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PHYSICAL REVIEW B 87, 201106(R) (2013)

Furtive quantum sensing using matter-wave cloaks

Romain Fleury and Andrea Alu*

Department of Electrical & Computer Engineering, The University of Texas at Austin, 1 University Station C0803,Austin, Texas 78712-1684, USA

(Received 13 March 2013; published 15 May 2013)

We introduce the concept of furtive quantum sensing, demonstrating the possibility of concealing quantumobjects from matter waves while maintaining their ability to interact and to get excited by the impinging particles.This is obtained by cloaking, with tailored homogeneous metamaterial layers, quantum systems having internaldegrees of freedom. The effect is a low-observable quantum sensor with drastically reduced elastic scatteringand optimized absorption levels, a concept that opens interesting venues in particle detection, high-efficiencyelectrical pumping, and quantum information processing.

DOI: 10.1103/PhysRevB.87.201106 PACS number(s): 78.67.Pt, 03.75.−b, 34.80.Gs, 85.35.Be

The enticing possibility of inducing invisibility with passivemetamaterial coatings has drawn considerable attention overthe past few years, leading to the formulation of differentapproaches for electromagnetic,1–13 acoustic,14–24 and morerecently, matter waves.25–33 Among these techniques, thetransformation-optics method redirects the impinging wavearound a concealed region in such a way that any form ofscattering is eliminated.1–5,14–21,25–30 As an alternative method,the scattering cancellation approach utilizes homogeneousand isotropic shells to suppress the dominant scatteringterms in the scattered field expansion, leading to substantialscattering reduction for moderately sized objects with arguablysimpler practical designs.6–13,22–24,31–33 Both methods recentlyhave been applied to matter waves,25–33 showing that itmay be possible to suppress elastic particle scattering off aquantum object by suitably coating it with metamaterials withspecifically tailored effective mass m and potential V .

Regardless of the cloaking technique, it is possible, inprinciple, to sustain the cloaking operation while allowing theimpinging field to enter the cloaking layer, enabling controlledinteraction with the hidden object that is not isolated by thecloak. By placing an absorptive object inside a plasmoniccloak, it has been theoretically suggested34 and recently provenexperimentally35 that it may be possible to extract some portionof the impinging energy while remaining undetectable andminimizing the perturbation on the measured field. In theelectromagnetic and acoustic scenarios, this has naturally ledto the idea of cloaked sensors, characterized by a dramaticreduction in their scattering cross section while preservingthe sensing capability.34–46 The concept has also later beenextended to a modified form of transformation-based cloakingand has opened exciting venues in near-field scanning opti-cal microscopy,39,40,42 low-observable antennas,44 improvedphotodetectors,35 and microphones.46

In this Rapid Communication, we apply this concept tothe quantum world, providing the possibility of controllingthe ballistic flow of particles through scattering centers andextending the reach of this technology to cold-atom physics,solid-state devices, nanophotonics, and quantum informationscience. With the development of new approaches to quantumelectronics dealing with increasingly smaller length scales,the control and monitoring without interference of ballisticelectron flows has become more and more challenging, and the

development of novel strategies for low-observable particleflow sensing is becoming increasingly important. Motivatedby these issues, we introduce the concept of furtive quantumsensing, showing that, by surrounding a small quantum systemwith a matter-wave cloak, it may be possible to cancel theelastic scattering while preserving the probability of inelastic-scattering events, opening the possibility of performing sens-ing operations on particle beams without sensibly disturbingthem.

Consider the geometry of Fig. 1(a) in which a quantumsystem with internal degrees of freedom, such as a two-levelquantum dot or a molecule, is used to detect the presence ofimpinging particles with defined momentum, such as electronsor cold atoms. Two different types of scattering events canbe distinguished. First, the particle can scatter elastically,conserving its energy but changing the momentum direction.Second, the particle can scatter through one of the possibleinelastic channels and can excite the system, in which case, itloses some energy. When this happens, we are able to detect theimpinging wave by, for instance, measuring the spontaneouslight emission from the quantum system.

