aa r X i v : . [ phy s i c s . c l a ss - ph ] S e p Magnetic noise around metallic microstructures
Bo Zhang and C. Henkel
Institut f¨ur Physik, Universit¨at Potsdam, 14469 Potsdam, Germany (Dated: 28 Aug 2007)We compute the local spectrum of the magnetic field near a metallic microstructure at finite temperature. Ourmain focus is on deviations from a plane-layered geometry for which we review the main properties. Arbitrarygeometries are handled with the help of numerical calculations based on surface integral equations. The mag-netic noise shows a significant polarization anisotropy above flat wires with finite lateral width, in stark contrastto an infinitely wide wire. Within the limits of a two-dimensional setting, our results provide accurate estimatesfor loss and dephasing rates in so-called ‘atom chip traps’ based on metallic wires. A simple approximationbased on the incoherent summation of local current elements gives qualitative agreement with the numerics, butfails to describe current correlations among neighboring objects.
PACS numbers: 44.40.+a Thermal radiation -05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
I. INTRODUCTION
Thermal motion of charge carriers in a metallic object cre-ates a randomly fluctuating magnetic field in the object’svicinity. These fields are relevant for many applications likehigh-precision measurements of biomagnetic signals , nu-clear magnetic resonance microscopy , and miniaturized trapsfor ultra cold atoms . The original purpose of Purcell’s influ-ential 1946 paper was to point out that these magnetic fieldshave a spectral density that by far exceeds the Planck law forblackbody radiation at low frequencies. In fact, if only black-body fields were present, magnetic dipole transitions betweenatomic or nuclear levels would never happen on laboratorytime scales. The near fields sustained by material objects (thatplay the roles of sources and cavity) give the dominant con-tribution. Indeed, these near fields contain non-propagating(evanescent) components that are thermally excited as welland that dominate over free space radiation . Phrased in an-other way, the dipole transition rate is enhanced because theelementary excitations in the metal provide additional decaychannels .We focus in this paper on accurate calculations of mag-netic field noise that are able to describe objects of arbi-trary shape. Such objects occur, for example, in magneticmicrotraps where complex networks of metallic wires createelectromagnetic potentials with typical scales in the micronrange . The behaviour of the field spectrum, as the metallicgeometry is changed, is far from intuitive.The spectral densityincreases with the material volume for small structures, butthis trend saturates as soon as the typical scale gets larger thanthe penetration length (skin depth) of the fields in the material.It has even been found that a thin metallic layer can produceless noise than a half-space, depending on the ratios betweenobservation distance, layer thickness, and skin depth .Experiments in the field of biomagnetism have shown signifi-cant changes when a metallic film is cut, at constant volume,into stripes . Calculations for these cases necessarily requirenumerical methods to describe the propagation of magneticfields both in vacuum and inside a metallic structure. This isthe main topic of this paper. We also discuss previously de-veloped approximations for planar structures and within themagnetostatic regime where analytical calculations are possi- ble. Our numerical methods are restricted here to two spatialdimensions (2D) where efficient solutions of Maxwell equa-tions can be found with the help of boundary integral equa-tions . Our approach can also be combined with anyother numerical method for field computations, permitting tocover three-dimensional cases as well.The results we find can be summarized as follows. A planarstructure (infinite lateral size) creates equal magnetic noise forall components of the magnetic field vector. This does not ap-ply in three dimensions, but is specific to the two-dimensionalsetting we focus on here. Finite metallic objects show a stronganisotropy: the noise occurs preferentially along ‘azimuthal’directions circling around the object. Increasing the amountof metallic material does not always give larger noise, in par-ticular when the geometrical size becomes comparable to theskin depth. We find qualitative agreement with measurementsof Ref.1 where thermal field fluctuations are reduced when ametallic object is split into disconnected pieces. The surfaceimpedance approximation, that provides an accurate descrip-tion of metallic reflectors for far-field radiation, is shown tobe not reliable for observation distances shorter than the skindepth. Finally, qualitative (albeit not quantitative) agreementis obtained between our numerical data and an approxima-tion based on the incoherent summation of fields generated bythermal current elements filling up the metallic volume. Thismethod has been used in the interpretation of previous exper-iments . Our results are, to our knowledge, the first quan-titative test of this approximation in a nontrivial geometry.The paper is organized as follows. We first review the linkbetween the thermal radiation spectrum and classical dipoleradiation (Sec.II). Planar structures are analyzed in Sec.IIIusing the angular spectrum representation. We demonstratein particular the isotropy of the magnetic noise spectrum anddiscuss the accuracy of the surface impedance approximation.Sec.IV is devoted to our numerical scheme and to the resultsfor single and multiple objects of rectangular shape. II. MAGNETIC NEAR FIELD NOISEA. Local noise power
