Isotropic transformation optics: approximate acoustic and quantum cloaking
IIsotropic transformation optics:approximate acoustic and quantum cloaking
Allan Greenleaf ∗ , Yaroslav Kurylev † Matti Lassas ‡ , Gunther Uhlmann §¶ Abstract
Transformation optics constructions have allowed the design ofelectromagnetic, acoustic and quantum parameters that steer wavesaround a region without penetrating it, so that the region is hiddenfrom external observations. The material parameters are anisotropic,and singular at the interface between the cloaked and uncloaked re-gions, making physical realization a challenge. We address this prob-lem by showing how to construct isotropic and nonsingular parametersthat give approximate cloaking to any desired degree of accuracy forelectrostatic, acoustic and quantum waves. The techniques used heremay be applicable to a wider range of transformation optics designs.For the Helmholtz equation, cloaking is possible outside a discreteset of frequencies or energies, namely the Neumann eigenvalues of thecloaked region. For the frequencies or energies corresponding to theNeumann eigenvalues of the cloaked region, the ideal cloak supportstrapped states; near these energies, an approximate cloak supports almost trapped states . This is in fact a useful feature, and we concludeby giving several quantum mechanical applications. ∗ Department of Mathematics, University of Rochester, Rochester, NY 14627 † Department of Mathematical Sciences, University College London, Gower Str, London,WC1E 6BT, UK ‡ Institute of Mathematics, Helsinki University of Technology, FIN-02015, Finland § Department of Mathematics, University of Washington, Seattle, WA 98195 ¶ Authors listed in alphabetical order. AG and GU are supported by US NSF, ML byAcademy of Finland and YK by UK EPSRC. a r X i v : . [ phy s i c s . op ti c s ] A ug Introduction
Cloaking devices designs based on transformation optics require anisotropicand singular material parameters, whether the conductivity (electrostatic)[27, 28], index of refraction (Helmholtz) [40, 19], permittivity and permeabil-ity (Maxwell) [47, 19], mass tensor (acoustic) [9, 15, 23, 46], or effective mass(Schr¨odinger)[54]. The same is true for other transformation optics designs,such as those motivated by general relativity [41]; field rotators [8]; concen-trators [43]; electromagnetic wormholes [20, 22]; or beam splitters [48]. Boththe anisotropy and singularity present serious challenges in trying to phys-ically realize such theoretical plans using metamaterials. In this paper, wegive a general method, isotropic transformation optics , for dealing with bothof these problems; we describe it in some detail in the context of cloaking, butit should also be applicable to a wider range of transformation optics-baseddesigns.A well known phenomenon in effective medium theory is that homogeniza-tion of isotropic material parameters may lead, in the small-scale limit, toanisotropic ones [44]. Using ideas from [45, 1, 11] and elsewhere, we showhow to exploit this to find cloaking material parameters that are at once bothisotropic and nonsingular, at the price of replacing perfect cloaking with approximate cloaking (of arbitrary accuracy). This method, starting withtransformation optics-based designs and constructing approximations to them,first by nonsingular , but still anisotropic, material parameters, and then bynonsingular isotropic parameters, seems to be a very flexible tool for creat-ing physically realistic theoretical designs, easier to implement than the idealones due to the relatively tame nature of the materials needed, yet essentiallycapturing the desired effect on waves.In ideal cloaking, for any wave propagation governed by the Helmholtz equa-tion at frequency ω , there is a dichotomy [19, Thm. 1] between generic val-ues of ω , for which the waves must vanish within the cloaked region D , andthe discrete set of Neumann eigenvalues of D , for which there exist trappedstates : waves which are zero outside of D and equal to a Neumann eigenfunc-tion within D . In the approximate cloaking resulting from isotropic transfor-mation optics that we will describe, trapped states for the limiting ideal cloak By singular , we mean that at least one of the eigenvalues goes to zero or infinity atsome points. almost trapped states for the approximate cloaks. The existenceof these should be considered as a feature, not a bug; we discuss this further inSec. 4.2 and give applications in [25].We start by considering isotropic transformation optics for acoustic (andhence, at frequency zero, electrostatic) cloaking. First recall ideal cloaking forthe Helmholtz equation. For a Riemannian metric g = ( g ij ) in n -dimensionalspace, the Helmholtz equation with source term is1 (cid:112) | g | n (cid:88) i,j =1 ∂∂x i (cid:18)(cid:112) | g | g ij ∂u∂x j (cid:19) + ω u = p, (1)where | g | = det( g ij ) and ( g ij ) = g − = ( g ij ) − . In the acoustic equation, forwhich ideal 3D spherical cloaking was described by Chen and Chan [9] andCummer, et al., [15], (cid:112) | g | g ij represents the anisotropic density and (cid:112) | g | thebulk modulus.In [19], we showed that the singular cloaking metrics g for electrostatics con-structed in [27, 28], giving the same boundary measurements of electrostaticpotentials as the Euclidian metric g = ( δ ij ), also cloak with respect to so-lutions of the Helmholtz equation at any nonzero frequency ω and with anysource p . An example in 3D, with respect to spherical coordinates ( r, θ, φ ),is ( g jk ) = r − sin θ θ
00 0 2(sin θ ) − (2)on B − B = { < r ≤ } , with the cloaked region being the ball B = { ≤ r ≤ } . This g is the image of g under the singular transformation( r, θ, φ ) = F ( r (cid:48) , θ (cid:48) , φ (cid:48) ) defined by r = 1 + r (cid:48) , θ = θ (cid:48) , φ = φ (cid:48) , < r (cid:48) ≤
2, whichblows up the point r (cid:48) = 0 to the cloaking surface Σ = { r = 1 } . The sametransformation was used by Pendry, Schurig and Smith [47] for Maxwell’sequations and gives rise to the cloaking structure that is referred to in [19]as the single coating . It was shown in [19, Thm.1] that if the cloaked regionis given any nondegenerate metric, then finite energy waves u that satisfythe Helmholtz equation (1) on B in the sense of distributions (cf. Sec. 2.2 B R denotes the central ball of radius R . Note that ∂∂θ , ∂∂φ are not normalized to havelength 1; otherwise, (2) agrees with [15, (24-25)] and [9, (8)], cf. [23]. r = 2, i.e., the same acousticboundary measurements, as do the solutions for the Helmholtz equation for g with source term p ◦ F . The part of p supported within the cloakedregion is undetectable at r = 2, while the part of p outside Σ appears to beshifted by the transformation F ; cf. [56]. Furthermore, on the inner sideΣ − of the cloaking surface, the normal derivative of u must vanish, so thatwithin B the acoustic waves propagate as if Σ were lined with a sound–hardsurface. Within B , u