MMultiple scattering induced negative refraction of matter waves
Florian Pinsker ∗ Department of Atomic and Laser Physics, University of Oxford, United Kingdom.
Starting from fundamental multiple scattering theory it is shown that negative refraction indicesare feasible for matter waves passing a well-defined ensemble of scatterers. A simple approach to thistopic is presented and explicit examples for systems of scatterers in 1D and 3D are stated that implynegative refraction for a generic incoming quantum wave packet. Essential features of the effectivescattering field, densities and frequency spectrum of scatterers are considered. Additionally it isshown that negative refraction indices allow perfect transmission of the wave passing the ensembleof scatterers. Finally the concept of the superlens is discussed, since it is based on negative refractionand can be extended to matter waves utilizing the observations presented in this paper. This pavesthe way for ‘untouchable’ quantum systems in analogy to cloaking devices for electromagnetic waves. a r X i v : . [ qu a n t - ph ] M a r Negative refraction of electromagnetic (EM) waves was first discovered by V. G. Veselago in [1] and in particularit has been noted that at the negative refractive material’s surface waves entering the material are bent towardsnegative angles. Decades later meta-materials with this property were developed and they were found to give rise tothe phenomenon of sub-wavelength localisation (superlensing). In addition to perfect localisation of the EM wavesthose materials imply complete transmission of the incoming wave passing through those so called superlenses [2–4].For a long time negative refraction was considered a mathematical curiosity rather than a feasible physical phenomenonshown in experiments, it is only due to the invention of artificial meta-materials that the necessary conditions couldbe provided experimentally [3–5] and by the means of those materials sub-wavelength localisation was shown [6]. Dueto the huge impact of sub-wavelength localisation of EM waves a natural question to ask is: Are negative refractivematerials feasible for matter waves as well and what is necessary to refract matter waves in a similar way as EM waves?To give answers to those questions in this work I will utilize multiple scattering theory, which has been widely appliedto derive refraction indices for matter waves [7–10]. It is worth noting that multiple quantum scattering theoryis an application of the full quantum many-body problem for fixed particle numbers that allows a more practicalviewpoint when single particle scattering on a cloud of potentially randomly positioned scatterers is of interest [10].The refraction indices n derived so far by this approach are however approximations and their validity is particularlyconfined to the case n >
0. So one may ask: Are matter waves restricted to refract positively, perhaps due to aproperty of the governing many-body Schr¨odinger equation? In fact, this work discusses the feasibility of negativerefraction in a way that is consistent with the quantum many-body problem and conditions for negative refraction arestated, i.e. scenarios of ensembles of scatterers implying negative refraction. Finally based on the concept of negativerefraction the concept of superlenses [3, 4] for matter waves is discussed. Although the presented theory is new Ipoint out that the topic of negative refraction for matter waves was considered before in [11] and a specific proposalin terms of the single particle Schr¨odinger equation including external magnetic fields was regarded as a candidatefor negative refraction, while the results presented in [11] were found to be inaccurate [12]. In addition only recentlyVeselago-type lensing was successfully implemented utilizing ultra cold atoms confined to optical lattices by switchingbetween the positive and negative energy branches of the many-body states [13]. In contrast to the proposal [11] andthe realisation [13] this work follows a different path showing that negative refraction is a property of certain closedquantum many-body systems (with fixed particle numbers) without the necessity to apply external fields, feasibleonly due to a proper arrangement and choice of scatterers.