The fluctuations of the thermal magnetic field B ( r , t ) arecharacterized by their local power spectral density (the Fouriertransform of the autocorrelation function) B ij ( r ; ω ) = Z d τ e i ωτ h B i ( r , t ) B j ( r , t + τ ) i . (1)Higher moments are not needed for our purposes, and the av-erage field (at frequency ω ) vanishes as is typical for thermalradiation. We assume the field to be statistically stationaryso that the spectrum (1) does not depend on t . The diago-nal tensor components B ii ( r ; ω ) give the spectrum for a givencartesian component B i ( r ) or polarization direction. Previ-ous work has shown a strong dependence on the position r near a metallic microstructure: power laws being the typi-cal behaviour in the frequency range where the wavelength λ = 2 πc/ω is much larger than the typical distances. For thetemperature dependence, see Eq.(3) below. Our parametersof interest are: normal metallic conductors with temperaturesabove a few K, r in the micron range and λ of the order ofcentimeters or larger ( ω/ π ≤
10 GHz ). This upper limit onfrequency corresponds to the strong magnetic dipole transi-tions in typical alkali atoms.) In this regime, the frequencydependence of the noise spectrum is weak and occurs via thematerial response (permittivity ε ( ω ) ). A characteristic lengthscale is the field penetration length (skin depth) δ defined by δ = 2 πλ Im p ε ( ω ) = q µ ωσ ( ω ) (2)where σ ( ω ) is the conductivity and µ the vacuum perme-ability. The second expression is based on the Hagen-Rubensapproximation Im ε ( ω ) ≈ σ ( ω ) / ( ε ω ) ≫ | Re ε ( ω ) | . Withinthe Drude model for a metal, this is verified at frequenciesmuch smaller than the charge carrier relaxation rate (in the s − range at room temperature). For highly conductingmaterials (Au, Ag, Cu), this results in a skin depth of the orderof µ m( ω/ π MHz) − / .To compute the magnetic correlation spectrum, we use thefluctuation-dissipation theorem which is valid at thermal equi-librium (temperature T ) B ij ( r ; ω ) = 2¯ h e ¯ hω/k B T − G ij ( r , r ; ω ) (3)where G ij ( r , r ′ ; ω ) is the Green function for the magneticfield, i.e., the field generated at r by a point magnetic dipolelocated at r ′ and oscillating at the frequency ω , B i ( r , t ) = G ij ( r , r ′ ; ω ) µ j e − i ωt + c . c . . This is actually a familiar result:the imaginary part of tr G ( r , r ; ω ) gives the local density ofmagnetic field modes, and the temperature dependent prefac-tor in Eq.(3) their average occupation number. The basic ben-efit of this formula is that it holds also for the full correlationtensor and even near material objects that absorb the field orgenerate thermal radiation. Generalizations to the nonequilib-rium case exist (fields produced by a ‘hot object’ surrounded by a ‘cold’ environment) , but are not needed for our pur-poses (see remarks in Sec.V). We also note that the temper-ature dependent prefactor in Eq.(3) reduces to k B T /ω for T ≥ . . In this limit, the order of field operators in thecorrelation function (1) becomes irrelevant. (The order wehave adopted yields the rate of a magnetic dipole transition i → f with energy difference E f − E i = ¯ hω .) B. Magnetic dipole radiation
We are thus led to solve the following electrodynamic prob-lem: find the complex magnetic field amplitude B ( r ; ω | µ ) created by a monochromatic point dipole µ ( t ) = µ e − i ωt +c . c . located at position r ′ . We then compute G ij ( r , r ′ ; ω ) = ∂B i ( r ; ω | µ ) ∂µ j (4)In the limit r → r ′ this field becomes the singular ‘self field’and requires a cutoff in wavevector space. Its imaginary partis cutoff-independent, however, and given by Im B ( r ′ ; ω ) = µ ω µ / (6 πc ) (in three-dimensional free space).The field B = B ( r ; ω | µ ) can be found from the vectorpotential A that solves the inhomogeneous Maxwell equation ∇ × ∇ × A − k ε ( r ) A = µ ∇ × µ δ ( r − r ′ ) , (5)where k = ω/c . The right-hand side is the current densitycorresponding to the magnetic dipole. There is no free chargedensity and we work in the gauge E = i ω A .We now focus on the following geometry (fig.1, right): theposition r ′ of the source (i.e., where the magnetic noise spec-trum is actually needed) is located in vacuum, and the metal-lic microstructures are filling a domain D where Im ε ( r ; ω ) = σ ( r ; ω ) / ( ε ω ) is nonzero (and large). metalvacuum:: j(r) B (r’) metal: B (r’) σ (r) µ DDD’ vacuum: D’ FIG. 1: Sketch of the considered geometry: (left) current fluctua-tions in a microstructure generate magnetic field fluctuations B ( r ′ ) at a position position r ′ outside it. (right) The magnetic noise spec-trum is calculated from the magnetic field radiated by a point mag-netic dipole µ located at r ′ . D and D ′ : domains where the conduc-tivity σ ( r ; ω ) is nonzero or zero, respectively. The outside domain is called D ′ . There, the vector po-tential satisfies an inhomogeneous Helmholtz equation withwavenumber k . All length scales we consider (distancedipole–microstructure d , object size) are much shorter thanthe wavelength so that k is actually very small and can beneglected in a first approximation. This is the magnetostaticregime. (The finite value of k is, of course, at the origin ofthe nonzero magnetic LDOS in free space.) We cannot makethe magnetostatic approximation in D because there, we havea wavenumber k p ε ( r ; ω ) = (1 + i) /δ ( r ) , and the (local)skin depth δ ( r ) is one of the characteristic length scales athand. The fields in the domains D and D ′ are connected bythe usual matching conditions: the components of A tangen-tial to the boundary are continuous, and B is continuous (thematerial is non-magnetic).Eq.