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Cloaked electromagnetic, acoustic, and amplifiers via transformation

Allan Greenleafa, Yaroslav Kurylevb, Matti Lassasc, Ulf Leonhardtd, and Gunther Uhlmanne,f,1

aDepartment of Mathematics, University of Rochester, Rochester, NY 14627; bDepartment of Mathematics, University College London, London WC1E 6BT, ; cDepartment of Mathematics and Statistics, University of Helsinki, 00014, Helsinki, Finland; dSchool of and , University of St. Andrews, St. Andrews, KY16 9SS, Scotland, United Kingdom; eDepartment of Mathematics, University of Washington, Seattle, WA 98195; and fDepartment of Mathematics, University of California, Irvine, CA 92697

Edited by George C. Papanicolaou, Stanford University, Stanford, CA, and approved April 5, 2012 (received for review October 23, 2011)

The advent of and has made the localization of the resulting waves, SH are neverthe- possible devices producing extreme effects on wave propagation. less consistent with the Heisenberg uncertainty principle. We Here we describe a class of invisible reservoirs and amplifiers for briefly describe possible implementations of Schrödinger hats. waves, which we refer to as Schrödinger hats. The unifying math- Highly oscillatory potentials are needed for QM Schrödinger ematical principle on which these are based admits such devices for hats, and we propose several realizations; one possible applica- any time harmonic waves modeled by either the Helmholtz or tion is for a near- scanning quantum . The less Schrödinger , e.g., polarized waves in , demanding acoustic and EM hats offer similar effects, but exist- acoustical waves and matter waves in quantum . Schrö- ing metamaterials (23–26) should make these Schrödinger hats dinger hats occupy one part of a parameter-space continuum of more immediately realizable, allowing physical verification and wave-manipulating structures which also contains standard trans- further exploration of the concept. formation optics based cloaks, resonant cloaks and cloaked sen- sors. Possible applications include near-field quantum microscopy. Outline of the Construction Schrödinger hats are formed from approximate cloaks surround-

field amplifiers ∣ invisibility cloaking ∣ improved cloaking ∣ imaging ing layers of barriers and wells. These can be implemented as APPLIED