can be any Neumann eigenfunction; if − ω is not aneigenvalue, then the wave must vanish on B , while if it is , then u can equalany associated eigenfunction there, leading to what we refer to as a trappedstate of the cloak.In Sec. 2 we introduce isotropic transformation optics in the setting of acous-tics, starting by approximating the ideal singular anisotropic density andbulk modulus by nonsingular anisotropic parameters. Then, using a homo-geneization argument [45, 1], we approximate these nonsingular anisotropicparameters by nonsingular isotropic ones. This yields almost, or approxi-mate, invisibility in the sense that the boundary observations for the resultingacoustic parameters converge to the corresponding ones for a homogeneous,isotropic medium.In Sec. 3 we consider the quantum mechanical scattering problem for thetime-independent Schr¨odinger equation at energy E ,( −∇ + V ( x )) ψ ( x ) = Eψ ( x ) , x ∈ R d , (3) ψ ( x ) = exp( iE / x · θ ) + ψ sc ( x ) , where θ ∈ R d , | θ | = 1, and ψ sc ( x ) satisfies the Sommerfeld radiation condi-tion. By a gauge transformation, we can reduce an isotropic acoustic equa-tion to a Schr¨odinger equation. In this paper we restrict ourselves to the casewhen the potential V is compactly supported, so that ψ sc ( x ) = a V ( E, x/ | x | , θ ) | x | d − · e iλ | x | + O (cid:16) | x | d (cid:17) , as | x | → ∞ . The function a V ( E, θ (cid:48) , θ ) is the s cattering amplitude at energy E of thepotential V . The inverse scattering problem consists of determination of V from the scattering amplitude. As V is compactly supported, this inverseproblem is equivalent to the problem of determination of V from boundarymeasurements. Indeed, if V is supported in a domain Ω, we define the4irichlet-to-Neumann (DN) operator Λ V ( E ) at energy E for the potential V as follows. For any smooth function f on ∂ Ω, we setΛ V ( E ) f = ∂ ν ψ | ∂ Ω where ψ is the solution of the Dirichlet boundary value problem( −∇ + V ) ψ = Eψ, ψ | ∂ Ω = f. (Of course, Λ V ( E ) = Λ V − E (0).) Knowing Λ V ( E ) is equivalent to knowing a V ( E, θ (cid:48) , θ ) for all ( θ (cid:48) , θ ) ∈ S d − × S d − . Roughly speaking, Λ V ( E ) can beconsidered as knowledge of all external observations of V at energy E [5].In Sec. 4 we also consider the magnetic Schr¨odinger equation with magneticpotential A and electric potential V ,( − ( ∇ + iA ) + V − E ) ψ = 0 , ψ | ∂ Ω = f, which defines the DN operator,Λ V,A ( E )( f ) = ∂ ν ψ | ∂ Ω + i ( A · ν ) f. There is an enormous literature on unique determination of a potential,whether from scattering data or from boundary measurements of solutions ofthe associated Schr¨odinger equation. In [51] it was shown that an L ∞ poten-tial is determined by the associated DN operator, and [39] and [7] extendedthis to rougher potentials. In dimension d = 2, it has been shown recentlythat uniqueness holds if V is merely in L p , p > d = 2 and each E >
0, there are continuous familiesof rapidly decreasing (but noncompactly supported) potentials which are transparent , i.e., for which the scattering amplitude a V ( E, θ (cid:48) , θ ) vanishes ata fixed energy E , a V ( E, θ (cid:48) , θ ) ≡ a ( E, θ (cid:48) , θ ) = 0 [29]. More recently, [32]described central potentials transparent on the level of the ray geometry.Recently, Zhang, et al., [54] have described an ideal quantum mechanicalcloak at any fixed energy E and proposed a physical implementation. Theconstruction starts with a homogeneous, isotropic mass tensor (cid:98) m and po-tential V ≡
0, and subjects this pair to the same singular transformation(“blowing up a point”) as was used in [27, 28, 47]. The resulting cloakingmass-density tensor (cid:98) m and potential V yield a Schr¨odinger equation that is5he Helmholtz equation (at frequency ω = √ E ) for the corresponding sin-gular Riemannian metric, thus covered by the analysis of cloaking for theHelmholtz equation in [19, Sec. 3]. The cloaking mass-density tensor (cid:98) m andpotential are both singular, and (cid:98) m infinitely anisotropic, at Σ, combining tomake such a cloak difficult to implement, with the proposal in [54] involvingultracold atoms trapped in an optical lattice.In this paper, we consider the problem in dimension d = 3. For each energy E , we construct a family { V En } ∞ n =1 of bounded potentials, supported in theannular region B − B , which act as an approximate invisibility cloak : forany potential W on B , the scattering amplitudes a V En + W ( E, θ (cid:48) , θ ) → n → ∞ . Thus, when surrounded by the cloaking potentials V En , the potential W is undetectable by waves at energy E , asymptotically in n . Furthermore, E either is or is not a Neumann eigenvalue for W on the cloaked region. Inthe latter case, with high probability the approximate cloak keeps particlesof energy E from entering the cloaked region; i.e., the cloak is effective atenergy E . In the former case, the cloaked region supports “almost trapped”states, accepting and binding such particles and thereby functioning as anew type of ion trap. Furthermore, the trap is magnetically tunable: appli-cation of a homogeneous magnetic field allows one to switch between the twobehaviors [25].In Sec. 4 we consider several applications to quantum mechanics of thisapproach. In the first, we study the magnetic Schr¨odinger equation and con-struct a family of potentials which, when combined with a fixed homogeneousmagnetic field, make the matter waves behave as if the potentials were almostzero and the magnetic potential were blowing up near a point, thus givingthe illusion of a locally singular magnetic field. In the second, we describe“almost trapped” states which are largely concentrated in the cloaked region.For the third application, we use the same basic idea of isotropic transfor-mation optics but we replace the single coating construction used earlier bythe double coating construction of [19], corresponding to metamaterials de-ployed on both sides of the cloaking surface, to make matter waves behaveas if confined to a three dimensional sphere, S .Full mathematical proofs will appear elsewhere [24]. The authors are gratefulto A. Cherkaev and V. Smyshlyaev for useful discussions on homogenization,to S. Siltanen for help with the numerics, and to the anonymous referees forconstructive criticism and additional references.6 Cloaking for the acoustic equation