(5) provides a unique solution subject to the boundarycondition that at infinity, the field behaves like an outgoingwave. In three [two] dimensions, this corresponds to a vec-tor potential proportional to e i k s /s [ e i k s / √ s ] in the freespace domain D ′ when the distance s = | r − r ′ | → ∞ tothe source becomes large compared to λ . In the magnetostaticlimit k → , the free space asymptotics is actually neverreached at finite distances. The relevant boundary condition isthen the same as for the scalar potential of an electric dipole:the vector potential goes to zero like /s [like /s ] in three[two] dimensions, respectively.Since we deal with a metallic object with | ε | ≫ , itis tempting to perform the calculation based on the surfaceimpedance boundary condition. The latter links the tangen-tial components of magnetic field and vector potential by B t = − i ωZA z , where ωZ = (1 + i) /δ . Note that this isa local relation that can only hold if the scale of variation ofthe fields on the object surface is much larger than the skindepth δ . In the present study, a point-like source illuminatesthe object with its near field [ A bulk ( r − r ′ ) in Eq.(7)], andthis field shows a typical extension of the order of the object-source distance d . The surface impedance approximation ishence expected to break down for d ≪ δ . We shall confirmthis explicitly for the planar structures discussed in the follow-ing Sec.III. III. RESULTS: LAYERA. Two dimensions
In this paper, we focus on a two-dimensional (2D) geome-try to simplify the numerical calculations described in Sec.IV.The magnetic moment is chosen in the computational plane(the xy -plane), as shown in Fig.1. Adapting the wave equa-tion (5) to two dimensions, we find that the vector potentialhas a single nonzero component that points out of the plane.We then work with a scalar function A ( r ) = A ( x, y ) thatsolves ∇ A + k ε ( r ) A = µ ( µ y ∂ x ′ − µ x ∂ y ′ ) δ ( r − r ′ ) (6)In a homogeneous medium (‘bulk’), the solution with the ap-propriate boundary conditions is A bulk ( r − r ′ ) = i µ µ y ∂ x ′ − µ x ∂ y ′ ) H ( k √ ε | r − r ′ | ) (7)where H is the Bessel function of the third kind (Hankelfunction), usually denoted H (1)0 = J + i Y . ¿¿From this, we get the magnetic field by taking the ‘curl’, B x = ∂ y A , B y = − ∂ x A . The resulting self field in free space is Im B ( r ′ | µ ) = µ k µ (8)provided the dipole µ is real. In the magnetostatic limit, thisfield is negligibly small. The bulk solution (7) then goes overinto A bulk ( r − r ′ ) ≈ − µ π ( x − x ′ ) µ y − ( y − y ′ ) µ x | r − r ′ | (9)This equation describes the field with which the dipole ‘il-luminates’ the sample. Note that it is scale-free: the typical‘spot size’ on the microstructure is only determined by thedistance d between dipole and top surface. B. Reflected field
In this section, we consider that the boundary of themedium is the plane y = 0 ; the field at the source point r ′ = (0 , d ) is then related to the (Fresnel) reflection coeffi-cients from the surface. We expand the solution to Eq.(6) inplane waves (wavevector k parallel to the boundary) and haveabove the medium ( y > ): A ( x, y ) = µ ( µ x ∂ y ′ − µ y ∂ x ′ ) + ∞ Z −∞ d k π e i k ( x − x ′ ) κ × (cid:16) e − κ | y − y ′ | + r ( k )e − κ ( y + y ′ ) (cid:17) (10)where κ = p k − k . (The square root is chosen such that Re κ ≥ .) The coefficient r ( k ) describes the reflection of thefield from the medium boundary. It is given by the Fresnelformula r ( k ) = r half space ( k ) ≡ κ − κ m κ + κ m , κ m = p k − /δ (11)for a medium with skin depth δ filling the half-space y < .For a layer (thickness h ) on top of a substrate, we have r layer ( k ) = r top + r bottom e − κ m h − r top r bottom e − κ m h (12)where r top = r half space is given by Eq.(11) and r bottom de-scribes the reflection from the layer–substrate interface. It isgiven by Eq.(11) with the replacements κ κ m , κ m κ s =( k − ε s k ) / where ε s is the substrate permittivity.All the relevant information for the magnetic noise poweris contained in the reflection coefficient r ( k ) . In fact, whenthe integral in (10) is performed and the imaginary part taken,it turns out that the reflected waves (second term) dominateover the free space contribution (first term) by at least a fac-tor λ δ/d ≫ . This is connected to the fact that the rel-evant wavenumbers k for our problem are of the order of / ( y + y ′ ) = 1 / (2 d ) which is much larger than k . We canhence apply the approximation κ ≈ | k | . The reflection coef-ficient (11) for the metallic half-space then depends only onthe parameter kδ . For the metallic layer geometry, we fo-cus for simplicity on a substrate whose conductivity is muchsmaller than in the metal. The influence of the substrate hasbeen studied in Ref.19: already a ratio of 10 to 100 betweenthe substrate and layer conductivities is sufficient to make thesubstrate behave like vacuum. We then have r bottom ≈ − r top in Eq.(12). C. Polarization dependence
Let us analyze first the dependence on the orientation ofthe source dipole. If µ is perpendicular to the medium (only µ y = 0 ), the reflected field is given by B y ( r | µ y ) = µ µ y ∞ Z −∞ d k π k κ r ( k )e i k ( x − x ′ ) e − κ ( y + y ′ ) (13)The limit r → r ′ yields an imaginary part Im B y ( r ′ | µ y ) = µ µ y Im Z d k π k κ r ( k )e − κd . (14)Repeating the calculation for a parallel dipole, we find for Im B x ( r ′ | µ x ) the same expression as Eq.(14), and conse-quently the noise spectrum is isotropic, B xx = B yy . This is aremarkable property of a laterally infinite structure in 2D. (In3D, the polarization perpendicular to a planar interface has anoise power twice as large as the parallel polarization .) Weshow below that a significant polarization anisotropy arisesabove a metallic wire of finite width. D. Wavevector dependence