follows, starting from the ideal 3D spherical transformation MATHEMATICS ransformation optics and metamaterials have made possible optics EM invisibility cloak (4). One subjects homogeneous, ε μ ‘ designs and devices producing effects on wave propagation isotropic 0 and permeability 0 to the blowing up T ’ ≔ F y not seen in , including invisibility cloaks for a point coordinate transformation (1, 2, 4), ð Þ¼ 1 y ∕2 y∕ y ; 0 y ≤ 2 (1, 2), electromagnetism (EM) (3–5), (6–8) and quan- ð þj j Þ j j for < j j , tum mechanics (QM) (9); devices inspired by jyj y (10, 11) and non-Euclidian geometry (12); field rotators (13); x ¼ FðyÞ¼ 1 þ ; for 0 < jyj ≤ 2; [1] EM wormholes (14); and illusion optics (15), among many others. 2 jyj See (16) for an overview. At nonzero , ideal (i.e., per- fect) cloaking produces a decoupling between the parts of the used to cloak the ball B1 of radius 1 centered at origin. This wave in the cloaked region and its exterior, and one has both works equally well in acoustics (7, 8, 27), and we now use the cloaking (the undetectability of the object within the cloak) and terminology from that setting. The resulting cloak consists of shielding (the inability of the wave to penetrate into the cloaked a spherically symmetric, anisotropic mass density M and bulk region) (17). In more realistic approximate cloaking, there is gen- modulus λ , both singular as r ≔ jxj → 1. For any 0 < ρ < 1, erically only a weak coupling between the regions; however, if the the ideal cloak can then be approximated by replacing M by (for acoustic or EM cloaks) or (for QM cloaks) the identity I and the bulk modulus by 1 in the shell 3 is an eigenvalue for the interior region, then there exist resonant BR − B1 ¼fx ∈ R : R>jxj ≥ 1g, where R ¼ 1 þ ρ∕2. This re- (or trapped) states which simultaneously destroy both cloaking sults in a nonsingular (but still anisotropic) mass density Mρ and and shielding (18, 19, 20). Near such a resonance, it is possible bulk modulus λρ, which converge to the ideal cloak parameters as to design the cloak parameters so that the flow of the wave from ρ → 0. Via homogenization theory, Mρ is (roughly speaking) the exterior into the cloak and vice versa are precisely balanced. approximable by isotropic mass densities mρ;ε, consisting of shells This restores and even improves cloaking, while allowing a mod- of small thickness having alternating large and small densities, erate penetration of the cloaked region by the incident wave (21), yielding a family of approximate cloaks (19), modeled by the leading to the possibility of transformation optics-based cloaked 2 Helmholtz equation ð∇ · mρ;ε∇ þ λρω Þu ¼ 0. One then obtains sensors. A different approach previously led to sensors in plasmo- an approximate QM cloak by applying the Liouville-gauge trans- nic cloaking (22). −1∕2 formation, substituting ψðxÞ¼mρ;ε ðxÞuðxÞ. Indeed, when u Purpose of Paper solves the Helmholtz equation, ψ satisfies the time independent 2 2 We show here that it is possible to go beyond the limited coupling Schrödinger equation, ð−∇ þ V c − EÞψ ¼ 0, where E ¼ ω is −1 1∕2 2 −1∕2 allowed by cloaked sensors and give designs, based on an over- the energy and V c ¼ð1 − mρ;ελρ ÞE þ mε ∇ ðmρ;ε Þ is the arching mathematical principle, for devices which we call Schrö- cloaking potential for the energy level E. dinger hats, acting as invisible reservoirs and amplifiers for waves and particles. Schrödinger hats (SH) exist for any wave phenom- enon modeled by either the Helmholtz or Schrödinger equation. Author contributions: A.G., Y.K., M.L., U.L., and G.U. have equally performed research and They are specified by either a mass density/bulk modulus pair (for wrote the paper. (Authors are listed in alphabetical order.) Helmholtz) or a potential (for Schrödinger), and come in families The authors declare no conflict of interest. which increasingly exhibit their characteristic properties as the This article is a PNAS Direct Submission. parameters ϵ and ρ go to zero. Freely available online through the PNAS open access option. A SH seizes a large fraction of a time harmonic incident wave, 1To whom correspondence should be addressed. E-mail: [email protected]. holding and amplifying it while contributing only a negligible This article contains supporting information online at www.pnas.org/lookup/suppl/ amount to ; see Fig. 1. In , despite doi:10.1073/pnas.1116864109/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1116864109 PNAS ∣ June 26, 2012 ∣ vol. 109 ∣ no. 26 ∣ 10169–10174 Downloaded by guest on September 28, 2021 where ψ em and ψ sh are solutions which coincide in jxj > 2.We show that, by appropriate choice of the design parameters, S may be made to take any prescribed positive value. For large values of S, the probability density of ψ sh is almost completely concentrated in the cloaked region.

