Schrodinger's Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics
The advent of transformation optics and metamaterials has made possible devices producing extreme effects on wave propagation. Here we give theoretical designs for devices, Schrödinger hats, acting as invisible concentrators of waves. These exist for any wave phenomenon modeled by either the Helmholtz or Schrödinger equations, e.g., polarized waves in EM, pressure waves in acoustics and matter waves in QM, and occupy one part of a parameter space continuum of wave-manipulating structures which also contains standard transformation optics based cloaks, resonant cloaks and cloaked sensors. For EM and acoustic Schrödinger hats, the resulting centralized wave is a localized excitation. In QM, the result is a new charged quasiparticle, a quasmon, which causes conditional probabilistic illusions. We discuss possible solid state implementations.
Schrödinger’s Hat: Electromagnetic, acoustic and
quantum amplifiers via transformation optics
Allan Greenleaf , Yaroslav Kurylev, Matti Lassas and Gunther Uhlmann
Dept. of Mathematics, University of Rochester, Rochester, NY 14627
Dept. of Mathematical Sciences, University College London, London, WC1E 6BT, UK
Dept. of Mathematics, University of Helsinki, FIN-00014, Finland,
Dept. of Mathematics, University of California, Irvine, CA 92697
Authors are listed in alphabetical order
(Version of July 22, 2011)
Transformation optics and metamaterials have made possible devices producing effects on wave propagation not seen in nature, including invisibility cloaks for electrostatics GLU1 (); GLU2 (), electromagnetism (EM) Le (); PSS1 (); SchurigEtAl (), acoustics CummerSchurig (); ChenChan (); CummerEtAl () and quantum mechanics (QM) Zhang (); field rotators ChenRotate (); EM wormholes GKLUWorm (); and illusion optics ChanIllusion (), among many others. The purpose of this paper is to give theoretical designs for devices, which we refer to as Schrödinger hats, acting as invisible concentrators, reservoirs and amplifiers for waves. Schrödinger hats exist for any wave phenomenon modeled by either the Helmholtz or Schrödinger equation, whether in EM, acoustics or QM. Schrödinger hats (SH) occupy one part of a parameter space continuum of wave-concentrating structures which also contains standard transformation optics based cloaks and cloaked sensors GKLUSensor (). For EM and acoustic SH, the resulting centralized wave is a localized excitation, which may be super-wavelength in scale; in QM, the SH produces a new quasiparticle, a quasmon. Acoustic and EM Schrödinger hats require negative index materials, while highly oscillatory potentials are needed for QM hats. A SH seizes a large fraction of an incident wave, holding and amplifying it as a quasmon, while contributing only a negligible amount to scattering. These devices are consistent with the uncertainty principle, and we illustrate the concept by a QM version of three card monte. While a quantum Schrödinger hat is invisible to one-particle scattering, we show by effective potential theory Parr () that a SH acts as an amplifier of two-particle Coulomb interactions. Such amplifiers may be useful for quantum measurement and information processing. The similar yet less demanding acoustic and EM hats offer equivalent effects, but existing metamaterials Shelby (); LiChan (); Lee (); ChenChanSurvey () make these designs more immediately realizable, allowing verification and further exploration of Schrödinger hats and quasmons.
There are a number of ‘paradoxes’ in which the laws of quantum mechanics imply results that conflict with our intuition paradox (). In this spirit, here we show that the behavior of matter waves, as governed by Schrödinger’s equation, combined with the virtual space/physical space paradigm of transformation optics, allows one to manipulate conditional probabilities in QM and create quantum illusions, in which observed locations of particles differ from their actual values.