Seen from the perspective of the impinging beam, inelasticscattering represents a loss of a portion of the impingingparticles, associated with the absorption cross section σabs, incontrast with the elastic cross section σel . Quantum sensorsare generally chosen such that not all impinging particlesparticipate in inelastic processes, in order to be able to furtherinteract with the matter-wave flow. Therefore, the bare sensorusually has a small σabs, leading to small absorption efficien-cies, defined as the ratio σabs/σel . Ideally, one would like tosuppress the elastic portion of the scattered beam for a givendesired level of absorption, increasing the overall efficiencyby avoiding unnecessary elastic-scattering contributions thatdivert a significant portion of the impinging beam in alldirections. We envision achieving this “ideal quantum sensor”in Fig. 1(b) by surrounding our system with a quantummetamaterial cloak. For this purpose, transformation-basedcloaks25 may be applied as envisioned for quantum amplifiersin Ref. 29, but here, we employ a homogeneous cloaking layerwith isotropic properties, designed applying the scatteringcancellation technique31–33 to suppress the overall elasticscattering at energy E of the impinging particles. Because thewave function enters the cloak, the sensing ability of the target

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FIG. 1. (Color online) (a) A bare quantum system with internaldegrees of freedom may be able to extract energy and may be ableto sense an impinging particle beam but will also perturb the matter-wave flow, becoming detectable by its surroundings and affectingits measurement; (b) a cloaked quantum sensor may still be able toextract energy but with much reduced elastic scattering, making italmost invisible to the particle flow.

may be preserved, but the sensor may become significantly lessdetectable. The absorption efficiency in this case may becomeextremely high, without the need for sacrificing absorptionand making the system considerably less intrusive than thebare sensor.

It may be wondered whether this counterintuitive func-tionality may even be allowed by the laws of physics. Toformally approach the problem, consider the partial-waveformulation of quantum scattering from a central field inwhich absorption losses are taken into account by consideringcomplex phase shifts δl or, equivalently, complex scatter-ing coefficients Sl = e2iδl . The scattered amplitude at anangle θ is fel(θ ) = i/(2k0)

∑+∞l=0 (2l + 1)(1 − Sl)Pl(cos θ ),47

providing σel = π/k20

∑+∞l=0 (2l + 1)|1 − Sl|2 and σabs = π/

k20

∑+∞l=0 (2l + 1)(1 − |Sl|2). The total, or extinction, cross

section σtot = σel + σabs quantifies the number of particlesdeviating from the direction �k0 of the incident beam, linked tothe forward elastic-scattering amplitude fel(0) by the opticaltheorem σtot = 4π/k0 Im fel(0).47

This fundamental relation tells us that absorption is neces-sarily accompanied by elastic scattering, and it is impossibleto totally suppress the elastic scattering of an absorptive target.This is expected since any transition in the cloak impliestaking energy out of one or more of the impinging particles,which would appear missing in the forward direction, fromthe point of view of the constant energy beam. A trade-off is,therefore, expected between absorption and scattering levels,limiting the total elastic-scattering reduction achievable witha cloaked quantum sensor given the desired absorption level,consistent with our related findings on cloaked electromagneticsensors.36–38 To quantify this fundamental bound, we noticethat, as a direct consequence of passivity (σabs � 0), eachcoefficient Sl necessarily belongs inside the unit circle in thecomplex plane and, by studying the parametric dependence ofthe cross sections on the scattering coefficients, it is possible toderive a specific range of achievable values for the pair formedby the partial absorption cross section σabs,l and the partialabsorption efficiency σabs,l/σel,l for any l. As an example, the

FIG. 2. (Color online) Effect of a core-shell isotropic potentialcloak on the electronic-scattering and absorption cross sections of theconsidered s resonator at E = 0.1 meV. The uncloaked s resonatorhas a poor absorption efficiency (green point). By tailoring the cloakproperties, it is possible to boost the absorption efficiency by upto 5 orders of magnitude (Vin = 0, red point) or reach any optimalabsorption level lying on the red line (Vin = −140 meV, red point).The red and purple lines represent fundamental physical bounds forpassive s resonators.

fundamental bound for the s wave (l = 0) is given by the twosolid lines segments in Fig. 2, which separate the allowedfrom the forbidden (shadowed) region, and meet at the pointof maximum (conjugate resonant) absorption σabs,l = σel,l .47

If we can accept an absorption level lower than the idealmaximum, as usual in quantum sensors, we may significantlyincrease the absorption efficiency by operating along the redline and, for reasonably large absorption values, we would stillbe capable of triggering transitions in our system, achievingfurtive quantum sensing with significantly reduced scattering.Similar bounds may be derived for any scattering order,essentially quantifying the fundamental limits of this concept.