Properties. Schrödinger hats have some remarkable effects on wave propagation. • They act as reservoirs, capturing, amplifying and storing energy from incident time harmonic acoustic or EM waves, or prob- ability density from incident matter waves in QM. We remark that potentials which, for some incident wave, produce a scat- tered wave which is exactly zero outside a bounded set, are said to have a transmission eigenvalue (30). In contrast, the scat- tered wave caused by a Schrödinger hat potential is approxi- Fig. 1. A quasmon inside a Schrödinger hat. The real part of the effective mately zero for all incident fields. ψ sh x; y; z z 0 effð Þ at the plane ¼ when a plane wave is incident • The amplification and concentration of a matter wave in the to a SH potential. By varying the design parameters, the concentration of cloaked region can be used to create probabilistic illusions. the wave inside the cloaked region can be made arbitrarily strong and For any L>2, consider (nonnormalized) wave functions the scattered field arbitrarily small. The matter wave is spatially localized, em sh ψ and ψ on BL, for empty space and a SH in B2, resp., but conforms to the uncertainty principle, with the large , visible B − B as the steep slope of the central peak, concentrating the in a which coincide in the shell L 2. Then for any region p ψ sh R ⊂ BL − B2 the conditional probability that the particle is ob- spherical shell in -space. For definition of eff and a video of time harmonic wave, see (SI Text and Movie S1). For comparison with non-Schrödinger hat served to be in R, given that it is observed in BL − B2, is the cloaks, see Fig. 2 and Fig. S1. The wave function with another incident wave is same for ψ em and ψ sh. However, by choosing the parameters of shown in Fig. S2. the SH appropriately, the probability that the particle ψ sh is in the cloaked region B1 can be made as close to 1 as wished. The particle ψ sh is like a trapped of the particle ψ em in that it For acoustic or EM cloaks constructed using positive index ma- is located in the exterior of the SH structure with far lower terials, resonances can allow large amounts of energy to be stored ψ em ψ sh B − B ‘ ’ probability than is, but when is observed in L 2, inside the cloaked region, but at the price of destroying the cloak- all of its time harmonic measurements coincide with those of ing effect (18, 19), particularly strongly in the near field. However, ψ em (see Fig. S3). here we show that inserting materials with negative bulk modulus • The highly concentrated part of the wave function inside the or permittivity within the cloaked region allows for the cloaked cloaked region of a QM Schrödinger hat is, as mentioned storage of arbitrarily large amounts of energy. A similar effect was above, a localized excitation which we refer to as a quasmon. described in two-dimensional superlenses, where nonradiating The strong concentration of the wave function in B1, without (i.e., cloaked) high-energy concentrations can appear due to anom- change to the wave function outside the ball B2 where the SH alous resonance (28, 29); see discussion in (SI Text). For brevity, potential is supported, is nevertheless consistent with the un- we describe Schrödinger hats primarily in the context of QM cloak- certainty principle: although the particle is spatially localized ing, where the analogous effect is concentration of probability within B1, the variation of its momentum is large, due to density. When the cloaking potential is augmented by a layered the large gradient of ψ sh on a spherical shell about the central internal potential consisting of N shells, alternating positive bar- peak; cf. Figs. 1 and 2 (red), and Fig. S3. riers and negative wells with appropriately chosen parameters, the • A quasmon excited within a QM Schrödinger hat has a well- probability of the particle being inside the cloaked region can be defined electric and variance of momentum, depending made as close to 1 as desired. For simplicity, we restrict ourselves on the parameters of the hat. The Schrödinger hat produces to N ¼ 2 for some of the discussion and simulations, but larger vanishingly small changes in the matter wave outside of the values of N allow for central excitations, or quasmons,witharbi- cloak, while simultaneously making the particle concentrate in- side the cloaked region. Thus, if the matter wave is charged, it trarily many sign crossings. More precisely, insert into B1 apiece- may couple via interaction with other particles or wise constant potential QðxÞ, consisting of two layers with values τ1 B τ B − B measurement devices external to the cloak. When a time har- in s1 , 2 in s2 s1 , and zero elsewhere. For suitable parameters ψ in ρ ϵ s τ Q monic incident field is scattered by the SH, the field is only , , j and j of the potential ,weobtain(SI Text) a Schrödinger perturbed negligibly outside of the support of the hat potential, V V Q E hat potential, SH ¼ c þ . Matter waves with energy that V ; there is essentially no scattering. However, the Schrödin- ’ SH are incident on the SH are modeled by Schrödinger sequa- ger hat concentrates the charge inside the cloaked region, pro- −∇2 V − E ψ 0 2 tion, ð þ SH Þ ¼ . portional to jψ inð0Þj , the square of the modulus of the value V The key feature of SH is that the resulting matter waves can which the incident field would have had at the center of BL in be made to concentrate inside the cloaked region as much as de- the absence of the SH. Due to the long range nature of the sired, while nevertheless maintaining the cloaking effect, quanti- Coulomb potential, this charge causes an which fied as follows. Assume that we have two balls of radius L>2, can be measured even far away from the SH. If the result is em sh in BL and BL , containing empty space and a Schrödinger hat, resp. zero, this indicates that ψ ð0Þ¼0; without disturbing the · · Let B1;B2 denote the balls of radii 1 and 2, resp., centered at 0, field, one determines whether the incident field vanishes at for ·¼ em or sh, and assume that matter waves ψ em and ψ sh on 0 (see SI Text). Generically, the nodal set for a complex-valued em sh BL , BL , resp., have the same boundary values on the sphere of wave function in 3D is a curve; however, if an radius L, corresponding to identical incident waves. Define the is real and there is no , then the real and ima- strength of the Schrödinger hat to be the dimensionless ratio ginary parts of the wave can be linearly dependent and the nodal set is a surface. In either case, a measuring device within Z 1 a Schrödinger hat can act as a non–interacting sensor, detect- S ψ sh x 2dx; [2] ¼ em 2 j ð Þj ing, for an ensemble of quantum systems, the nodal set on volðB1Þjψ ð0Þj B SH 1 which the incident matter wave vanishes, allowing for possible

10170 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1116864109 Greenleaf et al. Downloaded by guest on September 28, 2021 ( 3 y; for 2 < jyj < L; x ≔ FρðyÞ¼ [3] 1 jyj y ; ρ y ≤ 2 2.5 þ 2 jyj for < j j .