Our analysis is closely related to the inverse problem for electrostatics, orCalder´on’s conductivity problem. Let Ω ⊂ R d be a domain, at the boundaryof which electrostatic measurements are to be made, and denote by σ ( x ) theanisotropic conductivity within. In the absence of sources, an electrostaticpotential u satisfies a divergence form equation, ∇ · σ ∇ u = 0 (4)on Ω. To uniquely fix the solution u it is enough to give its value, f , on theboundary. In the idealized case, one measures, for all voltage distributions u | ∂ Ω = f on the boundary the corresponding current fluxes, ν · σ ∇ u , where ν is the exterior unit normal to ∂ Ω. Mathematically this amounts to theknowledge of the Dirichlet–Neumann (DN) map, Λ σ . corresponding to σ ,i.e., the map taking the Dirichlet boundary values of the solution to (4) tothe corresponding Neumann boundary values,Λ σ : u | ∂ Ω (cid:55)→ ν · σ ∇ u | ∂ Ω . (5)If F : Ω → Ω , F = ( F , . . . , F d ), is a diffeomorphism with F | ∂ Ω = Identity,then by making the change of variables y = F ( x ) and setting u = v ◦ F − ,we obtain ∇ · (cid:101) σ ∇ v = 0 , (6)where (cid:101) σ = F ∗ σ is the push forward of σ in F ,( F ∗ σ ) jk ( y ) = 1det[ ∂F j ∂x k ( x )] d (cid:88) p,q =1 ∂F j ∂x p ( x ) ∂F k ∂x q ( x ) σ pq ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = F − ( y ) . (7)This can be used to show thatΛ F ∗ σ = Λ σ . Thus, there is a large (infinite-dimensional) family of conductivities whichall give rise to the same electrostatic measurements at the boundary. This7bservation is due to Luc Tartar (see [37] for an account.) Calder´on’s in-verse problem for anisotropic conductivities is then the question of whethertwo conductivities with the same DN operator must be push-forwards ofeach other. There are a number of positive results in this direction, but itwas shown in [27, 28] that, if one allows singular maps, then in fact therecounterexamples, i.e., conductivities that are undetectable to electrostaticmeasurements at the boundary. See [36] for d = 2.¿From now on, for simplicity we will restrict ourselves to the three dimen-sional case. For each R >
0, let B R = {| x | ≤ R } and Σ R = {| x | = R } bethe central ball and sphere of radius R , resp., in R , and let O = (0 , , F : R − { O } → R − B , givenby x = F ( y ) := (cid:40) y, for | y | > , (cid:16) | y | (cid:17) y | y | , for 0 < | y | ≤ . (8)Letting σ = 1 be the homogeneous isotropic conductivity on R , F thendefines a conductivity σ on R − B by the formula σ jk ( x ) := ( F ∗ σ ) jk ( x ) , (9)cf. (7). More explicitly, the matrix σ = [ σ jk ] j,k =1 is σ ( x ) = 2 | x | − ( | x | − Π( x ) + 2( I − Π( x )) , < | x | < , where Π( x ) : R → R is the projection to the radial direction, defined byΠ( x ) v = (cid:18) v · x | x | (cid:19) x | x | , (10)i.e., Π( x ) is represented by the matrix | x | − xx t , cf. [36].One sees that σ ( x ) is singular, as one of its eigenvalues, namely the one corre-sponding to the radial direction, tends to 0 as | x | (cid:38)
1. We can then extend σ to B as an arbitrary smooth, nondegenerate (bounded from above andbelow) conductivity there. Let Ω = B ; the conductivity σ is then a cloak-ing conductivity on Ω, as it is indistinguishable from σ , vis-a-vis electro-static boundary measurements of electrostatic potentials, treated rigorously8s bounded, distributional solutions of the degenerate elliptic boundary valueproblem corresponding to σ [27, 28].The same construction of σ | Ω − B was proposed in Pendry, Schurig and Smith[47] to cloak the region B from observation by electromagnetic waves ata positive frequency; see also Leonhardt [40] for a related approach forHelmholtz in R . The cloaking conductivity σ above corresponds to a Riemannian metric g jk that is related to σ ij by σ ij ( x ) = | g ( x ) | / g ij ( x ) , | g | = (cid:0) det[ σ ij ] (cid:1) (11)where [ g jk ( x )] is the inverse matrix of [ g jk ( x )] and | g ( x ) | = det[ g jk ( x )]. TheHelmholtz equation, with source term p , corresponding to this cloaking met-ric has the form (cid:88) j,k =1 | g ( x ) | − / ∂∂x j ( | g ( x ) | / g jk ( x ) ∂∂x k u ) + ω u = p on Ω , (12) u | ∂ Ω = f. For now, g is allowed to be an arbitrary nonsingular Riemannian metric, g inn ,on B . Reinterpreting the conductivity tensor σ as a mass tensor ( which hasthe same transformation law (7) ) and | g | as the bulk modulus parameter,(12) becomes an acoustic equation, (cid:16) ∇· σ ∇ + ω | g | (cid:17) u = p ( x ) | g | on Ω , (13) u | ∂ Ω = f. This is the form of the acoustic wave equation considered in [9, 15]; see also[14] for d = 2, and [46] for a somewhat different approach. As σ is singularat the cloaking surface Σ := Σ = ∂B , one has to carefully define what onemeans by “waves”, that is by solutions to (12) or (13). Let us recall theprecise definition of the solution to (12) or (13), discussed in detail in [19].We say that u is a finite energy solution of the Helmholtz equation (12) orthe acoustic equation (13) if 9. u is square integrable with respect to the metric, i.e., is in the weighted L -space, u ∈ L g (Ω) = { u : (cid:107) u (cid:107) g := (cid:90) Ω dx | g | / | u | < ∞} ;2. the energy of u is finite, (cid:107)∇ u (cid:107) g := (cid:90) Ω dx | g | / g ij ∂ i u∂ j u < ∞ ;3. u satisfies the Dirichlet boundary condition u | ∂ Ω = f ; and4. the equation (13) is valid in the weak distributional sense, i.e., for all ψ ∈ C ∞ (Ω) (cid:90) Ω dx [ − ( | g | / g ij ∂ i u ) ∂ j ψ + ω uψ | g | / ] = (cid:90) Ω dx p ( x ) ψ ( x ) | g | / . (14)This last can be interpreted as saying that any smooth superposition ofpoint measurements of u satisfies the same integral identity as it would for aclassical solution. We also note that, since g is singular, the term | g | / g ij ∂ i u must also be defined in an appropriate weak sense.It was shown in [19, Thm. 1] that if u is the finite energy solution of theacoustic equation (13), then u ( x ) defines two functions v + ( y ) , y ∈ Ω, and v − ( y ) , y ∈ B , by the formulae u ( x ) = (cid:26) v + ( y ) , where x = F ( y ), for 1 < | x | < ,v − ( y ) , where x = y , for 0 < | x | < . (15)These functions satisfy the following boundary value problems:( ∇ + ω ) v + ( y ) = (cid:101) p ( y ) := p ( F ( y )) in Ω , (16) v + | ∂ Ω = f, and ( ∇ g inn + ω | g inn | / ) v − ( y ) = | g inn | / p ( y ) in B , (17) ∂ ν v − | ∂B = 010here ∂ ν u = ∂ r u denotes the normal derivative on ∂B .In the absence of sources within the cloaked region, (17) leads, as mentionedin the introduction, to the phenomenon of trapped states : If − ω is not aNeumann eigenvalue for ( B , g inn ), then v − ≡ B , the waves do not enter B , and cloaking as generally understood holds. On the other hand, if − ω is an eigenvalue, then v − can be any function in the associated eigenspace;indeed, one can have v + ≡
0, in which case the total wave u behaves as abound state for the cloak, concentrated in B ; for simplicity, we refer to thisas a trapped state of the ideal cloak. Next, consider nonsingular approximations to the ideal cloak, which aremore physically realizable by virtue of having bounded anisotropy ratio; see[49, 21, 36] for analyses of cloaking from the point of view of similar trun-cations. Studying the behavior of solutions to the corresponding boundaryvalue problems near the cloaking surface, as these nonsingular approximatelycloaking conductivities tend to the ideal σ , we will see that the Neumannboundary condition appears in (17) on the cloaked region B . At the presenttime, for mathematical proofs [24] of some of the results below we assumethat σ be chosen to be the homogeneous, isotropic conductivity, σ = κσ inside B , i.e., σ jk ( x ) = κδ jk , with κ ≥ − ω is not aNeumann eigenvalue on B . The first assumption is not needed for physicalarguments, but the second is.To start, let 1 < R < ρ := 2( R −