Consider now the specific scenario of a beam of quantumparticles scattered from an object with a size smaller thanthe de Broglie wavelength λ0 such that the s partial wavedominates the scattering. This may occur for sufficiently coldparticles or for sufficiently small quantum sensors. We modelthe sensor as a simple s resonator, characterized by its Breit-Wigner resonance line shape with resonance energy ER , elasticlinewidth �el , and inelastic linewidth �abs. The presence of thesensor may be rigorously modeled in its surrounding regionby imposing a suitable boundary condition on the logarithmicderivative of the radial wave function at an arbitrary sphericalradius a larger than the sensor.48 Our results for matter-wavecloaking33 show that the cloaking condition for s waves inthe long-wavelength limit only involves the cloak potential,whereas, its effective mass may be independently tailored tosuppress p-scattered waves. By controlling the potential usingmetamaterial layers, therefore, we may be able to significantlyreduce the elastic scattering from the s resonator. We assumea spherical cloak with homogeneous potential Vc between the

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FURTIVE QUANTUM SENSING USING MATTER-WAVE CLOAKS PHYSICAL REVIEW B 87, 201106(R) (2013)

radii ac1 and ac2 and a core potential Vin inside this shell. Bysolving the scattering equations and imposing proper boundaryconditions, it is possible to solve for the scattering coefficientsSl(Ref. 48) and to derive the cross sections.

As an example, Fig. 2 shows the effect, on the electronscattering, of varying the potential cloak for a quantum sensorwith ER = 0.15 meV, �el = 10−5, �abs = 10−6 eV, and a = 1 Aand a cloak design with ac2 = 1 and ac1 = 0.8 nm for anexciting beam with energy E = 0.1 meV. The cloak potentialVc is swept over a wide range of values for two fixed valuesVin = 0 and Vin = −140 meV, generating the two dashedlines. The green point corresponds to the uncloaked caseVc = 0 for which the absorption efficiency is relatively low anddefinitely not optimal (far from our derived bound), showinga strong probability of elastic-scattering events compared toinducing an inelastic transition. The value of absorption issignificantly lower than the maximum, allowing for largetuning of the absorption efficiency, according to our derivedbounds. Indeed, by using a cloak potential Vc with positive(blue dots in the figure) or negative values (black dots), it ispossible to tame the absorption cross section and efficiency,reaching any value between the purple and the red bounds.In particular, by introducing a positive Vc = 140 meV, wedramatically boost the absorption efficiency (over 5 orders ofmagnitude), moderately reducing the value of absorption but,at the same time, drastically suppressing the overall scattering.

FIG. 3. (Color online) Scattering and absorption cross sectionsversus electron energy for (a) the bare s resonator (green point inFig. 2), (b) a cloaked s resonator with lower absorption than thebare sensor (red point on black dashed line in Fig. 2), and (c) acloaked s resonator with higher absorption than the bare sensor(Vc = 1034 meV, blue dashed line in Fig. 2).

This represents the optimal design for this specific absorptionlevel, and it results in a low-observable nonintrusive furtivesensor at the design energy E. Different cloak designs canproduce different dispersions in Fig. 2, but for each design,an optimal value lying in the red portion of the absorptionefficiency bound may be found, providing large flexibility inchoosing the level of desired optimal absorption efficiency andthe corresponding level of absorption. As a second example,by adjusting the inside potential Vin = −140 meV (blue dottedline), we demonstrate strongly enhanced absorption efficiencyand even increased overall absorption levels compared to thebare sensor for Vc = 1034 meV.