Multiple scattering theory.–
The fundamental multiple scattering equations for quantum matter waves were firstintroduced by L. L. Foldy and extend the classical treatment based on the Boltzmann integro-differential equationswhen interference effects of the scattered particle become important [14]. The set of equations usually used todescribe the scenario for point scatterers where the strength of the scattered wave from a scatterer is proportional tothe external field acting on it read as follows [10, 14–16],Ψ( r ) = Ψ ( r ) + (cid:88) i G ( r − r i ) f i ξ i ( r i ) . (1)Here the l.h.s. is the wave function generated by multiple scattering processes that are induced by a system of N randomly arranged particles/obstacles and ξ i ( r i ) = Ψ ( r i ) + (cid:88) i (cid:54) = j G ( r j − r i ) f j Ψ( r j ) (2)describes the local field which generates the scattered wave at positions r i . Ψ ( r ) is the incident wave, i.e. the wavewithout the effects of scatterers, ξ i ( r i ) is the local field at r i where the i th scatterer e.g. a noble gas atom is locatedat a fixed position - importantly ξ i ( r i ) can be approximated by ξ i = exp( i k · r i ) when all multiple scattering processesat scatterer i are neglected [14], i.e. only the first order scattering from each scatterer i of the ensemble of multiplescatterers is considered. f i is the scattering amplitude/coefficient that relates the scattered wave to the scatterer andwhich in general is a function of the wave vector and position in space. In 3D the behavior of the particle close to the i th scatterer is assumed to be [10, 14, 15] G ( r ) = exp ( ± i | k | · | r | ) / (4 π | r | ), which is the Green’s function describingthe propagation of the wave scattered from the i th particle between the scattering particles. This form assumes aspherically symmetric scattering process (s-wave scattering) [14]. The corresponding Green’s function for a 2D systemis the Hankel function and for the 1D system it is a plane wave, which will be introduced later. It should be notedthat from (1) and (2) a plethora of derivations have been made particularly for refractive indices of matter waves andcomparison with experiments show excellent agreement with data e.g. in experiments with monatomic sodium gases[7–10, 14–16]. Statement of the problem.–
Let us assume Ψ and Ψ to approximately resemble plane waves, which can be regardeda valid assumption for waves far away from the scattering processes and they form a basis for general localised wavepackets. Given that and utilizing the mathematical notions from above we can state the scenario of interest as follows:For (cid:88) i G ( r − r i ) f i ξ i ( r i ) = exp ( i K · r ) − exp ( i k · r ) (3)do amplitudes f i = f i ( r i , k ), local fields ξ i ( r i , k ) and configurations of scatterers exist that extinguish the initial fieldand generate a new field which moves in opposite direction? Thus formally we are interested in states where theincoming wave with wave vector k implies an outgoing wave with wave vector K so that K = − k ≡ n k . (4)This equation defines the refraction index for the matter wave n [10], which by definition of the problem is negative.Note that although at first sight negative refraction with n = − R = 1, but in contrast waves refracted negatively occupy the space ‘behind’ thescatterers as well. As shown later there even is no loss of amplitude of the scattered wave after passing the scatterersfor certain stated parameters. The remainder of this work is dedicated to show which local fields and amplitudesare necessary to imply the refractive indices to be negative and of absolute value 1 and further properties of thosesystems. The motivation to investigate the case of negative refraction stems from [2], where meta-materials allow sub-wavelength localisation for electromagnetic matter waves and the argumentation is based on the extinction theoremfor matter waves [14]. Mathematically the l.h.s. of (3) consists of a set of homogeneous linear algebraic equations andto make statements about ξ i , f i etc. approximations can be made. Furthermore as plane waves are simplifications wewill also discuss wave packets, i.e. superpositions of plane waves, moving in opposite directions, such that for eachcomponent of the wave packet’s spectrum (4) is satisfied. Simplified state equation.–