The reflection coefficient (12) is plotted in Fig.2 for typi-cal layers. Consider first a thickness larger than a few skindepths. One observes a maximum value of its imaginary part(relevant for the magnetic LDOS) when the decay constant κ ≈ k is matched to /δ . This is confirmed by an asymptoticanalysis whose results are given in Table I. (See Refs.20,21for details on the asymptotic expansion.) One of the two lim-iting cases (namely k ≪ k ≪ /δ ) corresponds preciselyto the surface impedance approximation where the reflectioncoefficient (11) is approximated by r ( k ) ≈ − | k | δ (15)Here, the skin depth is much smaller than the lateral periodand the field barely penetrates into the material. Fig.2 andTable I show strong deviations in the opposite regime k ≫ /δ that is relevant at distances d ≪ δ .Consider now a layer much thinner than the skin depth.¿¿From Fig.2, different regimes can be read off that are sep-arated on the k -axis by the scales h/δ ≪ /h , as can beseen in Fig.2. It is worth noting that for small k , thin layers δ I m r( k ) thickthinh = δ / FIG. 2: Reflection coefficients (11, 12) for thin and thick metalliclayers. We plot the imaginary part only. The dashed lines representthe formulas of Table I. The wavenumber is scaled to the inverse skindepth /δ . For the thick layer, h = ∞ . We take the conductivity ofgold at room temperature and a frequency ω/ π ≈ . in allfigures. This leads to the value δ = 71 µ m and a vacuum wavelength λ/δ ≈ . × . k ≪ /δ /δ ≪ k Im r half space ( k ) kδ k δ k ≪ /δ h/δ ≪ k ≪ /h /h ≪ k Im r layer ( k ) Im − kδ kδ /h hkδ k δ TABLE I: Asymptotic approximations to the reflection coefficientsfrom a half-space and a layer. We distinguish between thin (thickness h ≪ δ ) and thick layers ( h ≥ δ , ‘half space’). The first and secondcolumns (thin layer) overlap in an intermediate k -range (see Fig.2).The magnetostatic limit k ≪ k is taken throughout. These formulasare plotted as dashed lines in Fig.2. show even larger losses [ Im r ( k ) ] than thick ones; the maxi-mum is shifted towards the smaller value k ∼ h/δ and hasa larger amplitude. This behaviour has been recognized be-fore in magnetic noise studies in the kHz range . In the in-frared range, it is also well known that the absorption by ametallic layer can be optimized at a specific thickness. (See,e.g., Ref.22 for incident far-field radiation where | k | ≤ k .)Conversely, for a given thickness h and dipole distance d , themagnetic noise power shows a maximum as the skin depth ischanged . This ‘worst case’ occurs when the characteristicwavevector /d is matched to h/δ . E. Distance dependence
The asymptotics in k -space translate into power laws forthe dependence of the magnetic power spectrum B ii ( d ; ω ) on distance d , as shown in Fig.3. In fact, the integrand inEq.(14) peaks around k ∼ / (2 d ) , and the result of the in-tegration is determined, to leading order, by the behaviourof r ( k ) in this range. We thus find the power laws summa-rized in Table II and visible in Fig.3. We use as convenient @ skin depth D - - no r m a li z edno i s epo w e r thin thick FIG. 3: Local magnetic noise power B ii ( d ; ω ) vs. distance frommetallic layer in double logarithmic scale (2D calculation). Topcurve [blue]: thick layer; bottom curve [red]: thin layer. The dashedlines give the leading order power laws of Table II. The thick curvesarise from the numerical integration of Eq.(10), the thin curves arean interpolation formula described in the text.The magnetic noise power is isotropic above a planar structure in2D (the perpendicular and parallel field components have the samepower). It is scaled to µ k B T / ( ωδ ) , and the distance is scaled tothe skin depth δ . Thin [thick] layer: h = 0 . δ [ δ ]. d ≪ δ δ ≪ d B ii, half space ( d ) log( δ/d )2 π δ πd d ≪ h ≪ δ h ≪ d ≪ δ /h δ /h ≪ d B ii, layer ( d ) log( h/d )2 π h πd δ πhd TABLE II: Power laws for the magnetic noise spectrum in two di-mensions above a half space and a thin metallic layer (dashed linesin Fig.3). The noise spectrum is given in units of µ k B T / ( ωδ ) . unit in all the plots the noise level µ k B T / ( ωδ ) . Normal-ized to blackbody radiation (in 2D free space), this level is (2 / ( k δ )) ∼ . × at . for gold at room temper-ature, a striking illustration of the Purcell effect . A commontrend is that the magnetic noise power increases as the metal-lic medium is approached. As the distance d is getting muchsmaller than the thickness h , thin and thick layers behave thesame, as expected. At larger distances, but still smaller thanthe skin depth, the noise power is proportional to the volumeof metallic material, hence to the layer thickness . Thistrend is reversed for d > δ p δ/ h where thin layers give alarger noise level than thick ones .A reasonably accurate approximation that interpolates be-tween these power laws can be found by performing the k -integral using the asymptotic formulas of Table I in their re-spective domains of validity. The result is a sum of incompletegamma functions Γ( n, x, x ′ ) ( n = 0 , , ) whose argumentsare, for example, x ∼ dh/δ , x ′ ∼ d/δ , or d/h (thin lines inFig.3 and Appendix A). We have checked that the asymptoticsof the gamma function reproduce the power laws summarizedin Table II. There are regimes where the sub-leading termsgive significant corrections, in particular in the transition re- gions between the power laws.Finally, the surface impedance approximation gives a mag-netic noise that is represented in Fig.3 by the dashed lineclose to the ‘thick layer’ for d > δ . The agreement withthe full calculation in this range is expected: the ‘illuminat-ing field’ is getting more and more uniform on the scale of theskin depth. At shorter distances, the surface impedance ap-proximation severely overestimates the noise level because itcannot describe properly the field variations on scales smallerthan δ . For the thin layer, the conventional surface impedanceapproach gives a wrong result even if d > δ because topand bottom surfaces do not decouple from each other. Thiscan be repaired using effective (thickness-dependent) surfaceimpedances, see, e.g., Ref.24 and citations therein. IV. RESULTS: FINITE SIZE OBJECTS
