A General Method for Computing Thermal Magnetic Noise Arising from Thin Conducting Objects

A general method for computing thermal magnetic noise arising from thin conducting objects

a, a a a a a Joonas Iivanainen ∗, Antti J. Mäkinen , Rasmus Zetter , Koos C.J. Zevenhoven , Risto J. Ilmoniemi , Lauri Parkkonen

aDepartment of Neuroscience and Biomedical Engineering, Aalto University School of Science, FI-00076 Aalto, Finland

Abstract Thermal motion of charge carriers in a conducting object causes magnetic ﬁeld noise that interferes with sensitive measurements nearby the conductor. In this paper, we describe a method to compute the spectral properties of the thermal magnetic noise from arbitrarily-shaped thin conducting objects. We model divergence-free currents on a conducting surface using a stream function and calculate the magnetically independent noise-current modes in the quasi-static regime. We obtain the power spectral density of the thermal magnetic noise as well as its spatial correlations and frequency dependence. We describe a numerical implementation of the method; we model the conducting surface using a triangle mesh and discretize the stream function. The numerical magnetic noise computation agrees with analytical formulas. We provide the implementation as a part of the free and open-source software package bfieldtools.

1. Introduction Here, we outline a direct approach to compute the quasi- static frequency-dependent magnetic noise from a conduct- Thermal agitation of charge carriers in a conductor causes ing object which can be considered as a surface with a small a fluctuating voltage and a current referred to as Johnson– but possibly non-constant thickness. We examine the inter- Nyquist noise [1, 2]. The thermal current fluctuations in nal coupling phenomena associated with the surface currents the conductor are associated with a magnetic field that in- in order to determine the independent modes of the John- terferes with nearby magnetically sensitive equipment and son current [15]. We use a stream-function formalism similar measurements. Thermal magnetic noise can, e.g., limit the to a previous analytical calculation on an infinite conducting performance of sensitive magnetometers operating in con- plane [10] and to a semi-analytical computation on a layered ducting shields (e.g., [3, 4, 5]), impose constraints on fun- grid of square conducting patches [16]. The cross-spectral damental physics experiments [6, 7] and cause decoherence density of the magnetic noise can be computed based on in atoms trapped near conducting materials [8] as well as in the current fluctuations of the individual modes described high-resolution transmission electron microscopy [9]. It is by a set of Langevin equations; the fluctuation amplitudes therefore important to estimate the magnetic noise contribu- are given by the equipartition theorem [17]. Examination of tion from nearby conductors when designing sensitive exper- the individual modes gives an intuitive picture on the physics iments and devices. that determine the field noise characteristics. Thermal magnetic noise from conductors can generally We present a numerical implementation of the approach be calculated either using direct approaches where the field which uses a discretization of the stream function on a tri- noise is computed from the modeled noise currents and their angle mesh representing the surface. The implementation statistics (e.g., [3, 4, 6, 10]) or with reciprocal approaches is applicable for any conducting surface, including curved where the noise is obtained by computing the power loss in- ones. We demonstrate computations in example geome- curred in the material by a known driving magnetic field (e.g., tries and, where possible, compare the results with analyt- [5, 11]). In simple geometries analytical expressions for the ical formulas for verification. The implementation is freely magnetic noise can be obtained using either of the two ap- available as a part of the open-source Python software pack- proaches (e.g., [3, 4, 5, 6, 10]). In more complicated geome- age bfieldtools (https://bfieldtools.github.io; [18,

arXiv:2007.08963v1 [physics.comp-ph] 17 Jul 2020 tries, the noise has to be computed numerically. Numeri- 19]). cal methods using the reciprocal approach have been used to compute the frequency-dependent magnetic noise (e.g., [9, 12, 13]), while a method using the direct approach has 2. Theory been suggested to compute the low-frequency noise arising We consider the magnetic noise in a frequency range from thin conductors [14]. where the macroscopic Johnson thermal noise current is divergence-free ( ~J 0). In other words, the macroscopic ∇ · = charge density does not ﬂuctuate, but the current ﬂuctua- ∗Corresponding author Email address: [email protected] (Joonas Iivanainen ) tions are due to microscopic thermal motion of charge [10].