Alternatively to solving eq. 1 directly one can proceed as follows. As discussed in detailby [14] eq. 1 can be simplified when considering the statistical average over all possible random positions of scatterers,which is our first assumption, while the case of fixed positions is trivially included. Then by further supposing thescatterers not to have internal degrees of freedom and to occupy a finite volume V one effectively considers (cid:104) Ψ( r ) (cid:105) = exp ( i k · r ) + (cid:88) i (cid:90) V G ( r − r i ) f i (cid:104) ξ i ( r i ) (cid:105) i d r i . (5)Here (cid:104) ξ i ( r i ) (cid:105) i is the scatterer induced field acting on the i th scatterer, which is averaged over all possible configura-tions of all the other scatterers [14]. This field can be approximately replaced by an averaged field, which is a fairapproximation when the number of scatterers N is large [14] and supposing that we substitute (cid:104) ξ i ( r i ) (cid:105) i → (cid:104) ξ ( r i ) (cid:105) . So(5) reduces to the simplified state equation (cid:104) Ψ( r ) (cid:105) = exp ( i k · r ) + (cid:90) V G ( r − r (cid:48) ) f ( r (cid:48) ) ρ ( r (cid:48) ) ξ ( r (cid:48) ) d r (cid:48) ≡ ψ ( r ) , (6)where ψ ( r ) is the coherent wave, i.e. the average matter wave function for all possible configurations of scatterers. ξ ( r ) denotes the average local field induced by the scatterers and f ( r ) the average amplitude - both functions aregiven by [14, 15] ξ ( r ) = (cid:88) i (cid:104) ξ i δ ( r − r i ) (cid:105) (7) f ( r ) = (cid:88) i (cid:104) f i δ ( r − r i ) (cid:105) (8)and ρ ( r ) is the number density of scatterers. As N → ∞ for scatterers close to each other compared to other lengthscales of the system, those sums can be approximated as well as integrals. Positive refraction.–
While this paper follows the aim to derive wave functions for negative refraction indices wegive a short overview how positive refraction can be derived easily from (6). The macroscopic one-body Schr¨odingerequation equation for the average wave function is [15] − (cid:126) m ∆ ψ ( r ) + v ( r ) = Eψ ( r ) , (9)where E is the incident particle energy and m its mass. In terms of eq. 9 (6) is a solution for the optical potential v ( r ) = (cid:40) − π (cid:126) ρf ξ ( r ) /m inside V0 outside V , (10)when assuming ξ = cψ i.e. to be proportional by a number c and for the incident wave energy given by E = (cid:126) k / (2 m )[15]. From (10) one derives the simple relation K (cid:48) = k + 4 πρf c and thus a refraction index n = K (cid:48) /k , with K (cid:48) denoting the wave vector of the generated wave. This derivation of the index of refraction, however cannot addressthe case of negative refraction as only a formula for the squares of the wave vectors is given. In order to extend thediscussion to negative refraction indices we may proceed as follows. Wave packets.–
To develop the formalism our starting point is a general scalar matter wave function describing aquantum particle in D spatial dimensions ψ ( r , t ) = (cid:90) R D d k · g ( k ) e i ( − k · r − ωt ) , (11)with g ( k ) being the distribution function of different frequency components of the wave. Correspondingly the deBroglie and Einstein relations which associate the wave properties with that of a massive ‘projectile’ are [17] p = m v = (cid:126) k (12) E = (cid:126) ω = hν. (13)Here p denotes the momentum of the projectile, v its velocity, ν the frequency and E the energy of the particle to bescattered. We consider the pair of initial and outgoing wave formally given by ψ ( r ) in , out = (cid:90) R D d k · g ( k ) e ± i k · r , (14)when incoming − and outgoing waves + are stationary. RESULTSNegative refraction in D We are interested in the scenario where the initial wave packet ψ ( r ) in is moving in opposite direction as the outgoingwave packet ψ ( r ) out and we first confine our considerations to D = 1. To model two counter-propagating waves weconsider by using eq. 6 the expression g ( k )( e ik · x − e − ik · x ) = (cid:90) V G ( x − x (cid:48) ) f ( k, x (cid:48) ) α ( x (cid:48) ) dx (cid:48) . (15)The superscript of the Green’s function indicates its spacial dimension. Here we have