2 For ρ ¼ 0 (that is, R ¼ 1), this is the singular transformation [1], leading to the ideal cloak, while for ρ > 0 (that is, R>1), Fρ is nonsingular and leads to a class of approximate cloaks (18, 19, 20, 1.5 32, 34). If we set I ¼ðδjkÞ denotes the identity matrix, then, for ρ ¼ 0, this pushes forward into an anisotropic singular mass ten- 1 sor, ðM0ÞjkðxÞ,onBL − B1, defined in terms of its inverse,

M−1 jk x F I jk y 0.5 ð 0 Þ ð Þ¼ðð 0Þ Þ ð Þ 1 3 ∂ F j ∂ F k ð 0Þ ð 0Þ pq 0 ≔ p ðxÞ q ðxÞδ ðxÞ : ∂F ∑ ∂x ∂x x F −1 y 0 p;q 1 ¼ 0 ð Þ det ∂x ðxÞ ¼ −0.5 M x B − B 0 0.5 1 1.5 2 2.5 3 3.5 4 0ð Þ is the matrix-valued function on 2 1 with elements 1 1 40 M x δ − P x x − 1 −2 x 2P x ; ð 0Þjkð Þ¼2 ð jk jkð ÞÞ þ 2 ðj j Þ j j jkð Þ 35 −2 where the matrix PðxÞ, having elements PjkðxÞ¼jxj xjxk, is the 30 projection to the radial direction. On the other hand, when ρ > 0, we obtain an anisotropic but nonsingular mass , MρðxÞ,on 25 BL − BR, given by

20 M−1 jk x F I jk y APPLIED ð ρ Þ ð Þ¼ðð ρÞ Þ ð Þ MATHEMATICS 15 1 3 ∂ F j ∂ F k ð ρÞ ð ρÞ pq ≔ p ðxÞ q ðxÞδ ðxÞ ; [4] ∂F ∑ ∂x ∂x −1 10 ρ p;q 1 x¼Fρ ðyÞ det ∂x ðxÞ ¼ 5 M x I B ∪ B − B ρ > 0 0 and ρð Þ¼ on R ð L 2Þ. For each , the eigenva- lues of Mρ are bounded from above and below; however, one of −5 them tend to ∞ as ρ → 0. Fixing an R0 < 1, we also define a sca- lar bulk modulus function λρðxÞ on BL, −10 0 0.5 1 1.5 2 2.5 3 3.5 4 8 η x ; x ∈ B ; < ð Þ R0 Fig. 2. The cloak–resonance–sensor–Schrödinger hat continuum. Scattering λ x 1 x 2 x − 1 −2; x ∈ B − B ; [5] ρð Þ¼: 8 j j ðj j Þ 2 R by potentials with different parameters, demonstrating four modes: cloak, 1; x ∈ B − B ∪ B − B resonance, sensor and Schrödinger hat. Graphs show the real parts of u~ (de- ð R R0 Þ ð L 2Þ fined in Thm. 1) for 0 ≤ r ≤ 4; note different vertical scales. Upper: Cloak B (black) allows little penetration of incident wave into cloaked region, 1. Re- where η is a layered combination of barriers and wells, sonant curve (blue) shows blowup (to off-chart values, see Fig. S1) within B1, and large near-field deviation from incident wave, indicating failure of cloak- N 1 ing. Sensor (purple) combines cloaking effect with moderate penetration η x η x; τ χ s ;s x : [6] ð Þ¼ ð Þ¼∑ ð j−1 jÞðj jÞ B B τj into 1. Schrödinger hat (red) obtains large value within 1, allowing for very j¼1 sensitive cloaked measurement of incident waves. Note that the cloak, sensor and SH solutions are essentially identical with the incident wave outside of Here τ ¼ðτ1; τ2; …; τN Þ, τj ∈ R are parameters which one can the cloaked region, while the resonant solution diverges, destroying cloaking vary, 0 ¼ s0 < sj < sN ¼ R0 are some fixed numbers, and (18, 19). Lower: Continuous parameter variation from sensor to Schrödinger χ r 1 s ;s ðsj−1;sjÞð Þ¼ on the interval ð j−1 jÞ and vanishes elsewhere. hat modes. Height of central excitation (quasmon) can be made arbitrarily B Thus, we have a homogeneous ball s0 coated with concentric large, and width of peak made arbitrarily small. For parameters used, see λ x λ x; τ (SI Text). homogeneous shells, sometimes writing ρð Þ¼ ρð Þ. While in acoustics λρ denotes the bulk modulus, in quantum mechanics −1 later, λρ will give rise to the potential. near-field scanning quantum microscopy, analogous to cloaked Next, consider in the domain BL the solutions of the boundary acoustic and EM sensors (21, 22) and near-field optical micro- value problem, scopes (31). −∇ M−1∇−ω2λ−1 u 0 B ;u h: [7] ð · ρ ρ Þ ρ ¼ in L ρjSL ¼ Details of the Construction. The two main ingredients of Schrödin-