Preprint submitted to arXiv July 20, 2020 This allows us to use stream-function expression for the sur- above, and e(t) gives the electromotive force (emf) that is face current. coupled to the patterns. Equation system (6) is analogous to that of coupled RL-circuits, where s contains the circuit cur- 2.1. Stream function and surface current rents. However, we note that circuit quantities such as M and We shortly introduce stream function expression of the sur- R depend on the normalization of the circuit basis functions ~ face current and describe how it relates to physical quantities ki , whereas energy quantities such as power dissipation and such as power dissipation and inductive energy. Specifically, inductive energy are free of this ambiguity [21]. we assume a thin surface S with conductivity σ(~r ) and thickness d(~r ). A divergence-free surface-current density on S can 2.2. Magnetic Johnson–Nyquist noise be expressed with a stream function Ψ (units A/m) as (e.g., Next, we investigate how to model the magnetic Johnson– [18, 20, 21]) Nyquist noise using the stream-function approach. The ~J(~r ,t) Ψ(~r ,t) ~n(~r ), (1) thermal current fluctuations are driven by the Johnson emf, = ∇k × which is proportional to a zero-mean Gaussian white noise where ~n(~r ) is the unit surface normal and is the tangential ∇k process [17]. In this context, equations (6) are coupled gradient on the surface. We further express the stream func- Langevin equations. tion as a linear combination Ψ(~r ,t) P s (t)ψ (~r ), resulting = i i i To determine the statistics of the current fluctuations, we in the current density apply the equipartition theorem to the system [17]. Accord- X X ing to the theorem, in a thermal bath with temperature T ~J(~r ,t) si (t) ψi (~r ) ~n(~r ) si (t)~ki (~r ), (2) = i ∇k × = i each independent degree of freedom of the system has an average energy of kBT /2, with kB being the Boltzmann con- where ~ki (~r ) ψi (~r ) ~n(~r ) represent spatial patterns of stant. The independent degrees of freedom of the system are = ∇k × surface-current density (units 1/m) and si (t) their time- given by the eigenvectors of M as they diagonalize the energy dependent amplitudes (units A). The magnetic field can be (5). computed from the patterns using the Biot–Savart law We thus look for independent patterns~κi (~r ) with diagonal M as linear combinations of~k (~r ). We further require that the Z j µ ~r ~r 0 ~ 0 ~ − patterns ~κi (~r ) diagonalize R so that also the Langevin equa- B(~r ,t) J(~r 0,t) 3 dS0 = 4π S × ~r ~r 0 tions (6) decouple. As the inductance and resistance matri- Z | − | X µ0 ~r ~r 0 X ces are symmetric positive-definite for an ordinary conduc- ~ − ~ si (t) ki (~r 0) 3 dS0 si (t)bi (~r ), = 4π S × ~r ~r = tor [21], these independent patterns can be found, for exam- i | − 0| i (3) ple, by solving a generalized eigenvalue equation [20, 22], i.e., finding an invertible matrix V such that ~ where µ0 is the vacuum permeability and bi (~r ) is the mag- ½ T ~ V RV diag(ri ,...,rN ) netic field from the pattern ki with a unit amplitude. RV MVΛ T = (7) = ⇔ V MV diag(li ,...,lN ), The instantaneous power dissipation between patterns ~ki = ~ and k j is [18, 21] where Λ diag(λ ,...,λ ) is a