introduced the abbreviation forthe unknown α ( x (cid:48) ) = ρ ( x (cid:48) ) ξ ( x (cid:48) ), which we determine as follows. Let us rewrite the r.h.s. of (15) as (cid:90) V G ( x − x (cid:48) ) f ( k, x (cid:48) ) α ( x (cid:48) ) dx (cid:48) = (cid:90) ∞−∞ G ( x − x (cid:48) ) f ( k, x (cid:48) ) α ( x (cid:48) ) dx (cid:48) , (16)and assume α ( x (cid:48) ) to be compactly supported on V , which is a fairly reasonable statement in physical termsparticularly as the density of scatterers is only supported in V . Now for a 1D system the Green’s function of theinhomogeneous Helmholtz equation ∂ x G ( x ) + k G ( x ) = − δ ( x ) with x ∈ R (17)corresponding to isotropic point scatterers at x = 0 is G ± ( x ) = e ± i | k || x | | k | , (18)which provides the behaviour of the particle close to the scatterer with incident wave number | k | = √ mE/ (cid:126) . Clearlylinear combinations of Green’s functions are Green’s functions as well and so we obtain by the above solutions e.g. G ( x ) = (cid:18) e i | k || x | | k | + e − i | k || x | | k | (cid:19) = cos( −| k || x | ) | k | , (19)i.e. a stationary wave solution and similarly one gets − i sin( − i | k || x | ) | k | .By combining (15) and (16) we obtain the condition for negative refraction in 1D when writing the two complexexponentials as sine, (cid:90) ∞−∞ f ( | k | ) · G ( x − x (cid:48) ) · α ( x (cid:48) ) dx (cid:48) ≡ ig ( k ) sin ( k · x ) . (20)We assume the amplitude f to be independent of x (cid:48) , i.e. the strength of the scattered wave is only proportional to theGreen’s function and the number density of scatterers. In principle scatterers that obey a position dependency of thescattered wave beyond that of the Green’s function of each scatterer, i.e. in f can be simplified by effectively choosingthe number distribution of scatterers ρ ( x ) accordingly. As α ( x ) is a product of the density of scatterers ρ ( x ) andthe local effective field ξ ( x ) there is in principle freedom to generate a variety of α ( x ) experimentally by arrangingscatterers appropriately, which fixes ρ ( x ). It is thus the choice of the experimenter first to use scatterers that indeedobey a position independency in f and secondly to effectively compensate this dependency by the arrangement ofscatterers. Note that the nuclear scattering amplitude has an expansion of the form [15] f ( k ) = − b i + ikb i + O ( k ) (21)with the bound scattering length b i . Examples for D scattering
Example .– In the following we will give explicit examples of scenarios yielding negative refraction indices: Forthe simplest example we set α ( x ) = δ ( x ) , (22)resembling scatterers at the position x = 0 with effective field ξ (0) = 1 = exp( i | k | G e ( x ) = (cid:40) − i sin( −| k || x | ) | k | for x ≥ i sin( −| k || x | ) | k | for x < , (23)which can be constructed from the elementary solutions of the Helmholtz equation with point source by taking thesuperpositions ∓ (cid:18) − e i | k || x | | k | + e − i | k || x | | k | (cid:19) = ± i sin( −| k || x | ) | k | . (24)So the material consists of scatterers inducing an anisotropic wave (23). The frequency dependent amplitude is set f ( | k | ) = − | k | g ( | k | ) . (25)Thus we have given an example for α ( x ) = ξ · ρ ( x ) and a condition on the frequency spectrum f ( | k | ) of the scatterers,which implies a negative refractive material for the generic incoming wave function defined by (20). Wave packetscan be obtained by integrating (20) over k with appropriate weighting factors. On the other hand for given f we canmodel wave functions via g that satisfy (25). Example : Anisotropic scatterers.– The above Green’s functions can be generalised to anisotropic scatterers, sothat scattering events depend on the direction of the incoming wave. In 1D the corresponding anisotropic Helmholtzequation for each scatterer is generalised to [18, 19] C ± ( x ) ∂ x φ + k φ = − δ ( x ) . (26)We assume C ± ( x ) = (cid:40) ± x ≥ − ± x < φ ± ( x ) = (cid:40) ± G ( x ) for ± x ≥ x | k | ) c + exp( − x | k | ) c for ± x < c , are chosen to satisfy continuity. Setting c = 0 we get a decaying solution for ± x <