We now describe numerical calculations that we have per-formed to assess the importance of the finite lateral size of themetallic structure. This is particularly relevant, for example,in atom chips where a continuous metallic layer is etched todefine wires that can be addressed with different currents .It is actually desirable to minimize the amount of metallic ma-terial, leaving just a few wires to create the fields for atomtrapping. In fact, it has been argued that the magnetic noisepower roughly scales with the metallic volume as long as thecharacteristic distances are smaller than the skin depth . Forlaterally finite structures, this claim as well as other calcu-lations have been based so far on approximate methods thatfail to reproduce even the planar layer to within a factor oftwo or three . The numerical results we describe hereare a first step towards an accurate estimate of magnetic noisepower near structures of finite size.
A. Boundary integral equations
Within the assumption of near field radiation being in equi-librium with the metallic object, we compute the noise powerfrom the magnetic Green function in Eq.(3). The magneticfield radiated by a point source and reflected by the objectsolves the wave equation (6). We reformulate the wave equa-tion in terms of boundary integral equations. This has beendescribed elsewhere , and we quote only the basic for-mulas here. Our unknowns are the nonzero component A ofthe vector potential and its normal derivative F = ∂A/∂n ≡ n · ∇ A on the object surface S , where n is the outward unitnormal to S . Both quantities are continuous (actually, F isequal to the tangential magnetic field) and can be found fromthe system of integral equations A ( r ) = 2 A bulk ( r − r ′ ) − I d a ( x ) (cid:20) G ( r − x ) F ( x ) − ∂G ∂n ( r − x ) A ( x ) (cid:21) (16) A ( r ) =2 I d a ( x ) (cid:20) G ε ( r − x ) F ( x ) − ∂G ε ∂n ( r − x ) A ( x ) (cid:21) (17)where A bulk ( r − r ′ ) is given in Eq.(7). Both the observationpoint r and the integration points x are taken on the objectboundary S here, d a ( x ) ) being the surface element. We usethe scalar Green functions [see Eq.(7)] G ε ( r ) = i4 H ( k √ ε | r | ) . (18)If we would take the magnetostatic limit, G ( r ) →− (2 π ) − log | r | , the Green functions in vacuum and in themedium would differ (in sub-leading order) by a constant,leading to inconsistencies. We avoid this by retaining the fi-nite value of k even for the vacuum Green function. Theintegrals (16, 17) are to be understood as principal values tohandle the singularities of the Green functions as x → r . Wediscretize them on a finite element decomposition of the objectboundary, as described in Ref.12. The resulting linear systemis solved with standard numerical tools. Once the fields A , F are known on the surface, the reflected field at the sourceposition ( r ′ ) can be found from A ref ( r ′ ) = − I d a ( x ) (cid:20) G ( r ′ − x ) F ( x ) − ∂G ∂n ( r ′ − x ) A ( x ) (cid:21) (19)Note that G and ∂G /∂n are both essentially real here (theimaginary parts scale with k ). The magnetic noise, via Im B ( r ′ ) , is thus determined by the imaginary parts of A and F on the object boundary. This is not surprising since theinduced current density is σE = i ωσA . B. Single wire
We have solved the integral equations for rectangular wiresof thickness h and width w . In a first step, we have validatedour numerical scheme by comparing flat, wide wires ( w ≫ h )to the infinite layer results of Sec.III. Typical plots are shownin Fig.4 where the magnetic noise power (symbols) is plot-ted vs. the distance d above the wire centre. Good agree-ment with the analytical results for an infinitely wide wire(solid lines) is only obtained at short distance, where for ge-ometrical reasons the wire appears wider. At distances above µ m , the deviations start to grow. In all the plots, we takea skin depth δ = 70 µ m . The slow convergence in the limit w → ∞ can be attributed to the long-range behaviour of thefields; this is more pronounced in two dimensions comparedto three. Note in particular the strong splitting between thetwo polarization directions for the thick wire that does not oc-cur above an infinitely wide wire in 2D (Sec.III C). Interest-ingly, the y -component shows more noise above a thick wirewhile this tendency is reversed above a thin wire. This polar-ization anisotropy could provide a tool to improve the lifetimein a magnetic trap: one orients the static trapping field paral-lel to direction of the strongest noise. (In fact, trap loss andspin flips are induced by magnetic fields perpendicular to thestatic trap field.) The choice of a trapping field along the weaknoise direction is favorable if one wants to reduce the dephas-ing rate of the trapped spin states (generated by fluctuationsof the Larmor frequency, see Ref. ). distance [ µ m] no r m a li z ed no i s e po w e r B ii thickthin x + wire (Bxx, Byy)layerincoh. summ. FIG. 4: Magnetic noise spectra B ii ( d ) vs. distance d above the centreof thin and thick metallic wires. Symbols + ( × ): numerical calcula-tion for the B yy ( B xx ) component (see Fig.1). Solid lines: infinitelywide wire (layer), as computed in Sec.III. Dashed lines: incoherentsummation (thin layer only, upper curve: B xx ), see Sec. IV C. Thinwire: width and thickness × µ m ; thick wire: × µ m .The skin depth is δ = 70 µ m . Another finite-size effect is shown in Fig.5 where the posi-tion is varied parallel to the top surface of a thin