Further insight into the cloaking mechanism may begrasped by inspecting the energy dependence of the scatteringand absorption cross sections of the system in the uncloakedand cloaked cases as shown in Fig. 3. As expected, a resonantline shape is observed for the bare s resonator [Fig. 3(a)]with a maximum peak for both scattering and absorption atenergy ER . Scattering is consistently larger than the absorptionwithin the considered range. When the sensor is surroundedby the optimized cloak with Vin = 0 and Vc = 140 meV[Fig. 3(b)], a drastic reduction in elastic cross section isobserved around the working energy, producing an invisiblequantum sensor, whose absorption remains comparable, albeitlower, than the bare sensor’s one. Panel (c) presents a designwith reduced scattering and larger absorption than the bare

FIG. 4. (Color online) Finite-element simulation for the elec-tronic quantum scattering for (a) the bare s resonator (green pointin Fig. 2) and (b) an optimally cloaked s resonator (red point onblack dashed line in Fig. 2) for E = 0.1 meV. The color contoursshow the real part of the wave function, and the arrows represent theroot-mean-square-averaged probability current distribution.

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sensor, obtained with Vin = −140 and Vc = 1034 meV. In bothcases, we achieve considerable scattering reduction, despitethe covered object being ten times larger than the bare sensor.The matter-wave cloak provides large flexibility in tailoringthe overall absorption at the energy of interest, allowing,at the same time, optimal absorption efficiency, consistentwith our derived bounds. These findings are analyticallysimilar, although physically different, than the electromagneticscenario outlined in Ref. 7.

We have confirmed our findings using three-dimensionalfinite-element simulations based on COMSOL Multiphysics,modeling the s resonator with a logarithmic derivative surfaceboundary condition. Figure 4 shows the vector distributionof the time-averaged probability current distribution and thereal part of the quantum wave function in the near field ofthe sensor for plane-wave incidence from the left. The baresensor significantly perturbs the matter-wave flow [Fig. 4(a)],but a properly designed cloak [Fig. 4(b)] restores the planar-wave fronts and the straight particle current lines. This resultsin a major elastic-scattering reduction without significantlyaffecting its ability to sense as demonstrated by the probabilityflow inside the cover. An observer outside the cloak would notbe able to detect the presence of the sensor, unless performingan extremely large number of quantum experiments based onparticle detection. The beam travels as if the sensor was notthere, except that, occasionally, one particle gives away someenergy, efficiently detected by the sensor, which is tailoredto suppress all the unnecessary elastic scattering in all otherdirections other than its shadow.

To conclude, we have explored the concept of furtivequantum sensing, the quantum counterpart of cloaked electro-magnetic and acoustic sensors. We have shown that, consistentwith these other wave scenarios, it is possible to manipulatequantum scattering and to significantly reduce the elastic crosssection of a quantum system with internal degrees of freedomwhile maintaining its ability to perform sensing via inelastic-scattering events. This is achieved using a homogeneouscloak that can cancel the portion of elastic scattering, usuallydominant, that is not strictly necessary to sustain the sensingoperation. We have derived the fundamental bounds forthis concept and have provided specific examples appliedto a subwavelength s resonator. Our theory may be readilyextended to larger sensors and more complex geometriesinvolving multiple partial waves by considering multilayeredcloaks.7 Realistic furtive quantum sensors may be obtainedby cloaking dyes in semiconductor core-shell nanoparticles,tailored to cancel the dominant s and p wave scattering, asdiscussed more extensively in Ref. 31. Our work extends thereach of sensor cloaking to quantum mechanics, providing newtools for extreme manipulation of matter waves and paving theway for furtive quantum sensing strategies for ballistic particlecontrol and detection at the nanoscale, of interest in efficientelectrical pumping, non intrusive particle detection, plasmadiagnostics, and quantum information processing.

This work has been supported by the AFOSR YIP AwardNo. FA9550-11-1-0009 and the DTRA YIP Award No.HDTRA1-12-1-0022.

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48See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.87.201106 for a detailed analytical derivationof the scattering problem for a cloaked two-level quantum system.

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