1) and introduce the coordinate trans-formation F R : R − B ρ → R − B R , x := F R ( y ) = (cid:40) y, for | y | > , (cid:16) | y | (cid:17) y | y | , for ρ < | y | ≤ . We define the corresponding approximate conductivity, σ R as σ jkR ( x ) = (cid:26) σ jk ( x ) for | x | > R,κδ jk , for | x | ≤ R. (18)Note that then σ jk ( x ) = (( F R ) ∗ σ ) jk ( x ) for | x | > R , where σ ≡ R >
1, the conductivity σ R is nonsingular, i.e., is bounded fromabove and below with, however, the lower bound going to 0 as R (cid:38)
1. Letus define g R ( x ) = det( σ R ( x )) = , for | x | ≥ , | x | − ( | x | − for R < | x | < ,κ , for | x | ≤ R, (19)cf. (11). Similar to (13), consider the solutions of( ∇· σ R ∇ + ω g / R ) u R = g / R p in Ω := B u R | ∂ Ω = f, As σ R and g R are now non-singular everywhere on D , we have the standardtransmission conditions on Σ R := { x : | x | = R } , u R | Σ R + = u R | Σ R − , (20) e r · σ R ∇ u R | Σ R + = e r · σ R ∇ u R | Σ R − , where e r is the radial unit vector and ± indicates when the trace on Σ R iscomputed as the limit r → R ± .Similar to (15), we have u R ( x ) = (cid:26) v + R ( F − R ( x )) , for R < | x | < ,v − R ( x ) , for | x | ≤ R, with v ± R satisfying( ∇ + ω ) v + R ( y ) = p ( F R ( y )) in ρ < | y | < ,v + R | ∂ Ω = f, and ( ∇ + κ ω ) v − R ( y ) = κ p ( y ) , in | y | < R. (21)Next, using spherical coordinates ( r, θ, ϕ ), r = | y | , the transmission condi-tions (20) on the surface Σ R yield v + R ( ρ, θ, φ ) = v − R ( R, θ, φ ) , (22) ρ ∂ r v + R ( ρ, θ, φ ) = κR ∂ r v − R ( R, θ, φ ) . p = 0, but also analyze the case p ( x ) = κ − (cid:88) | α |≤ N q α ∂ αx δ ( x ) , (23)where δ is the Dirac delta function at origin and q α ∈ C , i.e., there isa (possibly quite strong) point source the cloaked region. The Helmholtzequation (21) on the entire space R , with the above point source and thestandard radiation condition, would give rise to the wave u p ( y ) = N (cid:88) n =0 n (cid:88) m = − n p nm h (1) n ( κωr ) Y mn ( θ, ϕ ) , p nm = p nm ( ω ) , where Y mn are spherical harmonics and h (1) n ( z ) and j n ( z ) are the sphericalBessel functions, see, e.g., [13].In B R the function v − R ( y ) differs from u p by a solution to the homogeneousequation (21), and thus for r < Rv − R ( r, θ, ϕ ) = ∞ (cid:88) n =0 n (cid:88) m = − n ( a nm j n ( κωr ) + p nm h (1) n ( κωr )) Y mn ( θ, ϕ ) , with yet undefined a nm = a nm ( κ, ω ; R ). Similarly, for ρ < r < v + R ( r, θ, ϕ ) = ∞ (cid:88) n =0 n (cid:88) m = − n ( c nm h (1) n ( ωr ) + b nm j n ( ωr )) Y mn ( θ, ϕ ) , with as yet unspecified b nm = b nm ( κ, ω ; R ) and c nm = c nm ( κ, ω ; R ).Rewriting the boundary value f on ∂ Ω as f ( θ, ϕ ) = ∞ (cid:88) n =0 n (cid:88) m = − n f nm Y mn ( θ, ϕ ) , we obtain, together with transmission conditions (22), the following equationsfor a nm , b nm and c nm : f nm = b nm j n (3 ω ) + c nm h (1) n (3 ω ) , (24) a nm j n ( κωR ) + p nm h (1) n ( κωR ) = b nm j n ( ωρ ) + c nm h (1) n ωρ ) , (25) κR ( κωa nm ( j n ) (cid:48) ( κωR ) + κωp nm ( h (1) n ) (cid:48) ( κωR )) (26)= ρ ( b nm ω ( j n ) (cid:48) ( kρ ) + ωc nm ( h (1) n ) (cid:48) ( ωρ )) . ω is not a Dirichlet eigenvalue of the equation (13), we can find the a nm and c nm from (25)-(26) in terms of p nm and b nm , and use the solutionsobtained and the equation (24) to solve for b nm in terms of f nm and p nm .This yields b nm = 1 j n (3 ω ) + s n h (1) n (3 ω ) ( f nm − (cid:101) s n h (1) n (3 ω ) p nm ) ,c nm = s n b nm − (cid:101) s n p nm , (27) a nm = t n b nm − (cid:101) t n p nm where s n = κ R j n ( ωρ )( j n ) (cid:48) ( κωR ) − ρ ( j n ) (cid:48) ( ωρ ) j n ( κωR ) ρ ( h (1) n ) (cid:48) ( ωρ ) j n ( κωR ) − κ R h (1) n ( ωρ )( j n ) (cid:48) ( κωR ) ,t n = ρ j n ( ωρ )( h (1) n ) (cid:48) ( ωρ ) − ρ ( j n ) (cid:48) ( ωρ ) h (1) n ( ωρ ) ρ ( h (1) n ) (cid:48) ( ωρ ) j n ( κωR ) − κ R h (1) n ( ωρ )( j n ) (cid:48) ( κωR ) , (cid:101) s n = κ R h (1) n ( κωR )( j n ) (cid:48) ( κωR ) − κ R ( h (1) n ) (cid:48) ( κωR ) j n ( κωR ) ρ ( h (1) n ) (cid:48) ( ωρ ) j n ( ωR ) − κ R h (1) n ( ωρ )( j n ) (cid:48) ( κωR ) , (cid:101) t n = ρ h (1) n ( κωR )( h (1) n ) (cid:48) ( ωρ ) − κ R ( h (1) n ) (cid:48) ( κωR ) h (1) n ( ωρ ) ρ ( h (1) n ) (cid:48) ( ωρ ) j n ( κωR ) − κ R h (1) n ( ωρ )( j n ) (cid:48) ( κωR ) . Recalling that a nm , b nm and c nm depend on R , let us consider what happensas R (cid:38)
1, i.e., as ρ = 2( R − (cid:38)
0. We use the asymptotics j n ( ωρ ) = O ( ρ n ) , j (cid:48) n ( ωρ ) = O ( ρ n − ); (28) h (1) n ( ωρ ) = O ( ρ − n − ) , ( h (1) n ) (cid:48) ( ωρ ) = O ( ρ − n − ) , as ρ → , and obtain s n ∼ c ρ ρ n − + c ρ n c ρ ρ − n − + c ρ − n − ∼ c ρ n +1 , (29) t n ∼ c (cid:48) ρ ρ n ρ − n − + c (cid:48) ρ ρ n − ρ − n − c (cid:48) ρ ρ − n − + c ρ − n − ∼ c (cid:48) ρ n +1 , (30) (cid:101) s n ∼ c (cid:48)(cid:48) + c (cid:48)(cid:48) c ρ ρ − n − + c ρ − n − ∼ c (cid:48)(cid:48) ρ n +1 , (31) (cid:101) t n ∼ c (cid:48)(cid:48)(cid:48) ρ ρ − n − + c (cid:48)(cid:48)(cid:48) ρ − n − c (cid:48) ρ ρ − n − + c ρ − n − ∼ c (cid:48)(cid:48)(cid:48) , (32)14ssuming the constant c does not vanish. The constant c is the productof a non-vanishing constant and ( j n ) (cid:48) ( κω ). Thus the asymptotics (29)-(32)are valid if − ( κω ) is not a Neumann eigenvalue of the Laplacian in thecloaked region B and − ω is not a Dirichlet eigenvalue of the Laplacian inthe domain Ω. In the rest of this section we assume that this is the case.Since the system (24)-(26) is linear, we consider separately two cases, when f nm (cid:54) = 0 , p nm = 0, and when f nm = 0 , p nm (cid:54) = 0.In the case f nm (cid:54) = 0 , p nm = 0, we have b nm = O (1) , c nm = O ( ρ n +1 ) ,a nm = O ( ρ n +1 ) , as ρ → . The above equations, together with (28), imply that the wave v − R in theapproximately cloaked region r < R tends to 0 as ρ →
0, with the termassociated to the spherical harmonic Y mn behaving like O ( ρ n +1 ). As for thewave v + R in the region Ω − B R , both terms associated to the spherical harmonic Y mn and involving j n ( ωr ) and h (1) n ( ωr ), respectively, are of the same order O (1) near r = ρ . However, the terms involving h (1) n ( ωr ) decay, as r grows,becoming O ( ρ n +1 ) for r ≥ r > f nm = 0 , p nm (cid:54) = 0, we see that a nm ∼ − h (cid:48) n ( κωR ) j n ( κωR ) p nm = O (1) , as ρ → . Also, b nm = O ( ρ n +1 ) , c nm = O ( ρ n +1 ) , as ρ → . (33)These estimates show that v + R is of the order O (1) near r = ρ . However, itdecays as r grows becoming O ( ρ n +1 ) for r ≥ r > R →