ger hats are approximate cloaks and barrier/well layers. We start Since the matrix Mρ is nonsingular everywhere, across the inter- by recalling some facts concerning nonsingular approximations to nal interface SR we have the standard transmission conditions, ideal 3D spherical cloaks (18, 19, 20, 32, 33). For 0 ≤ ρ < 2, set R 1 1 ρ R → 1 ρ → 0 −1 −1 ¼ þ 2 , so that as ; as above, our asymptotics uρjS ¼ uρjS − ; er · ðMρ ∇uρÞjS ¼ er · ðMρ ∇uρÞjS − ; ¯ R þ R R þ R will be in terms of ρ.LetBR ¼fjxj < Rg, BR ¼fjxj ≤ Rg and [8] SR ¼fjxj¼Rg be the open ball, closed ball and sphere centered 0 R at the origin and of radius , resp. Introduce the coordinate where er is the radial unit vector and the indicates the the transformation Fρ∶BL − Bρ → BL − BR, boundary value on SR as r → R . In the physical space / virtual

Greenleaf et al. PNAS ∣ June 26, 2012 ∣ vol. 109 ∣ no. 26 ∣ 10171 Downloaded by guest on September 28, 2021 −1 2 space of transformation optics, here the physical space nent of the matrix Mρ , that is, σρðrÞ¼2ðr − 1Þ , for R < r < 2 is BL, and the virtual space is a disjoint union, ðBL − BρÞ ∪ BR. and σρðrÞ¼1 elsewhere. We solve the initial value problem for On the ball BL in physical space, one has [14] for r on the interval ½R0;L, and find the Cauchy data 0 0 du ðuðR0Þ;u ðR0ÞÞ at r ¼ R0, where u ≔ dr. Note that on the inter- þ −1 val ½R0;L, λρ does not depend on τ. Consider the case when s1 ¼ vρ ðFρ ðxÞÞ; x ∈ BL − BR; u x [9] R0∕2;s2 ¼ R0 and τ ¼ðτ1; τ2Þ, where τ1 and τ2 are constructed ρð Þ¼ − vρ ðxÞ; x ∈ BR; as follows: First, choose τ2 to be a negative number with a large absolute value. Then solve the initial value problem for [14]on 0 ½s1;R0 with initial data ðuðR0Þ;u ðR0ÞÞ at r ¼ R0. In particular, vρ where , on the virtual space, is the solution of 0 this determines the Cauchy data ðuðs1Þ;u ðs1ÞÞ at r ¼ s1. Sec- −∇2 − ω2 vþ y 0; y ∈ B − B ;vþ h; ondly, consider τ1; τ2, as well as ρ;R0, to be parameters, and ð Þ ρ ð Þ¼ L ρ ρ jSL ¼ solve the initial value problem for [14] on interval ½0;s1 with in- 0 itial data ðuðs1Þ;u ðs1ÞÞ at r ¼ s1. Denote the solution by and u r; τ ; ρ;R ; τ du r; τ ; ρ;R ; τ ð 1 0 2Þ and find the value dr ð 1 0 2Þjr¼0.For 2 2 −1 − ρ;R0, and τ2 given, find τ1 > 0 satisfying ð−∇ − ω λρ Þvρ ðyÞ¼0; y ∈ BR: [10] du r; θ; φ r; τ ; ρ;R ; τ 0 [15] With respect to spherical coordinates ð Þ, the transmission dr ð 1 0 2Þ ¼ . conditions [8] become r¼0