wire. Abovethe centre of wide wires, the noise levels are nearly constant(not shown). Beyond the wire edges, one observes a sharpdrop in B xx , with a characteristic scale fixed by the distance.The y -component shows a broad maximum near the edge thatis more pronounced for narrow wires. This is due to a gradu-ally changing direction of maximum noise that is ‘azimuthal’with respect to the object, as expected for magnetic fields gen-erated by currents flowing perpendicular to the computationalplane (see inset of Fig.5). We find the direction of maximumnoise by looking for the eigenvectors of the × matrix B ij .This matrix can be shown to be symmetric (reciprocity), andthe eigenvectors that are not orthogonal very near to the wire’scorner, are an artefact of our numerical method that convergesvery slowly at these points.In Figs.6, 7, the thickness of the wire is changed with theobservation point remaining above the centre. We observean approximately linear increase with the width that saturatesslowly. We also note that B xx (left) levels off faster than B yy (right). The difference between Fig.6 and Fig.7 is the distanceof observation: at short distance (Fig.6), the largest widthsshow a noise power fairly close to the planar layer limit (cf.the symbols at the right end). At distances comparable to theskin depth (Fig.7), the deviations from the planar layer limit(symbols) are still large. Note also that the noise has droppedin amplitude and that the increase with width is slower. C. Incoherent summation
This behaviour can be qualitatively understood using the‘incoherent summation’ approximation developed in Ref.23: lateral position [ µ m] no r m a li z ed no i s e po w e r B ii Bxx 20 µ m × µ mByy 20 µ m × µ mBxx 20 µ m × µ mByy 20 µ m × µ m edgeedge d = 10 µ m lateral position [ µ m] no r m a li z ed no i s e po w e r B ii Bxx 1 µ m × µ mByy 1 µ m × µ m d = 3 µ medge FIG. 5: Magnetic noise spectra B ii ( d ) vs. lateral position, at a fixeddistance d . The arrows mark the edges of the wires. Symbols × [ + ]:spectrum B xx [ B yy ] parallel [perpendicular] to the top face of thewire. Skin depth: δ = 70 µ m .Left panel: thickness and width are × µ m (wide wire) and × µ m (narrow wire). Distance d = 10 µ m . Right panel: thick-ness and width are × µ m , distance d = 3 µ m (see dotted line ofinset). Inset: illustration of anisotropic noise near the wire edge. Thecrosses are oriented along the polarization vectors that show max-imum and minimum noise, the ‘arm lengths’ being proportional tothe rms noise. The magnetic field noise is dominantly azimuthal,with field lines circling around the wire. The dotted line ( d = 3 µ m )shows the positions scanned through in the right panel. no r m a li z ed no i s e po w e r B xx width [ µ m]small distance (d < δ )thickthinxxx numericsincoh. summ. no r m a li z ed no i s e po w e r B yy width [ µ m]small distance (d < δ ) thickthin+++ numericsincoh. summ. FIG. 6: Magnetic noise spectra vs. the width of a rectangular wire.(left) x -polarization, parallel to the top face; (right) y -polarization.Symbols: numerical calculations; solid lines: incoherent summationapproximation (see Sec.IV C). The symbols on the right margin givethe values for an infinitely wide wire (layer). The observation pointis located above the wire centre, at a distance d = 10 µ m . Thewire thickness is µ m (thin) and µ m (thick). Skin depth: δ =70 µ m . width [ µ m] no r m a li z ed no i s e po w e r B ii large distance (d > δ ) thickthinxxx Bxx+++ Byy FIG. 7: Same as Fig.6, but at an observation distance d = 75 µ m .Results from the incoherent summation are not shown, as theystrongly deviate. distance [ µ m] no r m a li z ed no i s e po w e r B xx µ m × µ m20 µ m × µ m3 wires incoh.summ distance [ µ m] no r m a li z ed no i s e po w e r B yy µ m × µ m3 wires 20 µ m × µ mincoh.summ FIG. 8: Noise power generated by three wires, as a function of dis-tance (see inserted sketch, with the dashed line illustrating the ob-servation points). The wires have a quadratic cross section µ m × µ m and are separated by a gap of µ m . The noise is measuredabove the center of the central wire. Left panel: horizontal polar-ization, right panel: vertical polarization. Symbols: numerical re-sult; solid line: incoherent summation. For comparison is shown:a single wire of same cross section (dashed line) and a wide wire µ m × µ m with approximately the same volume (dash-dottedline). Skin depth δ = 70 µ m . the metallic volume is broken into mutually incoherent pointcurrent elements whose magnetic fields are computed withinmagnetostatics and neglecting the presence of the metallic ob-ject. We give the resulting formulas for two dimensions inAppendix B; the integrals are solved by special functions fora wire with rectangular cross section. The solid lines in Fig.6demonstrate that incoherent summation gives a reliable ap-proximation if the skin depth is the largest length scale (nottrue for the thick wire). The noise power always increaseswith the metallic volume within this approximation, however,and it may also happen that a wider wire produces a slightlyweaker noise ( x -polarized curve for a thin wire in Fig.6). Thisis qualitatively similar to the trend of Fig.3 where a thick layercan produce less noise than a thin one at distances larger thanthe skin depth. The polarization anisotropy is also qualita-tively reproduced by the incoherent summation method, al-though B xx is overestimated. In fact, due to damping on thescale of δ , not the entire volume of the thick layer contributesto the noise. The dashed lines in Fig.4 and further calculationsshow that the quantitative agreement is systematically betterfor the field component perpendicular to the nearest metal sur-face (here, B yy ). D. Multiple wires