1. More precisely, the solutions v ± R with converge to v ± , i.e.,lim R → v ± R ( r, θ, ϕ ) = v ± ( r, θ, ϕ ) , where v ± were defined in (15), (16), and (17). Equations (25),(27) and (33)show how the Neumann boundary condition naturally appears on the innerside Σ − of the cloaking surface. 15 .4 Isotropic nonsingular approximate acoustic cloak In this section we approximate the anisotropic approximate cloak σ R byisotropic conductivities, which then will themselves be approximate cloaks.Cloaking by layers of homogeneous, isotropic EM media has been proposedin [33, 10]; see also [17] for a related anisotropic 2D approach based onhomogenization. For general references on homogenization, see [3, 4, 16, 34];for some previous work on its application in the context of photonic crystals,see [30, 31, 55].We will consider the isotropic conductivities of the form γ ε ( x ) = γ ( x, rε )where r := r ( x ) = | x | is the radial coordinate, γ ( x, r (cid:48) ) = h ( x, r (cid:48) ) I ∈ R × and h ( x, r (cid:48) ) a smooth, scalar valued function to be chosen later that is periodic in r (cid:48) with period 1, i.e., h ( x, r (cid:48) +1) = h ( x, r (cid:48) ), satisfying 0 < C ≤ h ( x, r (cid:48) ) ≤ C .Let s = ( r, θ, φ ) and t = ( r (cid:48) , θ (cid:48) , φ (cid:48) ) be spherical coordinates corresponding totwo different scales. Next we homogenize the conductivity in the ( r (cid:48) , φ (cid:48) , θ (cid:48) )-coordinates. With this goal, we denote by e = (1 , , e = (0 , , e = (0 , ,
1) the vectors corresponding to unit vectors in r (cid:48) , θ (cid:48) and φ (cid:48) directions, respectively. Moreover, let U i ( s, t ) , i = 1 , , , be the solutions ofdiv t ( γ ( s, t )(grad t · U i ( s, t ) + e i ) = 0 , t = ( r (cid:48) , θ (cid:48) , φ (cid:48) ) ∈ R , (34)that are 1-periodic functions in r (cid:48) , θ (cid:48) and φ (cid:48) variables that satisfy, for all s , (cid:90) [0 , dt (cid:48) U i ( s, t (cid:48) ) = 0 , where, t (cid:48) = ( r (cid:48) , θ (cid:48) , φ (cid:48) ) and dt = dr (cid:48) dθ (cid:48) dφ (cid:48) .Define the two-scale corrector matrices [45, 1, 2] as P kj ( s, t ) = ∂∂t j U k ( s, t ) + δ kj . Then the homogenized conductivity is (cid:98) γ jk ( s ) = (cid:88) p =1 (cid:90) [0 , dt γ jp ( s, t ) P kp ( s, t ) = (cid:90) [0 , dt h ( s, r (cid:48) ) P kj ( s, t ) , (35)16nd satisfies C I ≤ (cid:98) γ ≤ C I .Since γ is independent of θ (cid:48) , φ (cid:48) , the above condition implies that U i = 0 for i = 2 ,
3. As for U , it satisfies ∂∂r (cid:48) (cid:18) h ( s, r (cid:48) ) ∂U ∂r (cid:48) (cid:19) = − ∂h ( s, r (cid:48) ) ∂r (cid:48) , with U being 1-periodic with respect to ( θ (cid:48) , φ (cid:48) ). These imply that U isindependent of ( θ (cid:48) , φ (cid:48) ) with ∂U ∂r (cid:48) = − Ch ( s, r (cid:48) ) . To find the constant C we again use the periodicity of U , now with respectto r (cid:48) , to get that C is given by the harmonic means h harm of h , C = h harm ( s ) := 1 (cid:82) dr (cid:48) h − ( s, r (cid:48) ) . (36)Let h a ( s ) denote the arithmetic means of h in the second variable, h a ( s ) = (cid:90) [0 , dr (cid:48) h ( s, r (cid:48) ) . Then the homogenized conductivity will be (cid:98) γ ( x ) = h harm ( x )Π( x ) + h a ( x )( I − Π( x )) , where Π( x ) is the projection (10). For similar constructions see, e.g., [11].If ( g R ( x ) − / ∇· γ ε ( x ) ∇ ) w ε = G on Ω , (37) w ε | ∂ Ω = f, applying results analogous to [45, 1] in spherical coordinates (see [24]), weobtain lim ε → w ε = w, in L (Ω) , (38)17here ( g R ( x ) − / ∇· (cid:98) γ ( x ) ∇ ) w = G on Ω , (39) w | ∂ Ω = f. The convergence (38) is physically reasonable; if we combine spherical layersof conducting materials, the radial conductivity is the harmonic average ofthe conductivity of layers and the tangential conductivity is the arithmeticaverage of the conductivity of the layers. Applying this, the fact that (cid:98) γ and γ ε are uniformly bounded both from above and below, and results from thespectral theory, e.g., [35], one can show [24] that if g R ( x ) − / ∇· γ ε ( x ) ∇ u ε + ω u ε = G on Ω , (40) u ε | ∂ Ω = f and ω is not a Dirichlet eigenvalue of the problem g R ( x ) − / ∇· (cid:98) γ ( x ) ∇ u + ω u = G on Ω , (41) u | ∂ Ω = f then lim ε → u ε = u, in L (Ω) . (42)To consider an explicit isotropic conductivity, let us consider functions φ : R → R and φ L : R → R given by φ ( t ) = , t < , t , ≤ t < , − (2 − t ) , ≤ t < , t ≥ , and φ L ( t ) = , t < ,φ ( t ) , ≤ t < , , ≤ t < L − ,φ ( L − t ) , t ≥ L − . γ ( r, rε ) = (cid:104) a ( r ) ζ ( rε ) − a ( r ) ζ ( rε ) (cid:105) , (43)where where a ( r ) is chosen positive and so that γ ( r, r (cid:48) ) > L , we define ζ j : R → R to be 1 − periodic functions , ζ ( t ) = φ L (cid:16) Lt (cid:17) , ≤ t < ,ζ ( t ) = φ L (cid:16) L ( t −
12 ) (cid:17) , ≤ t < . Temporarily fix an
R >
1; eventually, we will take a sequence of these (cid:38) (cid:98) γ is smooth enough, we piecetogether the cloaking conductivity in the exterior domain r > R and thehomogeneous conductivity in the cloaked domain in a smooth manner. Tothis end, we introduce a new parameter η > r theparameters a ( r ) ≥ a ( r ) ≥ (cid:90) dr (cid:48) [1 + a ( r ) ζ ( r (cid:48) ) − a ( r ) ζ ( r (cid:48) )] − = R − ( R − (1 − φ ( R − rη )) + κφ ( R − rη ) , if r < R, r − ( r − , if R < r < , , if r > , (cid:90) dr (cid:48) [1 + a ( r ) ζ ( r (cid:48) ) − a ( r ) ζ ( r (cid:48) )] = − φ ( R − rη )) + κφ ( R − rη ) , if r < R, , if R < r < , , if r > , thus obtaining a ( r ) and a ( r ) such that the homogenized conductivity is (cid:98) γ ( x ) = σ R,η ( x ) = π R (1 − φ ( R − rη )) + κφ ( R − rη ) , if r < R,π R , if R < r < , , if r > , (44)where π R = 2 R − ( R − Π( x ) + 2(1 − Π( x ))19nd Π( x ) is as in (10). (In (44), the term κφ ( R − rη ) connects the exteriorconductivity smoothly to the interior conductivity κ .)We denote the solutions by a R,η ( r ) and a R,η ( r ). Now when first ε →
0, then η → R →
1, the obtained conductivities approximate better andbetter the cloaking conductivity σ . Thus we choose appropriate sequences R n → η n → ε n → γ n ( x ) := (cid:20) a R n ,η n ( r ) ζ ( rε n ) − a R n ,η n ( r ) ζ ( rε n ) (cid:21) , r = | x | . (45)Note also that if a and a are constant functions then γ n = γ ( x , x/(cid:15) n ), sothat all γ n look the “same” inside the (cid:15) n period; this is the case in Figs. 1and 2. For later use, we need to assume that ε n goes to zero faster than η n ,and so choose ε n < η n ; we can also assume that all of the ε − n ∈ Z , whichensures that the function γ ( x, r ( x ) /ε n ) is C , smooth at r = 2. Denoting g n ( x ) := g R n ( x ), one can summarize the above analysis by: Isotropic approximate acoustic cloaking. If p is supported at the originas in (23), then the solutions of (cid:0) g n ( x ) − / ∇· γ n ( x ) ∇ + ω (cid:1) u n = p on Ω , (46) u n | ∂ Ω = f, tend to a solution of (13), as n → ∞ . This generalizes to the case when p is a general source, as long as its supportdoes not intersect the cloaking surface Σ, see [24]. This section is devoted to approximate quantum cloaking at a fixed energy,i.e., for the time-independent Schr¨odinger equation with the a potential V ( x ),( −∇ + V ) ψ = Eψ, in Ω .