0. For two differentspecies of scatterers positioned at the same points in space (each Green’s function approximately being solutions of aHelmholtz equation) we get the superposed effective Green’s function G e ( x ) = φ + ( x ) + φ − ( x ) . (29)Utilizing two species of scatterers is applicable for high | k | (cid:29) Extended scattering domain.–
While the δ distributed scatterers only resemble an approximation to realistic forma-tions of scatterers, we extend the 1D analysis to fields, α ( x ) = 1 √ πσ exp (cid:18) − x σ (cid:19) · exp ( ikx ) , (30)where the plane wave part is due to the spatial dependence of the effective field ξ = exp ( ikx ) and the remaindermodels an extended distribution of scatterers. Hence the l.h.s. (20) for a stationary Greens function solution (23) isgiven by, − f ( k ) | k | (cid:90) ∞−∞ sin( −| k | x ) 1 √ πσ · exp (cid:18) − ( x − x (cid:48) ) σ + ik ( x − x (cid:48) ) (cid:19) = f ( k ) | k | i √ π e − k σ sin (cid:18) k · x (cid:48) + 12 ik σ (cid:19) . (31)Further sin (cid:18) k · x (cid:48) + 12 ik σ (cid:19) = sin ( k · x (cid:48) ) cosh (cid:18) k σ (cid:19) + i cos ( k · x (cid:48) ) sinh (cid:18) k σ (cid:19) . (32)Again choosing the frequency spectrum f accordingly we can match the r.h.s. in (20) to obtain negative refractionfor σ → k → f we confine the consideration to appropriately related g . Negative refraction in D Next we turn to the case of negative refraction in 3D materials. Green’s functions of the 3D Helmholtz equationfor different frequencies | k | are G ( r ) = exp ( ± i | k | · | r | )4 π | r | (33)with r = ( x, y, z ) ∈ R , called respectively the outgoing and ingoing spherical waves solutions. Linear combinationsof the incoming and outgoing wave solutions for the 3D Helmholtz equation are e.g. the stationary waves G ( r ) = − cos( | k | · | r | )4 π | r | (34)and G ( r ) = − i sin( | k | · | r | )4 π | r | . (35)Similar as in the 1D case we set all scatterers to rest at the origin, but in contrast with a certain spatial distributiongiven by α ( r ) = 4 π | r | δ ( r ) , (36)while using the ξ ( r = (0 , , ± i k · r ) = 1 which neglects all additional multiple scattering processes at thisparticular scatterer [14]. W.l.og. we set k = ( k x , ,
0) and in addition we assume the scatterers to induce a Green’sfunction (35) G e ( r ) = − i sin( | k | · | r | )4 π | r | . (37)Starting from (6) we require as condition for negative refraction in 3D: (cid:90) V G ( r − r (cid:48) ) f ( k , r (cid:48) ) ρ ( r (cid:48) ) ξ ( r (cid:48) ) d r (cid:48) ≡ g ( k ) (exp ( − i k · r ) − exp ( i k · r )) = − ig ( k ) sin ( k x x ) . (38)We assume f ( k , r (cid:48) ) = f ( k ). On the other hand we get using (36) and (37) (cid:90) R f ( k ) · G ( r − r (cid:48) ) · α ( r (cid:48) ) d r (cid:48) = − if ( k ) sin( | k x | · | r | ) . (39)Next we apply the paraboloidal wave approximation, | r | (cid:39) x + z + y x that is valid for (cid:12)(cid:12) z + y x (cid:12)(cid:12) ≤
1. We are interestedin the far field where x → ∞ , so that the approximation is satisfied. Thus we obtain − if ( k ) sin( | k x | · | r | ) (cid:39) − if ( k ) sin( | k x | · x ) . (40)Choosing f ( k ) by comparison of (40) with (38) concludes the proof for the far field, i.e. in the x → ∞ limit. Implicit negative refraction
Scattering in D.–
While the previous examples provide a general guide to induce negative refraction, we now turnto a specific example feasible in systems satisfying the stated assumptions. For the sake of simplicity we considerthe 1D case. For each k there is a solution φ with no incoming wave from the right - a standard solution familiar incalculations for reflection and transmission from a generic scattering potential V ( x ) (with V = 0 as | x | > d ) modellinga single finite scatterer. While here we make specific choices note that the above general framework applies for anensemble of many (generally randomly distributed) scatterers. Now, we consider an incoming wave φ in ( x ) = e ikx + re − ikx x < − d (41)and consequently we have an outgoing wave φ out ( x ) = te ikx x > d, (42)where the reflection probability is R = | r | and the transmission probability T = | t | - for unitary scattering theconservation of probability applies, i.e. 1 − | r | = | t | . (43)We assume scatterers to act on finite range, d < ∞ . Using this generic framework we explicitly construct an examplewhere φ in ( x, k ) = φ out ( x, − k ) due to multiple scattering events and in contrast to perfect mirrors the solutions arenot confined to the area in front of the perfect mirror. Refracted waves in D.–