This is the generic situation in miniaturized magnetic traps(‘atom chips’) with wires being defined by etchings in a metal-lic layer. We consider three wires of identical cross sectionand smaller than the skin depth. We show in Fig.8 the depen-dence on the vertical distance, above the central wire. Ourresults interpolate smoothly between a single narrow wire( d ≪ w ) and a single wide wire ( d > ∼ w ), as could havebeen expected. In fact, the three geometries give nearly thesame noise in the azimuthal ( B x ) polarization. The incoher-ent summation approximation overestimates this noise com-ponent (similar to Fig.6). We attribute this to correlations be-tween the current fluctuations in neighboring wires that are not lateral position [ µ m] no r m a li z ed no i s e po w e r B xx µ m × µ m20 µ m × µ m incoh.summ 0 10 20 30 400.020.040.060.080.10 lateral position [ µ m] no r m a li z ed no i s e po w e r B yy µ m × µ m20 µ m × µ mincoh.summ FIG. 9: Same as Fig.8, but at fixed distance d = 10 µ m , scanningthe lateral position. x = 0 is above the center of the central wire). captured by incoherent summation. On the other hand, thisapproximation gives an excellent agreement for the weakernoise component B y .When we shift the observation point laterally, along the axisconnecting the wire centres, we get Fig.9. The stronger B x -polarization shows maxima of noise above each wire, as ex-pected. In the B y -polarization, a maximum occurs in the gapbetween the wires. This conforms to the general trend of ‘az-imuthal noise’ illustrated in Fig.5 (inset). It is also interestingthat above the central wire ( x = 0 ), three wires generate lessnoise than only one and also less than a single wide wire (ap-proximately a merger of the three). This observation goes intothe same direction as the experiments reported by Nenonenand co-workers where a reduction of thermal magnetic fieldswas achieved by cutting a metallic film into stripes. We at-tribute this behaviour to negative correlations between the cur-rents in neighboring wires brought about by the propagationof the magnetic field between them. In fact, the noise couldonly increase if the wire currents were strictly uncorrelated.The performance of the incoherent summation approxima-tion (solid lines) can be clearly seen, the trends being similarto Fig.8: good agreement for the B y -polarization, overestima-tion of the perpendicular case due to the neglect of correlationeffects. V. CONCLUSIONS
We have described in this paper numerical and analyticalresults for the thermal fields surrounding a two-dimensionalmetallic object of arbitrary cross-section. The role of the skindepth δ as a characteristic length scale has been highlighted.At distances smaller than δ , the spectral noise power roughlyscales with the volume of the metallic material (Figs. 3, 6, 7).We have reviewed a simple method based on this idea, the ‘in-coherent summation approximation’. It systematically overes-timates the noise power in one of the two field polarizations,but otherwise reproduces the main features as long as the skindepth is the largest scale. The strong polarization anisotropythat we have found suggests strategies to minimize loss or de-coherence due to thermal magnetic fields, as observed in re-cent experiments : this can be done by suitably choos-ing the direction of the static trapping fields. We have alsoshown that the noise power can be significantly non-additivewhen dealing with multiple objects. This could be relevant for the discrepancy between experiment and theory observedin Ref.19, although our method (restricted to 2D) do not per-mit quantitative predictions of trap lifetimes.We now comment on possible extensions of this analysis.Our framework is also able to provide an approximate de-scription of superconducting structures. In fact, since we dealwith magnetic field fluctuations at a finite frequency, there isalways some penetration into the superconductor or, equiv-alently, a finite resistivity. This can be attributed to a frac-tion of carriers in a normally conducting state. Calculationsfor superconductors of planar geometry have been reportedin Refs.10,29, with applications for miniaturized atom trapsin mind. More accurate descriptions require one to solve theLondon equations at finite frequency inside the superconduc-tor, using for example the two-fluid model .We recall that we use in this paper a local version of Ohm’slaw. For very pure metallic films, the ballistic transport ofcharge carriers implies a nonlocal response . This maybe particularly relevant for wires defined by doping a semi-conductor, but would require major changes for the numericalapproach of Sec.IV.Finally, a brief remark on non-equilibrium settings. Con-sider an isolated metallic object held at a temperature differentfrom its surroundings (materialized by the vacuum chamberwalls, for example). By applying the generalized Kirchhoffrelations (see, e.g., Ref.17), the radiation arriving at the ob-servation point can be split in two parts: one is proportionalto the product of the power a test dipole emits into the farfield and the temperature of the surroundings; the second partis proportional to the dipole radiation power absorbed by theobject and the metal temperature. At the sub-wavelength dis-tances of interest for this paper, one can show that the secondpart is dominant and that the error made in using the sametemperature for metal and surroundings is small. The equilib-rium calculation we have focused on here is then sufficient. Acknowledgements
This work has been supported by the European Union(project ACQP, contract IST-2001-38863) and Universit¨atPotsdam (graduate school ‘Confined Reaction and Interac-tions in Soft Matter’). We thank the quantum optics groupin Potsdam for creating a stimulating environment.