diagonal matrix with λ = 1 N i = r /l . The independent patterns are given by the columns of Z 1 i i Pi j (t) si (t)s j (t) ~ki (~r ) ~k j (~r )dS si (t)s j (t)Ri j , the invertible but generally non-unitary matrix V as ~κi (~r ) = σ(~r )d(~r ) · = = S P V ~k (~r ). (4) j ji j We can transform Eq. (6) to the new basis: where Ri j is the mutual resistance between the patterns. Sim- d T 1 T 1 T ilarly, the instantaneous inductive energy between the pat- V MVV− s(t) V RVV− s(t) V e(t) 0. (8) dt + − = terns is given by the mutual inductance Mi j [18, 21] 1 T By defining s˜(t) V− s(t) and e˜(t) V e(t), we obtain a set of Z Z ~ ~ = = 1 µ0 ki (~r ) k j (~r 0) decoupled Langevin equations Ei j (t) si (t)s j (t) · dSdS0 = 2 4π S S ~r ~r | − 0| d 1 s˜i (t) λi s˜i (t) e˜i (t)/li 0. (9) si (t)s j (t)Mi j . (5) dt + − = = 2 Effectively, we now have a number of independent RL- The amplitudes of the patterns evolve according to a cou- circuits with time constants τi li /ri 1/λi driven by emfs pled system of equations ([21]; Appendix A) = = e˜i (t). d The Johnson emf has a white noise (frequency- M s(t) Rs(t) e(t) 0, (6) independent) power spectral density (PSD) S that can dt + − = e˜i be used to solve the PSD of s˜i from the decoupled Langevin where s is a vector containing the pattern amplitudes s[i](t) equation [17] = si (t), M and R are the mutual inductance and resistance ma- Se˜i 1 Ss˜i (ω) 2 2 , (10) trices with elements M[i, j] Mi j and R[i, j] Ri j defined = r 1 (ω/λ ) = = i + i 2 where ω is the angular frequency. The average energy (5) of where ~rl are the Ni integration points of the sensor i and th the i independent degree of freedom is: w~ i (~rl ) are their vector weights. The CSD between measure- Z ments yi and yk is then 1 2 1 1 ∞ Ei (t) li s˜i (t) li Ss˜i (ω)dω ® 〈 〉 = = 2 〈 〉 = 2 2π CSDy ,y (ω) F {yi }∗F {yk } 0 i k = 1 Se˜i Se˜i Ni Nk # „ l λ , (11) X X i 2 i w~ (~r )∗ CSD (~r ,~r ,ω) w~ (~r ). (17) = 2 4r = 8ri i l B~ l h k h i = l 1 h 1 · · = = where the brackets denote the ensemble average. On the 〈·〉 other hand, according to the equipartition theorem the aver- 3. Implementation age energy is E 1 k T , which can be used together with 〈 i 〉 = 2 B In this Section, we briefly outline the numerical implemen- equation (11) to solve the Nyquist formula for the PSD of the tation of the magnetic noise computation. The implemen- Johnson emf: tation is a part of the bfieldtools Python software pack- S ˜ (ω) 4k Tr , (12) ei = B i age [19] and uses its stream-function discretization as well as where ri is associated with the average power dissipation numerical integrals and functions to compute the resistance P (t) r j˜ (t)2 . and inductance matrices. The theoretical and computational 〈 i 〉 = i 〈 i 〉 To compute the cross-spectral density (CSD) of the mag- aspects of the software are presented in detail in Ref. [18]. netic noise due to the Johnson current, we note that the In bfieldtools, the conducting surface is represented Fourier transform of the field from the independent patterns by a triangle mesh and the stream-function basis Ψ(~r ) P = is obtained as i si hi (~r ) consists of piecewise linear functions (or "hat functions") h (~r ). The hat function value is one at the vertex i and ©X ª