Ideal (perfect) 3D quantum invisibility cloaks at fixed energy are based on the behavior of solutions to Schrödinger equations, with specific potentials and singular, inhomogeneous and anisotropic mass density Zhang (). These are mathematically equivalent, via a Liouville gauge transformation, to Helmholtz equations which also allow for cloaking in scalar optics Le (); GKLU1 () and acoustics ChenChan (); CummerEtAl (); GKLUPreprint (). Realizing a QM cloak would be challenging, due to the extreme material parameters required Zhang (). We have previously described approximate QM cloaks, avoiding extreme and anisotropic parameters but nevertheless acting with arbitrary cloaking effectiveness GKLU3 (); GKLU_JST (). If a matter wave is incident to such a potential, the scattered wave can be made as small as desired. Analysis of approximate QM cloaks revealed a difficulty: the wave vanishes inside the cloak unless the cloak supports an almost trapped matter wave (or resonance), whose existence destroys the cloaking phenomenon and makes the ‘cloaked’ region in fact detectable. However, approximate cloaks can be tuned with a precise choice of parameters, close to but not at resonance; the flow of the wave from the exterior into the cloak and from the cloaked region out into the exterior are balanced, and the cloaking effect is not destroyed, but rather greatly improved GKLUSensor (). We point out similar but surface-plasmon based effects AE () and other subwavelength plasmonics related to sensing Zul (); Fran ().
An approximate QM cloak can be implemented as follows, starting from the ideal 3D spherical transformation optics EM invisibility cloak PSS1 (). This is based on the ‘blowing up a point’ coordinate transformation GLU1 (); GLU2 () ,
This works equally well in acoustics, forming a cloak with a spherically symmetric singular anisotropic mass density and singular bulk modulus ChenChan (); CummerEtAl (); GKLUPreprint (). Consider the case where the anisotropic mass density, , is the identity matrix, and the inverse of the bulk modulus, , is outside the layer ; the cloaked region is the ball of radius 1 centered at origin. For an arbitrary choice of , the ideal cloak is then approximated, replacing both the mass density and bulk modulus by 1 in the shell (or layer) . This gives a non-singular mass density and non-singular bulk modulus , which approach the ideal cloak parameters as . Via homogenization theory, the anisotropic mass density is approximable by isotropic mass densities , consisting of shells of thickness having alternating large and small densities, yielding a family of approximate cloaks GKLU_JST (). One then obtains a QM cloak by applying the Liouville-gauge transformation , so that the Helmholtz equation becomes the time independent Schrödinger equation, , where is the energy and is the cloaking potential for the energy level .
For acoustic or EM cloaks constructed using positive index materials, resonances can allow large amounts of energy to be stored inside the ‘cloaked’ region, but at the price of destroying the cloaking effect GKLU3 (); GKLU_JST (). However, inserting negative index materials within the cloaked region allows for the cloaked storage of arbitrarily large amounts of energy; for simplicity, we describe this primarily in the context of QM cloaking, where the analogous effect is concentration of probability mass. When the cloaking potential is augmented by an internal potential consisting of a series of shells, alternating positive barriers and negative wells with appropriately chosen parameters, the probability of the particle being inside the cloaked region can be made as close to 1 as desired. More precisely, insert into a piecewise constant potential , consisting of two shells, with values , in , resp., and zero elsewhere. For suitable parameters and of the potential , we obtain, in the Supplement, a Schrödinger hat potential, . Matter waves incident on the SH are modeled by Schrödinger’s equation,
The key feature of is that the matter waves governed by (2) can be made to concentrate inside the cloaked region as much as desired, while nevertheless maintaining the cloaking effect, quantified as follows. Assume that we have two balls of radius , one () empty space and another () containing a Schrödinger hat. Let denote the central balls of radii 1 and 2, resp., for or , and assume that matter waves and on , , resp., have the same boundary values on the sphere of radius L, corresponding to identical incident waves. Define the strength of the Schrödinger hat to be the ratio
where and are solutions which coincide in . We show that, by appropriate choice of the design parameters, may be made to take any prescribed positive value. For large values of , the probability mass of is almost completely concentrated in the cloaked region. For , the wave is rapidly oscillating, and so we also consider the effective wave, , which is obtained as the limit of (in a suitable weak sense discussed in the Supplement). This is distinct from the ‘mirage effect’ for standard cloaks, which makes a source within the cloaking layer appear to be in a different position due to the chain rule ZGNP (); GKLU1 ().