20 standard gauge transformation converts the equation (46) to such a Schr¨odingerequation. Assuming that u n satisfies equation (46) with ω = E , and defining ψ n ( x ) = γ / n ( x ) u n ( x ) , (47)with γ n as in (45), we then have that γ − / n ∇· γ n ∇ ( γ − / n ψ n ) = ∇ ψ n − V n ψ n , where V n = γ − / n ∇ ( γ / n ) .ψ n thus satisfies the equation,( −∇ + V n − Eγ − n g / n ) ψ n = 0 in Ω , which can be interpreted as a Schr¨odinger at energy E by introducing theeffective potential V En ( x ) : = V n ( x ) − Eγ − n g / n + E, (48)so that ( −∇ + V En ) ψ n = Eψ n in Ω . (49)We will show that for generic E the potentials V En function as approximateinvisibility cloaks in quantum mechanics at energy E (recall the discussion inthe Introduction of the ideal quantum mechanical cloaking of [54]), while fora discrete set of E , the approximate cloaks support almost trapped states.Let us next consider measurements made on ∂ Ω. Let W ( x ) be a boundedpotential supported on B , let Λ W + V En ( E ) be the Dirichlet-to-Neumann (DN)operator corresponding to the potential W + V En , and Λ ( E ) be the DNoperator, defined earlier, corresponding to the zero potential.The results for the acoustic equation given in Sec. 2 yield the following result,constituting approximate cloaking in quantum mechanics; for mathematicaldetails of the proof, see [24]. Approximate quantum cloaking.
Let E ∈ R be neither a Dirichlet eigen-value of −∇ on Ω nor a Neumann eigenvalue of −∇ + W on B . Then,the DN operators (corresponding to boundary measurements at ∂ Ω of matter aves) for the potentials W + V En converge to the DN operator correspondingto free space, that is, lim n →∞ Λ W + V En ( E ) f = Λ ( E ) f in L ( ∂ Ω) , for any smooth f on ∂ Ω .Since convergence of the near field measurements imply convergence of thescattering amplitudes [5], we also have lim n →∞ a W + V En ( E, θ (cid:48) , θ ) = a ( E, θ (cid:48) , θ ) . Note that the V En can be considered as almost transparent potentials atenergy E , but this behavior is of a very different nature than the well-knownresults from the classical theory of spectral convergence, since the V En do not tend to 0 as n → ∞ . (On the contrary, as we will see shortly, they alternateand become unbounded near the cloaking surface Σ as n → ∞ .) Moreimportantly, the V En also serve as approximate invisibility cloaks for two-body scattering in quantum mechanics. We note the following fundamentaldichotomy: Approximate cloaking vs. almost trapped states.
Any potential W supported in B , when surrounded by V En , becomes, for generic E , unde-tectable by matter waves at energy E , asymptotically in n . Furthermore,the combination of W and the cloaking potential V En has negligible effect onwaves passing the cloak. On the other hand, for a discrete set of energies E ,the potential W + V En admits almost trapped states . This means that, if E is an eigenvalue of W inside B , there are E n close to E which are eigen-values of W + V En in Ω , and the corresponding eigenfunctions are heavilyconcentrated in B ; see Sec. 4.2 for details. As all measurement devices have limited precision, we can interpret this assaying that, given a specific device using particles at energy E , one can designa potential to cloak an object from any single-particle measurements madeusing that device. 22 .2 Explicit approximate quantum cloak We now make explicit the structure of the potentials V En , obtaining analyticexpressions used to produce the numerics and figures below. Recall that thepotential V n when γ n is given by (43), with L > d dt φ L ( t ) = , if t < ≤ t < L − L ≤ t, , if 0 ≤ t < L − ≤ t < L − , if 1 ≤ t < L − ≤ t < L − V n = γ − / n ∇ ( γ / n ) (50)= ε − n a R n ,η n ( r ) χ n ( rε n ) − a R n ,η n ( r ) χ n ( rε n )1 + a R n ,η n ( r ) ζ ( rε n ) − a R n ,η n ( r ) ζ ( rε n ) + O ( ε − n )where χ n ( r ) = , if r ∈ (0 , /L ) + Z and R < rε n < , − , if r ∈ (1 /L, /L ) + Z and R < rε n < , , if r ∈ (( L − /L, ( L − /L ) + Z and R < rε n < , − , if r ∈ (( L − /L,
1) + Z and R < rε n < , , otherwise , and χ n ( r ) = χ n ( r − ).We then see that the V n are centrally symmetric and can be considered asbeing comprised of layers of potential barrier walls and wells that becomevery high and deep near the inner surface Σ R n . Each V n is bounded, but as n → ∞ , the height of the innermost walls and the depth of the innermostwells goes to infinity when approaching the interface Σ from outside. Thesesame properties are then passed from V n to V En by (48).23 Figure 1:
One radial cell of conductivity.
The isotropic conductivity γ n ( x ) is of the form γ n ( x ) = h ( x, | x | ε n ), where the function r (cid:48) (cid:55)→ h ( x, r (cid:48) )with a fixed value of x has period 1 in variable r (cid:48) . The horizontal axis is r (cid:48) = | x | ε n ∈ [0 , h ( x, r (cid:48) ) as in (43), with the a = 2 and a = 0 .
8. 24
Figure 2:
One radial cell of potential.
Potential V n corresponding via(50) to conductivity γ n in Fig 1. The term of order ε − n , with the values a = 2 and a = 0 .