i F {B~(~r )}(ω) F s (t)~b (~r ) = i i zero at other vertices with linear interpolation on the triangle i faces. Each of these basis functions represents an elemen- ©X ª X F s˜ (t)β~ (~r ) F {s˜ }(ω)β~ (~r ), (13) = i i = i i tary current pattern which circulates around the correspond- i i ing vertex i. The magnetic field is obtained from the stream function s with a linear map (Eq. (3)). For example, the z- where β~i (~r ) denotes the magnetic field from~κi . The CSD be- i component of the field at N evaluation points is tween magnetic field components at ~r and ~r 0 along unit vectors ~n and ~n 0 is given by b Cs, (18) z = D E ~n F {B~(~r )}∗F {B~(~r 0)} ~n 0 where C is an N M matrix mapping the M vertex-circulating · · × D E currents (s[i] si ) to field component amplitudes at the eval- ¡X ~ ¢¡X ~ ¢ = ~n F {s˜i }∗βi (~r ) F {s˜k }βk (~r 0) ~n 0 uation points. = · · i k The resistance matrix R (with surface conductivity ³XX D E ´ ~n β~ (~r ) F {s˜ }∗F {s˜ } β~ (~r 0) ~n 0 σ(~r )d(~r ) discretized as constant on the triangles) and induc- = · i i k k · i k tance matrix M of the elementary current patterns can be # „ computed using the software. In the case of an open mesh, ~n CSDB~(~r ,~r 0,ω) ~n 0, (14) = · · the boundary conditions of the stream function are set as # „ P P described in Ref. [18]. Multiple separate conductors can where we defined CSD β~ (~r ) F {s˜ }∗F {s˜ } β~ (~r 0) as B~ = i k i 〈 i k 〉 k the CSD tensor of the magnetic field. be handled by computing the inductances between all the The CSD tensor can be simplified by noting that the am- patterns and by forming a block resistance matrix comprising the resistance matrices of the individual conductors. plitudes s˜i are independent: their temporal cross-correlation R We decouple the elementary circuits by solving the gener- is s˜i (t)s˜ (t t 0)dt 0 for i , k. For i k, the auto- k + = = correlation with exponential decay is given as the Fourier alized eigenvalue equation (7) for eigenvalues Λ and eigenvectors V using SciPy [23]. We then evaluate the CSD matrix transform of the PSD of Eq. (10). The CSD of s˜i and s˜k is Σb of the magnetic field component at ω using equation (18) thereby F {s˜i }∗F {s˜k } Ss˜ (ω)δik and the CSD tensor of the 〈 〉 = i as magnetic noise is T® T® T T T T Σb bz b CV s˜s˜ V C CVΣs˜V C , (19) # „ = z = = X ~ ~ where s Vs˜ and is a diagonal matrix with elements CSDB~(~r ,~r 0,ω) βi (~r )Ss˜i (ω)βi (~r 0). (15) Σs˜ = i = Σs˜[i,i] Ss˜i (ω) (Eq. (10)). = T We next describe how to compute the CSD between field We model the measurement yi in Eq. (16) as yi wi bi , T = measurements by an array of sensors. We approximate the where wi is a row vector comprising the sensor weights and b C s is a column vector of the magnetic noise along the measurement of the ith sensor yi (t) as a weighted sum of the i = i magnetic field over the spatial extent of the sensor directions of the vector weights at the integration points. The elements of the measurement CSD matrix can then be com- N Z Xi puted as follows yi (t) w~ i (~r ) B~(~r ,t)dV w~ i (~rl ) B~(~rl ,t), (16) = · ≈ · T T® T T T l 1 Σ [i,k] w b b w w C VΣ˜V C ,w . (20) = y = i i k k = i i s k k 3 A Mesh Noise-current patterns 1st 5th 10th 50th 100th 250th