We next describe some remarkable properties of Schrödinger’s hat. To start with, the highly concentrated part of the wave function which the SH and the incident wave produce inside the cloaked region can be considered as a quasiparticle, which we call a quasmon. A quasmon has a well-defined electric charge and variance of momentum depending on the parameters of . Secondly, the amplification and concentration of a matter wave in the cloaked region can be used to create probabilistic illusions. Consider (non-normalized) wave functions and which coincide in , i.e., exterior to the cloaking structure. Then for any region in the conditional probability that the particle is observed to be in , given that it is observed in , is the same for and . However, by choosing the parameters of the SH appropriately (see the Supplement), the probability that the particle is in the cloaked region can be made as close to 1 as wished. Roughly speaking, the particle is like a trapped ghost of the particle in that it is located in the exterior of the cloaking structure with far lower probability than is, but when is observed in , all measurements coincide with those of . When a particle is close to a , with a large probability it is grasped by the hat and bound into storage within the cloaked region as a quasmon. This is nevertheless consistent with the uncertainty principle: although the particle is spatially localized within , the expected value of the magnitude of its momentum is large, due to the large gradient of on a spherical shell about the central peak; cf. Figs. 1 & 3(center, right).
The Schrödinger hat produces vanishingly small changes in the matter wave outside of the cloak, while simultaneously making the particle concentrate inside the cloaked region. Thus, if the matter wave is charged, it may couple via Coulomb interaction with other particles or measurement devices external to the cloak. When an incident field is scattered by the SH, the wave field is not perturbed outside of the support of the hat potential ; there are no changes in scattering measurements. However, the cloak concentrates the charge inside the cloaked region, proportional to the square of the modulus of the value which the incident field would have had at the center of in the absence of the SH. Due to the long range nature of the Coulomb potential, this charge causes an electric field which may be strong even far away from the SH. If one measures the electric field and the result is zero, then this indicates that ; without disturbing the field, one determines whether the incident field vanishes at a given point. A measuring device within a Schrödinger hat thus acts as a non-interacting sensor, detecting the nodal curves or surfaces on which the incident matter field vanishes, an effect analogous to cloaked acoustic and EM sensors AE (); GKLUSensor () and near-field scanning optical microscopes AE2 (). As described in the Supplement, a Schrödinger hat potential also amplifies the interaction between two charged particles.
The behavior of Schrödinger hats and quasmons can be illustrated by means of a quantum variant of Three Card Monte, the classic game of chance in which a coin is hidden under one of three bowls and the player guesses where the coin is. Consider first a preliminary version of the game, played by Alice, who runs the game, and Bob, who makes the guesses. In place of bowls, they play the game using empty balls, each a copy of , and the coin is replaced by a QM particle. The surface of each ball is made of material representing an infinite potential wall, so that a particle within cannot escape; this corresponds to the Dirichlet boundary condition on the boundary. In the game, Alice inserts one particle into one of the balls, after which she mixes the balls randomly and asks Bob to guess in which ball the particle is. Bob chooses one and makes internal measurements near the boundary of . Bob, wishing to determine whether the particle is in a region , , measures the value of an observable , which is 1 if the particle is observed () and 0 otherwise; the value of is also 0 if the particle is not in the chosen ball. The expected value of is . Here, is the probability that Bob chose the ball that into which Alice inserted the particle, and , where , , and is the empty space wave function on .
To make the game more interesting, Alice and Bob make a wager: they agree that Bob will pay € to Alice in advance of each turn, but if he then observes a particle will receive 1 € back from Alice. With these rules, the game is fair, with expected profit 0 for both Alice and Bob.
Now suppose that, before play commences, and unbeknownst to Alice and Bob, a third player (the Cloaker) replaces each of the empty balls with a ball equipped with a Schrödinger hat. The expectation of is now , where , where , , and is the wave function on .
Since a Schrödinger hat is an effective cloak, outside of the ball which contains , and so . On the other hand, the presence of the Schrödinger hat amplies the wave function in and so ; hence , cf. Fig. 3. When the game is played many times, Bob’s expected chance of observing the particle in the ball which he chose is smaller than it was before the Schrödinger hats were inserted. In other words, after the Cloaker’s intervention the particles start to disappear from Bob’s observations and Bob starts to lose; Alice is unknowingly ‘cheating’. The game can be made as unfair as one wishes by choosing parameters so that is very large, using general Schrödinger hat potentials as described in the Supplement.
