calculation which used an infinite conductor.In general, this method provides a sufficiently precise es-timation of the impact of JNN on the measurements bythe CsM. d ( cm ) B z C s M ( d , m s )( p T ) FIG. 11: Comparison between RMS normal noiseamplitude integrated over 70 ms, calculated from thenoise spectrum (blue line), and the average-field noisemeasured by a CsM estimated with the finite-elementmethod (orange points).The sensitivity goal for n2EDM translates to a maxi-mum RMS noise of 2.7 pT in 70 ms for the CsM [23]. Theupper limits of the noise for CsM at various distances alllie below the sensitivity limit. In addition, for an nEDM-measurement cycle of 200 s (cid:29)
70 ms, the magnetic noiseseen by a CsM will be averaged out to a much lower value;hence, we confirm that JNN from the electrodes is negli-gible for the design and placement of all CsM within theexperiment.
V. CONCLUSION
This paper reports on a finite-element study ofJohnson-Nyquist noise (JNN) originating from the bulkmetal electrodes in the n2EDM experiment being con-structed by the nEDM collaboration at PSI. In the firstpart, we revisited the theoretical noise spectra [6, 8], andcompared them to the measurements on a thin aluminumsheet using a superconducting quantum interference de-vice (SQUID).Next, we derived for a given frequency bandwidth ex-pressions for the root-mean-square normal noise ampli-tudes of averages over a two-dimensional disk and a cylin-0der of finite volume. These are important in understand-ing the spatial correlation of JNN and are necessary forpractical purposes. Numerical results from the analyti-cal derivation were computed with the method of MonteCarlo integration and demonstrate good agreement withthe calculation performed in the literature [8].Using a discretization of the electrodes into a finitenumber of magnetic white-noise dipoles, we calculatedtemporal and spatial magnetic fields generated by JNN.By averaging these magnetic fields over time and vol-ume, we obtained the mean magnetic field sensed by pre-cessing ultracold neutrons (UCN) and mercury (
Hg)atoms. The standard deviation of 3000 randomly pro-duced configurations for UCN and mercury is 3.8 fT,which we consider as small enough for next-generationneutron electric-dipole-moment (nEDM) searches. Withthe same method, we found that for the cesium (
Cs)vapor magnetometers, the maximum RMS noise observedwithin a measurement time of 70 ms is approximately0.6 pT, which lies below the sensitivity goal of 2.7 pTfor n2EDM. Thus, we confirm that the precision of thecesium magnetometers will not be constrained by JNNfrom the aluminum electrodes.Additionally, by computing the average-field differenceobserved by UCN and mercury, we found that the noiseis sensed highly correlated and mostly cancels out by us-ing a co-magnetometer to normalize the UCN measure-ments. That is, the impact of JNN is negligible for nEDMsearches down to a sensitivity of 4 × − e · cm for asingle 200 s measurement. Assuming a projected experi-ment of 500 days with ∼
280 cycles per day, this resultsin a factor of 374 smaller limit, which is sufficiently smallfor our planned nEDM search using a co-magnetometer concept.
VI. ACKNOWLEDGMENTS
We would like to thank A. Crivellin and M. Spirafor helpful discussions. We are grateful for the tech-nical support from P. H¨ommen and R. K¨orber withthe material measurements in BMSR-2, PTB, Berlin.The material measurements inside BMSR-2 were sup-ported by the Core Facility “Metrology of Ultra-LowMagnetic Fields” at PTB funded by Deutsche Forshungs-gemeinschaft (DFG) through funding codes: DFG KO5321/3-1 and TR408/11-1. The swiss members ac-knowledge the financial support from the Swiss Na-tional Science Foundation through projects 157079,163413, 169596, 188700 (all PSI), 181996 (Bern), 172639(ETH), and FLARE20FL21-186179. This work hasalso been supported by the Cluster of Excellence “Pre-cision Physics, Fundamental Interactions, and Struc-ture of Matter” (PRISMA + EXC2118/1) funded byDFG within the German Excellence Strategy (ProjectID 39083149) from Johannes Gutenberg UniversityMainz. This work is also supported by Sigma Xi grants [1] J. H. Smith, E. M. Purcell, and N. F. 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