N = 630

N = 1844

N = 5418

B 101 C 102 D E N = 630 z/R = 0.05 101 N = 1844 z/R = 0.10 1 10 N = 5418

100 100 101 100 휏 (ms) 1 noise (fT/rHz) 10 z B Relative error (%) N = 630 10 1 Relative error (%) N = 1844 10 2 Analytic 100 N = 5418 N = 5418 10 1 101 103 10 1 100 101 10 1 100 101 0 1000 2000 Mode index Distance (z/R) Distance (z/R) Number of modes

Figure 1: Comparison of the numerical solution of magnetic thermal noise of a circular conducting disk with radius R 1.0 m (centered on the x y-plane) to = an analytical formula. A: Three meshes with different numbers of triangles (N) representing the disk and exemplary contours of the numerically solved noise- current patterns. Blue and red contours depict current ﬂows in opposite directions. B: The time constants τ of the modes computed using the meshes. C–D: Comparison between the numerical and analytical solution of the low-frequency magnetic noise (Bz ) on the z-axis. C: Comparison between the numerical solution obtained using the densest mesh and the analytical formula. D: The relative errors of the numerical solutions to the analytical formula. E: Relative error as a function of number of current modes for the densest mesh.

In practice, we compute the cross-spectral densities using • B noise at the center of a spherical conducting surface multidimensional NumPy-arrays [24] and by summing over as a function of the sphere radius the relevant dimensions of the arrays. This way, we can, e.g., compute the cross-spectral density of the magnetic field in • B noise at the center of a cylindrical conducting surface 300 observation points at 100 frequencies and store the result along the long axis of the cylinder. in an array with dimensions of 300 300 3 3 100. × × × × The analytical formulas for these three cases can be found in the paper by Lee and Romalis [5]. Besides validation, we present other example computations. Unless stated other- 4. Validation and numerical results 7 1 1 wise, we used d 1 mm and σ 3.8 10 Ω− m− , corre- = = × sponding to aluminium at room temperature T 293 K. We first computed special case examples that allowed com- = paring our numerical computation of the magnetic noise to 4.1. Validation cases analytical formulas at the low-frequency limit. Specifically, we investigated the following: Figure 1 presents the computation of the low-frequency Bz noise along the z-axis due to a disk with a radius R 1.0 m = • Bz noise along the z-axis due to a uniform conducting centered on x y-plane. The disk was modeled with three dif- disk centered on the x y-plane ferent meshes with 630, 1 844 and 5 418 triangles. Fig. 1A 4 2 2 10 10 2 Analytic 10 z Numerical

101

0 R noise (fT/rHz) 10 noise (fT/rHz) z 1

z 10 B Infinite plane B 101 Disc

10 1 3-dB cutoff frequency (Hz) 101 102 10 1 100 Frequency (Hz) Distance (z/R) 0.2 0.4 0.6 0.8 1.0 R (m) Figure 3: The spectral density of the thermal magnetic noise Bz on the z- axis due to a circular conducting disk with radius R 1.0 m centered on the z 60 Analytic = x y-plane. Left: Spectral density a function of frequency and distance. The Numerical curves with different colors present the noise with different relative distances 50 2 m from the mesh (ranging from 0.05R to R). Right: The frequency at which the noise power has decreased by 3 dB from its zero-frequency value. The 40 solid line gives the cutoff frequencies for an inﬁnite plane calculated using an analytical formula.

noise (fT/rHz) 30 z

B A B 1 m 20 Bx

800 800 By /Hz) 2 Bz 1.0 0.5 0.0 0.5 1.0 600 600 /Hz)

z (m) 2 400 Figure 2: Numerical computation of low-frequency magnetic noise in- 400 side spherical and cylindrical conducting surfaces (represented as triangle PSD (fT Bx 200 meshes) and comparison to analytical formulas. Top: Magnetic noise in the 200 By center of the sphere with different sphere radii R. Bottom: Noise in the ﬁeld CSD to x = 0 m (fT Bz 0 component along the long axis (z) of the cylinder. The analytical formula 0 only applies in the center of the cylinder (z 0 m). 1 0 1 1 0 1 = x (m) x (m) C D shows examples of the stream-function contours of the nu- Near DC Bx By 800 50 Hz Bx Bz merically computed patterns of the noise current, while /Hz) 100 2 100 Hz By Bz Fig. 1B shows their time constants. At a relative distance of 600 500 Hz /Hz) z 0.05R, the relative error of the numerical solution of B 2 = z 0 noise to the analytical formula is 2.7% when the densest mesh 400 is used; with a larger distance, the relative error is smaller. CSD (fT 200 Higher-order modes contribute to the noise when the dis- 100 CSD to x = 0 m (fT tance is small (Fig. 1E). 0 Figure 2 shows the numerical results for the low-frequency 1 0 1 1 0 1 magnetic noise inside a closed sphere (2 562 vertices; 5 120 x (m) x (m) triangles) and a cylinder (3 842; 7 680). The computation and Figure 4: Magnetic noise cross-spectral density (CSD) along the x-axis (z = analytical formula agree in both cases, with relative errors of 0.1R) due to a a circular conducting disk with radius R 1.0 m centered on = 0.06% and 0.03% in the case of the sphere and the cylinder, the x y-plane. A: Low frequency noise power spectral density along the x- axis. B: Low-frequency noise CSD to x 0 m. C: Noise cross-spectral density respectively. = of Bz to x 0 m at different frequencies. The inset shows the amplitude- = normalized cross-spectral density. D: Noise CSD between different compo- 4.2. More examples nents of the magnetic ﬁeld.