We conclude by describing one possible path, discussed in more detail at the end of the paper, towards a solid state realization of a quantum Schrödinger hat, utilizing a sufficiently large heterostructure of semiconducting materials. By homogenization theory, the SH potential can be approximated using layered potential well shells of depth and wall shells of height . By rescaling the coordinate we can make the values smaller (note that in such scaling the size of the support of the SH potential grows and becomes smaller). This sequence of spherical potential walls and wells can be implemented using a heterostructure of semiconducting materials. In such a structure the wave functions of electrons with energy close to the bottom of the conduction bands can be approximated using Bastard’s envelope function method Bastard (). Choosing the materials and thickness of the spherical layers suitably, the envelope functions then satisfy a Schrödinger equation whose solutions are close to those corresponding to the SH potential.
In this supplement, we provide the rigorous analysis needed to confirm the existence and behavior of Schrödinger hats, specify the parameters used in the Figures, and detail the proposed solid-state implementation.
I Analysis of Quantum cloaks and Schrödinger hat potentials
i.1 An approximate acoustic cloak
Below we will use the approximate cloaks modeled by the Helmholtz (or the Schrödinger) type equation
where is a parameter corresponding to the effectiveness of the cloak, is the frequency, and and are the coefficient functions defined below. Let denote the outward unit normal vector of . Measurements on the boundary are mathematically modeled by the Dirichelet-to-Neumann operator defined by
describing the response of the system, i.e., the Neumann boundary value , when the Dirichlet data is posed on the boundary. In the theory of the approximate cloaks the coefficient functions and are constructed in such a way that as the Dirichlet-to-Neumann operators approach to the Dirichlet-to-Neumann operator for the boundary value problem
modeling empty space. In practical terms, this means that when the parameter is close to 1, for the approximative cloak all boundary observations on are close to the observations on made when the domain is filled with a homogeneous, isotropic medium.
Approximate cloaks are the basis of our construction of Schrödinger hat potentials. We start by recalling some facts concerning nonsingular approximations to ideal 3D spherical cloaks KSVW (); GKLU3 (); GKLU_JST (); GKLU_NJP (); KOVW (); RYNQ (). For , let and be the open ball and sphere, resp., centered at the origin and of radius in three-space. Moreover, let be the closed ball. For , set , so that as , and introduce the coordinate transformation ,
For (), this is the singular transformation of GLU1 (); GLU2 (); PSS1 (), leading to the ideal transformation optics cloak, while for (), is nonsingular and leads to a class of approximate cloaks RYNQ (); GKLU3 (); GKLU_JST (); KSVW (); KOVW (). Thus, if denotes the homogeneous, isotropic mass tensor tensor, then, for , the transformed tensor becomes an anisotropic singular mass tensor, , on , defined in terms of its inverse,
This means that in the Cartesian coordinates is the matrix with elements
where the matrix , having elements , is the projection to the radial direction.
On the other hand, when , we obtain an anisotropic but nonsingular mass tensor, , on , given by
For each , the eigenvalues of are bounded from above and below; however, two of them as . We define an approximate mass tensor everywhere on by extending it as an identity matrix,
In sequel, we use the notation also for . We define a scalar function on ,
Here , are parameters which one can vary, are some fixed numbers, and is the indicator function of the interval . This means that we have a homogeneous ball coated with homogeneous shells. Sometimes we denote . Note that in acoustics has the meaning of inverse of bulk modulus; later, in quantum mechanics, it gives rise to the potential.
Below, we consider what happens as . In fact, for rigorous mathematical analysis we should modify the above definition of by replacing it, e.g., by so that for all the quadratic form corresponding to operator becomes smaller (for any fixed ) as decreases. However, in order to compute solutions explicitly and to present considerations in a simplified way, we will consider the case when is defined as above. Mathematical proofs will be presented elsewhere.
Next, consider in the domain the solutions of the Dirichlet problem,
Since, for , the matrix is nonsingular everywhere, across the internal interface we have the standard transmission conditions,
where is the radial unit vector and indicates the trace on as .
In the physical space one has
with in the virtual space, which consists of the disjoint union , satisfying
With respect to spherical coordinates , the transmission conditions (12) become
where are the standard spherical harmonics. Then equations (11) give rise to a family of boundary value problems for the . For our purposes, the most important one is the lowest harmonic term (the -mode), , i.e., the radial component of , which is independent of . This is studied in the next section.
i.2 Spherical harmonic coefficients
The lowest harmonics. For , consider the Dirichlet problem on the ball ,
We will express asymptotics in terms of the quantity as .