APPENDIX A: UNIFORM APPROXIMATION
The integral over k in Eq.(14) can be performed analyti-cally if the power-law approximations of Table I are used forthe reflection coefficients. We split the integration range at thecrossing points between the power laws and sum the contribu-tions. The full expression is cumbersume, and we quote hereonly the most complicated case, the thin layer in the range k ≪ k ≪ /δ (first column of Table I). The integral can behandled with the formula k Z k d k ( − kδ ) k e − kd kt = (1 − i) δ td Γ(2 , k d, k d ) (A1) + f d Γ(1 , k d, k d )+ i ft e − d/t Γ(0 , k d − d/t, k d − d/t ) where t = δ /h and f = (1 + i) δ /t + i /t . In this case, k = k and k = ( tδ ) − / . The incomplete gamma functionis defined by Γ( n, b, a ) = Γ( n, b ) − Γ( n, a ) (A2) Γ( n, a ) = ∞ Z a d t t n − e − t (A3)Using the asymptotics of the gamma function , we get thepower laws in Table II. The logarithmic behaviour arises from a ≪ , a ) ≈ − log a. (A4) APPENDIX B: INCOHERENT SUMMATION
We outline here the adaptation of the incoherent summationmethod of Ref.23 to two dimensions. The thermal spectrum ofthe current density is given at low frequencies ( ¯ hω ≪ k B T )by h j ∗ ( x ; ω ) j ( x ′ ; ω ′ ) i = 2 πδ ( ω − ω ′ )2 k B T σ ( x ; ω ) δ ( x − x ′ ) (B1)This spectrum is already integrated over a unit length in the z -direction (parallel to the current) along which the currentdensity is assumed to be uniform (two-dimensional geome-try). In this formulation, σ is (the real part of) the 3D con-ductivity that we assume local, as reflected by the spatial δ -correlation. We only take into account currents parallel to the z -direction. Each current element generates a magnetic field in the xy -plane that we compute in the magnetostatic approx-imation and ignoring the presence of the embedding metal.The latter point is the key approximation made. This gives amagnetic noise spectrum (integrated over a unit length along z ) of the order of S B = µ k B Tωδ , (B2)with cross correlations given by B ij ( x ; ω ) = S B π ( δ ij (tr Y ) − Y ij ) (B3) Y ij ( x ) = Z V d x ′ d x ′ ( x i − x ′ i )( x j − x ′ j ) | x − x ′ | (B4)where V is the volume occupied by the metal. The ‘geomet-rical tensor’ Y ij is dimensionless (a specific 2D property) anddepends only on the ratio of observation distance and objectsize. It does not involve the skin depth, of course.For a microstructure with rectangular cross section, an ob-server located above the center of the structure sees a noisepower B xx ( d ; ω ) S B = 12 π "(cid:20) arctan( x ′ y − y ′ ) (cid:21) w x ′ = − w y ′ = − h + 12 π "(cid:20) Im Li ( i x ′ y − y ′ ) (cid:21) w x ′ = − w y ′ = − h (B5) B yy ( d ; ω ) S B = 12 π "(cid:20) arctan( y − y ′ x ′ ) (cid:21) w x ′ = − w y ′ = − h + 12 π "(cid:20) Im Li ( i x ′ y − y ′ ) (cid:21) w x ′ = − w y ′ = − h where Li n ( · ) is the polylogarithm and we have used the nota-tion h [ f ( u, v )] bu = a i dv = c ≡ f ( a, c ) − f ( a, d ) − f ( b, c ) + f ( b, d ) (B6) J. Nenonen, J. Montonen, and T. Katila, Rev. Sci. Instr. , 2397(1996). J. A. Sidles, J. L. Garbini, W. M. Dougherty, and S.-H. Chao,Proc. IEEE , 799 (2003). R. Folman, P. Kr¨uger, J. Schmiedmayer, J. H. Denschlag, and C.Henkel, Adv. At. Mol. Opt. Phys. , 263 (2002). J. Fort´agh and C. Zimmermann, Rev. Mod. Phys. , 235 (2007). E. M. Purcell, Phys. Rev. , 681 (1946). G. S. Agarwal, Phys. Rev. A , 230 (1975). T. Varpula and T. Poutanen, J. Appl. Phys. , 4015 (1984). R. R. Chance, A. Prock, and R. Silbey, in
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