8, is shown as a function of r (cid:48) = | x | /ε n , as this variesthrough the period [0 , .3 Enforced boundary conditions on cloaking surface As described in Sec. 2.2,, the natural boundary condition for the Helmholtzand acoustic equations with perfect cloak, including those with sources withinthe cloaked region B , is the Neumann boundary condition on Σ − . How-ever, the above analysis of approximate cloaking for the Schr¨odinger equationmakes it possible to produce quantum cloaking devices which enforce moregeneral boundary conditions on Σ − , e.g., the Robin boundary conditions,which may be a useful feature in applications.To describe this, let χ e ε ( | x | ) = (cid:26) , if 1 − (cid:101) ε < r < , , otherwise , with α = α ( (cid:98) x ) , (cid:98) x = x/ | x | a function on Σ = ∂B .Introduce an extra potential wall inside B close to the surface Σ, namely,take W ( x ) in the form W ( x ) = Q e ε ( α ; | x | ) = α ( (cid:98) x ) χ e ε ( | x | ) (cid:101) ε , and then consider the boundary value problem,( −∇ − E + V En + Q e ε ) v = p in B , (51) v | ∂ Ω = f. As n → ∞ the solution v = v n e ε to (51) tends, inside B , to the solution ofthe equation ( −∇ − E + Q e ε ) v e ε = p in B , (52) ∂ r v e ε | Σ = 0 . Now, as (cid:101) ε (cid:38)
0, we see that Q e ε → αδ ( r − , (53) For analysis of ideal cloaking, allowing various boundary conditions, as long as theyare consistent with von Neumann’s theory of self-adjoint extensions, see Weder [53].
26o that the functions v e ε tend to the solution of the boundary value problem (cid:16) − ∇ − E + αδ ( r − (cid:17) v = p in B , (54) ∂ r v | Σ = 0 . Note that to give the precise meaning of the above problem and its solution,we should interpret (54) in the weak sense. Namely, v is the solution to (54),if for all ϕ ∈ C ∞ ( B ) (cid:90) B dx [ ∇ u · ∇ ϕ − Euϕ ] + (cid:90) Σ dS ( x ) αuϕ = (cid:90) B dx pϕ, which may be obtained from (54) by a (formal) integration by parts andutilizing (53). However, the above weak formulation is equivalent to theboundary value problem, ( −∇ − E ) v = p in B , ( ∂ r v − αv ) | Σ = 0 . Thus, the Neumann boundary condition for the Schr¨odinger equation at theenergy level E has been replaced by a Robin boundary condition on Σ − , andthe same holds for ideal acoustic cloaking.Returning to approximate cloaking, this means that if, for ε, (cid:101) ε very small,with ε << (cid:101) ε , we construct an approximate cloaking potential with layers ofthickness ε and height ε − , and augment it by an innermost potential wall ofwidth (cid:101) ε and height (cid:101) ε − , then we obtain an approximate quantum cloak withthe wave inside B behaving as if it satisfies the Robin boundary condition.It is clear from the above that the boundary condition appearing on thecloaking surface is very dependent on the fine structure of the approximatelycloaking potential. Physically, this boundary condition may be enforced byappropriate design of this extra potential wall (rather than being due to thecloaking material in B − B ), so that we refer to this as an enforced bound-ary condition in approximate cloaking, as opposed to the natural Neumanncondition that occurs in ideal cloaking.27 .4 Approximation of V En with point charges One possible path to physical realization of these approximate quantum me-chanical cloaks would be via electrostatic potentials, approximating (again!)the potentials V En by sums of point sources. Indeed, solving the equation V En ( x ) = (cid:90) B R ∞ dy − f En ( y )2 π | x − y | , x ∈ R , R ∞ >> . for f En is an ill-posed problem, but using regularization methods one couldfind approximate solutions; the resulting f En ( x ) could then be approximatedby a sum of delta functions, giving blueprints for approximate cloaks imple-mented by electrode arrays. We use the analytic expressions found above to compute the fields for a planewave with E in ( x ) = Ae ikx · (cid:126)d . The computations are made without referenceto physical units; for simplicity, we use E = 0 . κ = 2 and amplitude A = 1.Unless otherwisely stated, the cloak has parameters ρ = 0 .
01, i.e. R = 1 . × , and η = 0 . R = { r = R } . This means that (cid:15) insidethe cloaking surface is η/
20 and outside the cloaking surface (2 − R ) / W ( x ) = v in χ [0 , . ( r ) . To illustrate approximate cloaking, we used v in = −
98; to obtain an almosttrapped state, v in = 2 . ψ n and u n we use the ap-proximation that L >>
1. This implies that the cloaking conductivity γ R is piecewise constant, and correspondingly, the cloaking potential V En is aweighted sum of delta functions, and their derivatives, on spheres. In thenumerical solution of the problem, we represent the solution u n of the equa-tion ∇· γ n ∇ u + g / n ω u = 0 in terms of Bessel functions up to order N = 14in each layer where the cloaking conductivity is constant. The transmission28ondition on the boundaries of these layers are solved numerically by solvinglinear equations. After this we compute the solution ψ n of the Schr¨odingerequation using formula ψ n ( x ) = γ n ( x ) / u n ( x ).Below we give the numerically computed coefficients of spherical harmonics Y n in the case when v in = −
98 and ρ = 0 .
01, in which we do not have aneigenstate inside the cloaked region. The result are compared to the casewhen we have scattering from the potential W without a cloak. Table 1. coefficients of scattered waves for v in = − and ρ = 0 . n c n with cloak and W c n with W but no cloak0 − . − . i +0 . i . − . i − . i . . i − . i − . . i − . i − . − . i +0 . i . − . i +0 . i . . i +0 . i ( (cid:80) | c n | ) / . Figure 3:
The magnitudes of the far fields.
The far-fields θ (cid:55)→ | a ( θ, ϕ ) | with ϕ = 0 are shown: Black curve: scattering from W without thecloak. Blue curve: scattering from W surrounded by cloak, ρ = 0 . Red curve: scattering from W surrounded by cloak, ρ = 0 . Scattering from cloak.
Left: The real part of ψ when a planewave scatters from an approximate cloak in the case when E is not an in-terior eigenvalue. Due to limited resolution, ψ is sparsely sampled in radialdirection; in reality, ψ oscillates in the cloak more than is shown. Right: Adetail with finer resolution in the cloaking layers, in polar coordinates. In this section, we consider three examples of the results and ideas above toquantum mechanics. Further discussion of applications is in [25].