We have shown elsewhere GKLU_JST () that
For a specific value of the parameter , denoted , there is a blow-up effect, or interior resonance, destroying cloaking. This corresponds to the case when is such that there equation (17) has a non-zero radial solution with . In this case the solution grows very much inside the cloaked region as . his means that the inside of the cloak is in resonance and the wave tunnels outwards through the cloak, so that this resonance is detected by boundary measurements outside of the cloak.
For another specific value of , denoted , the cloak acts as an approximate cloak and inside the cloaked region the solution is proportional to the value which the field in the empty space would have at the origin. This corresponds to the case when the equation (17) has a radial solution which satisfies and , or equivalently, for .
Due to the transmission condition (12) we see that the values and are close and .
We now explain in detail how to choose : First, fix and , and choose . Consider the ordinary differential equation corresponding to the radial solutions of the equation (17), that is,
and pose the Cauchy data (i.e. initial data) at , , . Here, is the -component of the matrix , that is, , for and , elsewhere. Then we solve the initial value problem for the the ordinary differential equation (18) on interval and find the Cauchy data at . Note that on the interval , does not depend on . Consider next the case when
where and are constructed in the following way:
First, we choose to be a negative number with a large absolute value. We then solve of the initial value problem for (18) on interval with initial data at . In particular, this determines the Cauchy data at .
Secondly, consider , as well as , to be parameters, and solve the initial value problem for (18) on interval with initial data at . Denote the solution by and find the value . Then, for given , and , we find satisfying
We choose to be the smallest value for which (20) holds, and denote this solution by . Summarizing the above computations, we have obtained a cloak at the frequency , that is, for the energy , such that its radial solution satisfies for . Moreover, when is large, this solution grows exponentially fast on the interval , as becomes smaller, while on the interval it satisfies , so that defines a smooth spherically symmetric solution of (17).
In the context of QM cloaks below, the construction above can be considered as follows: Inside the cloak there is a potential well of the depth , enclosed by a potential wall having the height . The parameters and are chosen so that the solution is large inside the cloak due to the resonance there. Moreover, the choice of the parameters is such that the flow associated to the wave function from the outside into the cloaked region and from the cloaked region to the outside are in balance. The cloaked region is thus well-hidden even though the solution may be very large inside the cloaked region. In a scattering experiment, with high probability the potential captures the incoming particle, but due to the chosen parameters of the cloak, external measurements cannot detect this.
Using the implicit function theorem, one can show that for generic values of and , there is a limit We note that the solution of (18) has limit , where and an eigenfunction of the boundary value problem
where and is normalized so that .
Higher order harmonics. Let . As was chosen to be the smallest solution of (20) we have that an eigenfrequency of the problem
and is pure imaginary. Here etc. are yet undefined coefficients. Note that the terms with are absent near since has no singularity at .
Now, for ,
with as yet unspecified and .
Expand the boundary value on in surface spherical harmonics as
As shown in the previous section, , , and both and can be solved for using the transmission conditions, which determines the coefficients for .
Next we consider the higher-order coefficients, for . To simplify notations, denote by the solution of (14) for which for . Then the coefficients in (23) can be written in the form Observe that there exist the limit
and due to the way the coefficient was chosen, for generic values of we have
Next we assume that is such that (26) holds.
By the transmission condition (12),
Together with the transmission conditions on and , this implies that
The above considerations for and for can be summarized as follows: As , the solutions converge in to the solution corresponding to the homogeneous virtual space,
and, in the domain , to the solution in empty space,
Next we consider the implications of this for quantum mechanics.
i.3 Approximate isotropic cloaks in quantum mechanics - Schrödinger’s hat potential
The approximate anisotropic cloak can be further approximated by an isotropic cloak , where is a smooth isotropic (i.e., scalar-valued) mass density, which we denote by lowercase to distinguish it from the anisotropic mass tensor denoted by . It satisfies , and leads to an approximate cloak equation,
We will use isotropic mass densities which, for , are of the form
Here, are bounded non-negative smooth functions with period one such that near integer values, while near integer values. For each we choose a sequence of as such that is an integer. As for and , we take . It is possible to choose and as , so that
The are smooth functions in ;
The approximate as and then . Namely, the operators converge (as described below) to as and . Note that these are independent of .
Below we use the shorthand notation instead of . Denote