þ − We choose τ1 to be a value for which [15] holds, and denote this vρ ðρ; θ; ϕÞ¼vρ ðR; θ; ϕÞ; [11] solution by τ1ðρ;R0; τ2Þ; choosing higher values for τ1 leads to 2 þ 2 − ρ ∂rvρ ðρ; θ; ϕÞ¼R ∂rvρ ðR; θ; ϕÞ: quasmons with larger numbers of sign changes. Set τshðρÞ ≔ ðτ1ðρ;R0; τ2Þ; τ2Þ. Summarizing the above computations, we have 2 Since Mρ; λρ are spherically symmetric, cf. [4, 5], we can separate obtained a cloak at frequency ω, that is, for the energy E0 ¼ ω , variables in [7], writing uρ as with radial solution uðxÞ satisfing uðxÞ¼j0ðωjxjÞ for jxj ∈ ½2;L. Moreover, when τ2 is large, this solution uρðrÞ grows exponen- ∞ n s ;s r nm m tially fast on the interval ½ 1 2,as becomes smaller, while uρðr; θ; ϕÞ¼ uρ ðrÞY n ðθ; ϕÞ; [12] 0 ∑ ∑ on the interval ½0;s1 it satisfies u ð0Þ¼0, so that uðrÞ defines n¼0 m¼−n a smooth spherically symmetric solution of [13]. m In the context of QM cloaks below, the construction above can where Y n are the standard spherical harmonics, giving rise to a nm be considered as follows: Inside the cloak there is a potential well family of boundary value problems for the uρ . The most impor- of depth −τ1, enclosed by a potential barrier of height τ2. The tant term, which we now analyze, is the lowest harmonic (the s- τ τ u0;0 u parameters 1 and 2 are chosen so that the solution is large in- mode), ρ , i.e., the radial component of ρ. side the cloak due to the resonance there. The cloaked region is thus well–hidden even though the solution may be very large in- The Lowest Harmonic. Consider the asymptotics as ρ → 0 of the side the cloaked region. In a fixed-energy scattering experiment, Dirichlet problem, with high probability the potential captures the incoming particle, −1 2 −1 but due to the chosen parameters of the cloak, external measure- ð−∇ · Mρ ∇−ω λρ Þuρ ¼ 0 in BL;uρjS ¼ hðxÞ: [13] L ments cannot detect this. Using the implicit function theorem, one can show that for We have shown (19) that for a specific value of the parameter generic values of τ2 and R0, limρ→0τ1ðρ;R0; τ2Þ ≔ τ1ðR0; τ2Þ τ ∈ RN , denoted τ ¼ τresðρÞ, such that the equation 13 has a non- exists. We note that the solution uρðrÞ ≔ uðr; τ1ðρ;R0; τ2Þ; ρ; τ2Þ zero radial solution with h ¼ 0, there is an interior resonance, of [14] has limit limρ→0uðr;τ1ðρ;R0; τ2Þ; ρ; τ2ÞÞ ¼ cΦðrÞ;r< 1, destroying cloaking. The field uρ grows larger inside the cloaked c ∈ C Φ r ≢0 ρ → 0 where and ð Þ is an eigenfunction of the boundary region as , and this resonance is detectable, both by (near- value problem, field) boundary measurements outside of the cloak and (far-field) scattering data. 2 2 −1 ð∇ þ ω λ ðy; τÞÞΦðjyjÞ ¼ 0; ∂rΦðjyjÞjr 1 ¼ 0; [16] On the other hand, for another value of τ, denoted τ ¼ τshðρÞ, 0 ¼ the cloak acts as an approximate cloak and inside the cloaked where τ ¼ðτ1ðR0; τ2Þ; τ2Þ and Φ is normalized so that region the solution is proportional to the value which the field ‖Φ‖L 2 B ¼ 1. This finishes the analysis of the lowest harmonic; in the empty space would have at the origin. This corresponds ð 1Þ 13 u for the higher order harmonics, see (SI Text). to the equation having a radial solution ρ which satisfies ρ → 0 B − B 0 In summary: As , in the region L 2 the solutions ∂ruρðLÞ¼ωj0ðωLÞ and uρðLÞ¼j0ðωLÞ, or equivalently, uρðxÞ converge to the solution u corresponding to the homoge- uρðxÞ¼j0ðωjxjÞ for x ∈ BL − BR. Due to the transmission con- neous virtual space boundary problem, dition [8] we see that the values τresðρÞ and τshðρÞ are close, sh res with limρ→0τ ðρÞ − τ ðρÞ¼0. 2 2 ð−∇ − ω Þu ¼ 0 in BL;ujS ¼ hðxÞ [17] We now explain how to find τ ¼ τshðρÞ; note that τshðρÞ de- L pends on ω, ρ > 0, and 0 < R0 < 1, but these parameters are B omitted in the notation below. For simplicity, we with and, in the cloaked region 1, to the solution in empty space, N ¼ 2. Consider the ODE corresponding to the radial solutions limuρðxÞ¼βuð0ÞΦðxÞ [18] uðrÞ of the equation 13, i.e., ρ→0 1 d d − r 2σ r u r − ω2λ−1 r u r 0; [14] where ΦðyÞ¼ΦðrÞ is the radial solution of the equation 16, 2 dr ρð Þ dr ð Þ ρ ð Þ ð Þ¼ 1 r β ¼ Φ 1 and uð0Þ is the value of the solution of [17] at the origin. ð Þ 3 One can replace the ball BL with an arbitrary domain Ω ⊂ R and pose the Cauchy data (i.e., initial data) at r ¼ L, containing BL, and show that adding the bulk modulus [6] inside 0 uðLÞ¼j0ðLωÞ, ∂ruðLÞ¼ωj0ðLωÞ. Here, σρðrÞ is the rr-compo- the cloaked region improves the cloaking effect (see SI Text)