We first construct a system consisting of a fixed homogeneous magnetic fieldand a sequence of electrostatic potentials, the combination of which produceboundary or scattering observations (at energy E ) making it appear as if themagnetic field blows up near a point.The magnetic Schr¨odinger equation with a magnetic potential A (for mag-netic field B = ∇ × A ) and electric potential V is of the form − ( ∇ + iA ) ψ + V ψ = Eψ, in Ω , ψ | ∂ Ω = f, (55)where we have added the Dirichlet boundary condition on ∂ Ω. Take now V = V En and denote the corresponding solutions of (55) by ψ n . Let u n := γ − / n ψ n ; then these u n satisfy, cf. (47)–(49), − g − / n ∇ A · γ n ∇ A u n = Eu n , u n | ∂ Ω = f, ∇ A := ∇ + iA . Similar to the considerations above, we see that if n → ∞ , then u n → u , where u is the solution to the problem − g − / ∇ A · σ ∇ A u = Eu, u | ∂ Ω = f. Letting w ( y ) = u ( x ) with x = F ( y ) , y ∈ B \ { O } , x ∈ Ω \ B , we have that w is the solution to the magnetic Schr¨odinger equation, at energy E , with 0electric potential and magnetic potential (cid:101) A − ( ∇ + i (cid:101) A ) w − Ew = 0 , in B . Since magnetic potentials transform as differential 1 − forms, we see that,briefly using subscripts for the coordinates, (cid:101) A j ( y ) = (cid:88) k =1 A k ( x ) ∂x k ∂y j Now take the linear magnetic potential A = (0 , , ax ), corresponding tohomogeneous magnetic field B = ( a, , (cid:101) A = A in B − B , while in B (cid:101) A ( y ) = a (cid:18) | y | (cid:19) y | y | (cid:0) − y y , − y y , ( y ) + ( y ) + | y | / (cid:1) . ¿From this we see that (cid:101) A ( y ) blows up near y = 0 as O ( | y | − ) so that thecorresponding magnetic field (cid:101) B ( y ) blows up near y = 0 as O ( | y | − ).Consider now the Dirichlet-to-Neumann operator for the magnetic Schr¨odingerequation (55) with V = V En , i.e., the operator Λ V n ,A that mapsΛ V n ,A : ψ | ∂ Ω (cid:55)→ ∂ ν ψ | ∂ Ω . Then the above considerations show that, as n → ∞ , Λ V n ,A f → Λ , e A f . Inother words, as n → ∞ , the boundary observations at energy E , for themagnetic Schr¨odinger equation with a linear magnetic potential A , in thepresence of the large electric potentials V En , appear as those of a very largemagnetic potential (cid:101) A blowing up at the origin, in the presence of very smallelectric potentials. 32 .2 Case study 2: Almost trapped states concentratedin the cloaked region. Let Q ∈ C ∞ ( B ) be a real potential. The magnetic Schr¨odinger equa-tion (55) with potential V = Q + V En is, after a gauge transformation,cf. Sec. 3.1, closely related to the operator D n u = − g n ( x ) − / ∇ A · γ n ∇ A u + Qu, with domain { u ∈ L (Ω) : D n u ∈ L (Ω) , u | ∂ Ω = 0 } . We also define theoperator D , Du = − g ( x ) − / ∇ A · σ ∇ A u + Qu, which is a selfadjoint operator in the weighted space L g (Ω) with an appro-priate domain related to the Dirichlet boundary condition u | ∂ Ω = 0. Theoperators D n converge to D (see [24] for details) so that in particular for allfunctions p supported in B lim n →∞ ( D n − z ) − p = ( D − z ) − p in L g , (56)if z is not an eigenvalue of D .Assume now that E is a Neumann eigenvalue of multiplicity one of the op-erator −∇ A + Q in B but is not a Dirichlet eigenvalue of operator −∇ e A inΩ = B . Using formulae (15)–(17), one sees that then E is a eigenvalue of D of multiplicity one and the corresponding eigenfunction φ is concentrated in B , that is, φ ( x ) = 0 for x ∈ Ω \ B . Assume, for simplicity, that κ = 1, andlet p be a function supported in B that satisfies a p = (cid:90) B dx p ( x ) φ ( x ) = (cid:90) Ω dx g / ( x ) p ( x ) φ ( x ) (cid:54) = 0 . If Γ is a contour in C around E containing only one eigenvalue of D , then12 πi (cid:90) Γ dz ( D − z ) − p = a p φ. (57)However, by (56),12 πi (cid:90) Γ dz ( D − z ) − p = lim n →∞ πi (cid:90) Γ dz ( D n − z ) − p. Plane wave and approximate cloak. Re ψ when E is not aninterior Neumann eigenvalue: matter wave passes cloak almost unaltered.The Moir´e pattern is an artifact. 34igure 6: Almost trapped state.
A noncentral almost trapped state:Re ψ for potential W ( x ) = v in χ [0 , . ( r ), v in = − .
92 and E = 0 .
5, surroundedby the approximate cloak described in Sec 3.5.35y standard results from spectral theory, e.g., [35], this implies that if n issufficiently large then there is only one eigenvalue E n of D n inside Γ, and E n → E as n → ∞ . Moreover, a p φ = lim n →∞ a n,p φ n , where φ n is the eigenfunction of D n corresponding to the eigenvalue E n and a n,p is given as a n,p = (cid:90) Ω dx g / n ( x ) p ( x ) φ n ( x ) = (cid:90) B dx p ( x ) φ n ( x ) . This shows, in particular, that, when n is sufficiently large, the eigenfunctions φ n of D n are close to the eigenfunction φ of D and therefore are almost 0 inΩ − B .Applying the gauge transformation (47), we see that the magnetic Schr¨odingeroperator −∇ A + ( V E n n + Q ) has E n as an eigenvalue, −∇ A ψ n + ( V En + Q ) ψ n = E n ψ n , where ψ n = g n ( x ) − / φ n . It follows from the above that this eigenfunction ψ n is close to zero outside B . This means that the corresponding quantumparticle is mostly concentrated in B , which we may think of as an almosttrapped state located in B . S quantum mechanics in the lab The basic quantum cloaking construction outlined above can be modified tomake the wave function on B behave (up to a small error) as though itwere confined to a compact, boundaryless three-dimensional manifold whichhas been “glued” into the cloaked region. Mathematically, this could be anymanifold, M , but for physical realizability, one needs to take M to be thethree-sphere, S , topologically, but not necessarily with its standard metric, g std . By appropriate choice of a Riemannian metric g on S , the resultingapproximately cloaking potentials can be custom designed to support anessentially arbitrary energy level structure.As the starting point one uses not the original cloaking conductivity σ (the single coating construction), but instead what was referred to in36igure 7: Schematic:
Constructing an S approximate quantum cloak.3719, Sec. 2] as a double coating. This is singular (and of course anisotropic)from both sides of Σ, and in the cloaking context corresponds to coating bothsides of Σ with appropriately matched metamaterials. Here, we denote adouble coating tensor by σ (2) . The part of such a σ (2) inside B is specifiedby (i) choosing a Riemannian metric g on S , with corresponding conductiv-ity σ ij = | g | g ij ; (ii) a small ball (cid:101) B δ about a distinguished point x ∈ S ; (iii)a blow-up transformation T : S − { x } → S − (cid:101) B δ similar to the F usedin the standard single coating construction; and (iv) a gluing transformation T : S − (cid:101) B δ → B , identifying the boundary of (cid:101) B δ with the inner edge ofthe cloaking surface, Σ − . Then, σ (2) is defined as T ∗ ( T ∗ σ ) on B , and anappropriately matched single coating on B − B as before. This correpsondsto a singular Riemannian metric g (2) on B , with a two-sided conical singu-larity at Σ. One can show [19, Sec. 3.3] that the finite energy distributionalsolutions of the Helmholtz equation ( ∇ g (2) + ω ) u = 0 on B split into directsums of waves on B − B , as for σ , and waves on B which are identifiablewith eigenfunctions of the Laplace-Beltrami operator −∇ g on the compact,boundaryless Riemannian manifold ( S , g ), with eigenvalue ω .If one takes g to be the standard metric on S , then the first excited energylevel is degenerate, with multiplicity 4, while a generic choice of g yields allenergy levels simple. On the other hand, it is known that, by suitable choiceof the metric g , any desired finite number of energy levels and multiplicitiesat the bottom of the spectrum can be specified arbitrarily [12], allowing ap-proximate quantum cloaks to be built that model abstract quantum systems,with the energy E having any desired multiplicity. References [1] G. Allaire: Homogenization and two-scale convergence,
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