Next, we examined the magnetic noise and its frequency dependence using a simple conductor. We computed the Bz an infinite planar conductor [3] is also shown. At small rela- noise on the z-axis as well as the magnetic noise CSD along tive distances to the disk (z 0.1R), the numerical 3-dB fre- < the x-axis due to a circular conducting disk centered on the quencies scale as those for an infinite plane. At distances x y-plane (R 1 m; mesh with 5 418 triangles, Fig. 1A). comparable to the radius z R, the 3-dB frequency is con- = ≈ The spectral density of Bz noise due to the disk is shown in stant suggesting contribution from a single mode with the Fig. 3. The same figure also shows the estimated frequency largest time constant. Fig. 4 shows examples of cross-spectral at which the PSD is reduced by three decibels from the zero- density of magnetic noise calculated on the x-axis. 1 frequency value. The 3-dB cutoff frequency (4µ0σdz)− for We then investigated the magnetic noise due to a planar 5 conductor with a star shape (1 442 vertices, 2 702 triangles). Conflicts of interests Fig. 5 illustrates the noise-current patterns on the conductor and the magnetic noise spectral density at different per- The authors declare no conflicts of interest. pendicular distances from the conductor. At small relative distances, the magnetic noise spectral density has a spatial References structure that resembles the shape of the conductor. At larger distances, the magnetic noise loses the structural detail, re- [1] J. B. Johnson, “Thermal agitation of electricity in conductors,” Physical flecting different fall-off distances of the field noise compo- review, vol. 32, no. 1, p. 97, 1928. nents that correspond to the noise-current modes with dif- [2] H. Nyquist, “Thermal agitation of electric charge in conductors,” Phys- ical review, vol. 32, no. 1, p. 110, 1928. ferent levels of detail. [3] T. Varpula and T. Poutanen, “Magnetic field fluctuations arising from Last, as a practical example, we computed the low- thermal motion of electric charge in conductors,” Journal of Applied frequency magnetic noise CSD seen by a superconducting Physics, vol. 55, pp. 4015–4021, 1984. quantum interference device (SQUID) array (102 magne- [4] J. Nenonen, J. Montonen, and T. Katila, “Thermal noise in biomag- netic measurements,” Review of scientific instruments, vol. 67, no. 6, tometers; MEGIN Oy, Helsinki, Finland). We investigated two pp. 2397–2405, 1996. geometries where the magnetometer array was either near [5] S.-K. Lee and M. V. Romalis, “Calculation of magnetic field noise from an aluminium plate or inside a closed cylindrical aluminium high-permeability magnetic shields and conducting objects with sim- shield (Fig. 6). The aluminium had a thickness of 5 mm and ple geometry,” Journal of Applied Physics, vol. 103, no. 8, p. 084904, 2008. was at room temperature in both cases. The magnetometers [6] S. Lamoreaux, “Feeble magnetic fields generated by thermal charge were modelled as point-like for simplicity. fluctuations in extended metallic conductors: Implications for electric- The computed low-frequency noise spectral CSD in the dipole moment experiments,” Physical review A, vol. 60, no. 2, p. 1717, 1999. SQUID array is presented in Fig. 6. Compared to the intrinsic [7] C. T. Munger Jr, “Magnetic johnson noise constraints on electron elec- noise level of these commercial SQUID sensors ( 3 fT/ pHz), tric dipole moment experiments,” Physical Review A, vol. 72, no. 1, ∼ the magnetic noise is significant in both cases. p. 012506, 2005. [8] C. Henkel, “Magnetostatic field noise near metallic surfaces,” The Euro- pean Physical Journal D-Atomic, Molecular, Optical and Plasma Physics, vol. 35, no. 1, pp. 59–67, 2005. 