10172 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1116864109 Greenleaf et al. Downloaded by guest on September 28, 2021 8 Via homogenization, the waves in such a Schrödinger hat with > wðxÞ; in BL − B2; M < anisotropic ρ are well-approximated by waves governed by the −1 lim ψn ¼ ν x w F x ; B − B ; Helmholtz equation with isotropic mass densities mρ;ε as ε → 0. n→∞ > ð Þ ð ð ÞÞ in 2 1 :> One can then use the classical Liouville gauge transformation to w 0 Φ x ; B1; 2 ð Þ ð Þ in obtain potentials V ρ;ε such that, at energy E ¼ ω , the solutions 8 2 > w x 2; B − B ; of the Schrödinger ð−∇ þ V ρ;ε þ QρÞψ ¼ Eψ, admit <> j ð Þj in L 2 2 −1 2 solutions with similar behavior. See the theorems below and lim jψnj ¼ θðxÞjwðF ðxÞÞj ; in B2 − B1; n→∞ > (SI Text). : 2 jwð0ÞΦðxÞj ; in B1; Scattering by a Schrödinger Hat. Now consider the effect of Schrö- 3 dinger hats on scattering experiments in R . In free space, a wave L2 B Ψ weakly in ð LÞ and in the sense of distributions, resp., function satisfies We note that θðxÞ above is not equal to ηðxÞ2. Due to these the- orems, it is natural to define ψ sh ðxÞ ≔ θðxÞ1∕2u~ðxÞ, whose absolute −∇2 − E Ψ 0; R3; [19] eff 2 ð Þ ¼ in value squared equals limn→∞jψnðxÞj . Ψ x Ψin x ≔ eiωe·x; e 1; ω2 E and we choose ð Þ¼ ð Þ j j¼ ¼ , a plane Implementations. We briefly describe possible physical realizations wave in the direction e. We compare these with the wave func- of Schrödinger hats for a variety of wave phenomena, first for 3 tions in R scattered by the SH potential, acoustics and EM using materials with negative bulk modulus or permittivity, and then for QM via highly oscillatory potentials. 2 3 ð−∇ þ V ρ;ε þ Qρ − EÞΨρ;ε ¼ 0; in R ; (The lossy nature of presently available metamaterials will make [20] Ψ Ψin Ψsc ; effective implementations of Schrödinger hats technically chal- ρ;ε ¼ þ ρ;ε lenging.) Details will appear elsewhere.