5. Conclusion and outlook [9] S. Uhlemann, H. Müller, J. Zach, and M. Haider, “Thermal magnetic field noise: Electron optics and decoherence,” Ultramicroscopy, We presented a method to compute the (cross) spectral vol. 151, pp. 199–210, 2015. [10] B. J. Roth, “Thermal fluctuations of the magnetic field over a thin con- density of magnetic thermal noise due to an arbitrarily ducting plate,” Journal of Applied Physics, vol. 83, no. 2, pp. 635–638, shaped conductor that can be considered a surface, i.e., thin 1998. compared to the distance to the observation points. We [11] J. Clem, “Johnson noise from normal metal near a superconducting have made the implementation openly available (and con- SQUID gradiometer circuit,” IEEE Transactions on Magnetics, vol. 23, no. 2, pp. 1093–1096, 1987. tributable) as a part of the open-source Python software [12] J.-H. Storm, P. Hömmen, D. Drung, and R. Körber, “An ultra-sensitive package bfieldtools. The numerical approach allows vi- and wideband magnetometer based on a superconducting quantum sualization of the noise-current patterns, providing an intu- interference device,” Applied Physics Letters, vol. 110, no. 7, p. 072603, itive perspective on the physics. We validated the numerical 2017. [13] J. Storm, P. Hömmen, N. Höfner, and R. Körber, “Detection of body implementation by comparing the results to analytical for- noise with an ultra-sensitive SQUID system,” Measurement Science and mulas, and found agreement within 1%. The accuracy in- Technology, vol. 30, no. 12, p. 125103, 2019. ∼ creased with the number of triangles in the discretized sur- [14] H. J. Sandin, P. L. Volegov, M. A. Espy, A. N. Matlashov, I. M. Savukov, and L. J. Schultz, “Noise modeling from conductive shields using Kirch- face. hoff equations,” IEEE transactions on applied superconductivity, vol. 21, We presented examples where we calculated the noise no. 3, pp. 489–492, 2010. from a single conducting surface, but we note that the com- [15] J. Harding and J. Zimmerman, “Quantum interference magnetometry putations apply similarly for a system comprising multiple and thermal noise from a conducting environment,” Physics Letters A, vol. 27, no. 10, pp. 670–671, 1968. separate conductors that are inductively coupled. Skin ef- [16] A. Tervo, Noise optimization of multi-layer insulation in liquid-helium fects are not currently included, but could be possibly incor- cryostat for brain imaging. Master’s thesis, Aalto University, 2016. porated by dividing the conductor to multiple inductively- [17] D. T. Gillespie, “The mathematics of Brownian motion and Johnson coupled layers each of which has a thickness smaller than the noise,” American Journal of Physics, vol. 64, no. 3, pp. 225–240, 1996. [18] A. J. Mäkinen, R. Zetter, J. Iivanainen, K. C. Zevenhoven, L. Parkkonen, skin depth as in Ref. [26]. and R. J. Ilmoniemi, “Magnetic-field modeling with surface currents: Physical and computational principles of bfieldtools,” arXiv preprint arXiv:2005.10060, 2020. Acknowledgments [19] R. Zetter, A. J. Mäkinen, J. Iivanainen, K. C. Zevenhoven, R. J. Il- moniemi, and L. Parkkonen, “Magnetic-field modeling with surface currents: Implementation and usage of bfieldtools,” arXiv preprint This work has received funding from the European Union’s arXiv:2005.10056, 2020. Horizon 2020 research and innovation programme under [20] G. N. Peeren, “Stream function approach for determining optimal grant agreement No. 820393 (macQsimal). The content is surface currents,” Journal of Computational Physics, vol. 191, no. 1, pp. 305–321, 2003. solely the responsibility of the authors and does not necessar- [21] K. C. Zevenhoven, S. Busch, M. Hatridge, F.Öisjöen, R. J. Ilmoniemi, and ily represent the official views of the funding organizations. J. Clarke, “Conductive shield for ultra-low-field magnetic resonance 6 1 m z = 0.1 mz = 0.2 mz = 0.3 mz = 0.4 mz = 0.5 m