sc where Ψρ;ε satisfies the Sommerfeld condition. (Note i. Equation S23 with isotropic mass mρ;ε and bulk modulus λρ in that the SH potential is of compact support, since it vanishes out- (SI Text) describes an approximate acoustic cloak. The nega- τ Q side B2). tive values of 2 required for the potential ρ correspond to a Since scattering data for these problems is equivalent to material with negative bulk modulus; such materials have al- APPLIED MATHEMATICS Dirichlet-to-Neumann operators on ∂BL (35), one can use the ready been proposed (36). There are many designs and reali- – results above to analyze scattering by a SH potential. We see, zations for acoustic cloaks (6 8, 17, 37); acoustic Schrödinger using [S16], [S21], and [S24] from (SI Text), that when ρ and ε hats could be implemented by placing layered negative bulk in modulus material inside such a cloak. Possible implementa- are small enough the solution Ψρ;ε is close to Ψ outside B2, close −1∕2 in −1 in tions of acoustic cloaks, as well as potential difficulties, are to mρ;ε ðxÞΨ ðF ðxÞÞ in B2 − BR, and close to Ψ ð0ÞΦðxÞ in discussed in (SI Text). Similarly, for electromagnetic waves, the cloaked region. Thus, we see that scattering observations, one can consider a cylindrical EM cloak (5) and insert in it i.e., observables depending on the far field patterns of the solu- material with negative permittivity to implement a structure ρ ε tions, are almost the same for the Schrödinger hat (when and similar to the SH potential. The analysis related to such cylind- are small) as for empty space. rical cloaks with parameters suitably chosen to create a SH potential for incident time harmonic TM-polarized waves is Theorems. The effects of Schrödinger hats on incident time similar to the arguments here for the 3D spherical cloak, harmonic waves are encapsulated in the following statements. S9 2 although with different asymptotics for [ ]. Assume that E ¼ ω is not a Dirichlet eigenvalue in BL and ii. Invisible plasmons. Surface plasmons are localized electro- 1 2 2 h ∈ H 2ðSLÞ.Letw be the unique solution of ð∇ þ ω Þw ¼ 0; magnetic waves that can exist at the boundaries between ma- wjS ¼ h. We assume that the radially symmetric, isotropic mass terials with positive and negative electric permittivity such as L 1 density mρ;ε can be written as mρ;εðxÞ¼mðjxj; ε jxjÞ where m is and metals (38). The localized wave inside the SH periodic function in the variable with period 1. Below, ν resembles a surface plasmon; however, in contrast to ordin- −1∕2 −1 and θ are the limits of mρ;εðxÞ and mρ;εðxÞ , resp., in the ary surface plasmons, it is hidden. These cloaked plasmons sense of distributions, as ε → 0, ρ → 0. This means that θðxÞ is could be produced using techniques to those in 2D plasmonic the averagepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of r 0 ↦ mðjxj;r0Þ and νðxÞ is the average of cloaking (39), where the electromagnetic wave of the plas- r 0 ↦ mðjxj;r0Þ over the interval 0 ≤ r 0 ≤ 1. Below, Φ is the mon propagates on a surface covered by a gold layer L2-normalized eigenfunction satisfying [16]. with concentric rings of polymer on top. The widths and the radii of the rings can be chosen so that the structure im- Theorem 1. For any R0;R1 and τ2 there are parameters τ1ðnÞ, n ∈ plements an invisibility cloak for a wave localized on the sur- Z ρ ; ρ 0 ε → 0 n → ∞ face plane. In 2D plasmonic cloaking (39), such a structure þ and sequences n n, n as , such that the solutions u 7 m has been used to implement a nonmagnetic optical cloak n of the acoustic equation corresponding to coefficients ρn;εn 2 (40), an approximate cloak for which the rays are trans- and λρ 0 converge weakly in L ðBLÞ, with n formed according to the transformation [3]. Including two 8 > w x ; B − B ; additional metal- rings in the interior of the cloak- <> ð Þ in L 2 ing structure, we can then implement a SH potential (see SI −1 ue ≔ lim un ¼ w F x ; B − B ; Text and Fig. S4). In this case the plasmon is not only con- n→∞ > ð ð ÞÞ in 2 1 :> fined to the surface but becomes strongly localized in the SH, w 0 Φ x ; B : ð Þ ð Þ in 1 while remaining invisible. iii. Hidden matter waves. It is also possible that a SH can be pro- V sh m Let n be the SH potential corresponding to coefficients ρn;εn and duced for matter wave cloaks (9). Here one can exploit the λ 0 E ρn and energy . fact that the normal roles of light and matter can be reversed: light acts like a refractive index for matter waves, see e.g., (41). Theorem 2. The solutions ψn of the Schrödinger equation in BL with Light fields can thus be used to generate the effective index sh the SH potentials V n , the energy E and the Dirichlet boundary distributions required for the implementation of a SH, see ψ h value njSL ¼ satisfy (42). For cold confined in 2D by a light sheet, one could

Greenleaf et al. PNAS ∣ June 26, 2012 ∣ vol. 109 ∣ no. 26 ∣ 10173 Downloaded by guest on September 28, 2021 produce the SH potentials by illuminating the sample ortho- concentric potential wells and barriers can be implemented gonally to the light sheet with a light beam, generated by a with a heterostructure of semiconducting materials. In such hologram, that carries the intensity distribution required for a structure the wave functions of with energy close the SH. The result would be a highly localized matter wave to the bottom of the conduction bands can be approximated that nevertheless remains hidden. using Bastard’s envelope function method (43). Choosing the iv. Another possible path towards a solid state realization of a materials and thickness of the spherical layers suitably, the en- quantum Schrödinger hat utilizes a sufficiently large hetero- velope functions satisfy a Schrödinger equation whose solu- structure of semiconducting materials. By homogenization tions are close to those for a SH potential. theory, the SH potential can be approximated using layered potential well shells of depth −V − and barrier shells of height V x V ACKNOWLEDGMENTS. AG is supported by US NSF; YK by UK EPSRC; ML by þ. By rescaling the coordinate we can make the values Academy of Finland; UL by EPSRC and the Royal Society; and GU by US smaller; in such scaling the size of the support of the SH po- NSF, a Walker Family Endowed Professorship at UW, a Chancellor Professor- tential grows and E becomes smaller. This layered family of ship at UC, Berkeley, and a Clay Senior Award.

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