1.16 m y Bx x

By Magnetic noise (fT/rHz)

Figure 5: Magnetic thermal noise due to a star-shaped planar conductor. Left: The triangle mesh representing the conductor and the numerically computed thermal current patterns on the conductor. The time constant of the pattern decreases from left to right, top to bottom. Blue and red contours depict current ﬂows in opposite directions. Right: Low-frequency noise spectral density at different vertical distances to the conductor. The plot limits are the same as the size of the conductor shown in A.

A B

100 cm

200 cm

40 cm 40 cm 100 cm

Noise CSD (fT2/Hz) Noise spectral density Noise CSD (fT2/Hz) Noise spectral density (fT/rHz) (fT/rHz)

18.0 32.8 16.0 32.0 14.0 31.2 12.0 30.4 10.0 29.6 8.0 28.8 6.0 28.0

Figure 6: Thermal magnetic noise CSD at low frequencies in a SQUID array helmet comprising 102 magnetometers (red arrows) in two geometries: the helmet near an aluminium plate (A) and the helmet inside a closed cylindrical aluminium shield (B). The noise spectral density is plotted as topographic 2D projection of the sensor-array geometry (generated using the MNE software [25]). The aluminium is at room temperature and has a thickness of 5 mm in both cases.

imaging: Theory and measurements of eddy currents,” Journal of ap- et al., “SciPy 1.0: fundamental algorithms for scientiﬁc computing in plied physics, vol. 115, no. 10, p. 103902, 2014. Python,” Nature methods, vol. 17, no. 3, pp. 261–272, 2020. [22] G. H. Golub and C. F.Van Loan, Matrix computations, vol. 3. JHU press, [24] T. E. Oliphant, A guide to NumPy, vol. 1. CreateSpace Independent Pub- 2012. lishing Platform, 2015. [23] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, [25] A. Gramfort, M. Luessi, E. Larson, D. A. Engemann, D. Strohmeier, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, C. Brodbeck, L. Parkkonen, and M. S. Hämäläinen, “MNE software for

7 processing MEG and EEG data,” Neuroimage, vol. 86, pp. 446–460, 2014. [26] H. Sanchez Lopez, F.Freschi, A. Trakic, E. Smith, J. Herbert, M. Fuentes, S. Wilson, L. Liu, M. Repetto, and S. Crozier, “Multilayer integral method for simulation of eddy currents in thin volumes of arbitrary geometry produced by MRI gradient coils,” Magnetic Resonance in Medicine, vol. 71, no. 5, pp. 1912–1922, 2014.

Appendix A. Matrix equation

Here, we briefly present the derivation of the matrix equation (6) using the stream-function representation of the surface- current density (2). We divide the electrical surface current ~J(~r ,t) to two components as ³ ´ ~J(~r ,t) σ(~r )d(~r )E~(~r ,t) σ(~r )d(~r ) E~ (~r ,t) E~ (~r ,t) = = F + s ³ ∂~A(~r ,t) ´ σ(~r )d(~r ) E~s (~r ,t) , (A.1) = − ∂t + where E~F is the Faraday-inductive field given as the negative time derivative of the magnetic vector potential ~A and E~s is the source field. The source field represents the active components responsible for the currents in the conductor, while the inductive electric field is due to the magnetic field generated by the currents and represents their inductive coupling. The source field can be, e.g., due to an external time-varying magnetic field (E~ ∂~A /∂t) or due to a combination of mi- s = − s croscopic thermal current fluctuations ~J f and their associated macroscopic electric field (E~ ~J /σd E~ ). s = f + f By reordering the terms and expressing the vector potential using the current density, Eq. (A.1) reads

Z ∂ µ0 ~J(~r 0,t) ~J(~r ,t) dS0 E~s (~r ,t) 0. (A.2) ∂t 4π S ~r ~r + σ(~r )d(~r ) − = | − 0| We consider a frequency range where the charge density does not ﬂuctuate ( ~J 0); the current density can be expressed ∇ · = with the stream function: Z ~ ~ ∂ X µ0 kk (~r 0) X kk (~r ) sk (t) dS0 sk (t) E~s (~r ) 0. ∂t 4π S ~r ~r + σ(~r )d(~r ) − = k | − 0| k (A.3) ~ By taking a dot product with kl (~r ) and integrating over the surface, we have

Z Z ~ ~ ∂ X µ0 kl (~r ) kk (~r 0) sk (t) · dSdS0 ∂t 4π S S ~r ~r k | − 0| Z ~ ~ Z X kl (~r ) kk (~r ) ~ sk (t) · dS kl (~r ) E~s (~r ,t)dS 0, + k S σ(~r )d(~r ) − S · = (A.4) from which the resistance and inductance matrix elements can be identiﬁed to get the equation system

X ∂ Mlk sk (t) Rlk sk (t) el (t) 0, (A.5) k ∂t + − = where e (t) R ~k (~r ) E~ (~r ,t)dS is the source emf coupled to l = S l · s pattern l.