To further elucidate the feasibility of negative refraction in an ensemble of scatterers weproceed as follows. We consider the incoming wave moving from the l.h.s. towards the scatterers which then scatterson the meta-material spanning from a to c , φ in ( x ) = e ikx + r LA e − ikx x < a. (44)The induced wave function between two scatterers, say A and B at positions a and b , given there is another scatterer C at c with a < b < c and subsequent scatterers and scattering events are neglected, is φ A,B ( x ) (cid:39) t LA e ikx + t LA r LB e − ikx + t LA r LB r RA e ikx + t LA t LB r LC t RB e − ikx a < x < b, (45)where t ki ∈ C and r ki ∈ C with i = A, B, C and k = L, R are the corresponding transmission or reflection coefficientsfrom the left or right (
L/R ) of the three scatterer system. At the same time each pair of transmission and reflectioncoefficients has to conserve probability (43) and if | r RA | < φ in ( x, k ) → φ in ( x, k ) = e ikx + ( r LA + t LA r LB t RA ) e − ikx . (46)To get the scattering behavior that induces a negatively refracted wave between A and B , i.e. φ in ( x, k ) = φ out ( x, − k ) = φ A,B ( x ), as implied in (3) we find using (45) and (46) the necessary conditions( r LA + t LA r LB t RA ) = t LA (1 + r RA r LB ) (47)and 1 = t LA ( r LB + t RB t LB r LC ) . (48)Note that the setup trivially extends to N scatterers, A j denoting the j th scatterer. By letting the first A atposition a → −∞ and the last scatterer A N at a N → ∞ we have generated a scenario consistent with (20) whenchoosing t LA accordingly. Formula (45) is an approximation considering only the first and second order reflections andtransmissions - iteratively one would obtain the exact form including all reflection and transmission processes. Numerical example.–
Let us consider an electron of mass m = 9 . × − kg with velocity v = 1 m/s corresponding to a wave vector (12) k = p/ (cid:126) = 8 , /mm . Furthermore we consider the three scatterers A, B, C andspecify their scattering properties by choosing t RA = (cid:112) / r RA = − (cid:112) / r LA = 1 / t LA = √ / r LB = 0 . t LB = 0 . t RB = 1, r LC = 0 .
89 to satisfy (47) and (48). Then using the above characteristics we obtain the wave functions φ A,B ( x ) (cid:39) . e i , x + e − i , x ≡ φ out ( x, .
7) = φ in ( x, − .
7) (49)where [ x ] = mm and note that the wave is independent of the actual positions a, b and c . Complete tunnelling in negative refraction media
Next we analyse the transmittance of a matter wave passing a negative refraction index material. We assume thematerial with refraction index n to span from x = 0 to x = L . Therefore the wave function before, within and afterthe material is ψ in ( x ) = e ikx + re − ikx x < ψ n < ( x ) = Ae κ x + Be − κ x ≤ x ≤ L (51) ψ out ( x ) = te iKx L < x (52)correspondingly, where κ = i ( k + χ + iσ ) = in ( k ) k is in accordance to [7–9]. Here some attenuation σ of the wavewithin the material and a change in phase χ has been introduced and the transmission amplitude is again t . At thepoints x = 0 and L continuity of the wave function requires (cid:40) r = A + BAe κ L + Be − κ L = te ikL (53)and continuity of its first derivative (cid:40) ik (1 − r ) = κ ( A − B ) κ (cid:0) Ae κ L − Be − κ L (cid:1) = ikte ikL . (54) Re( n ) T FIG. 1. Transmittance T as a function of Re( n ) for decay σ = 0 . c = 1 .
94 showing amplification of the matter wave forcertain negative refraction indices n . Consequently the transmission amplitude is t = 2 ne − ikL n cos( knL ) − i (1 + n ) sin( knL ) (55)with κ = ink . Further using n ( k ) = 1 + k ( φ + iσ ) and writing c = kL we obtain the transmittance T = | t | = 4 | n | | n cos( c · n ) − i (1 + n ) sin( c · n ) | . (56)In Fig. 1 the transmittance is shown as a function of the real part of the refraction index, i.e. Re( n ) = 1 + φ/k .For the given parameters we observe that a negative refraction medium for matter waves allows their amplificationwithin the material in analogy to the concept of electromagnetic waves [3]. However the conservation of probabilityprovides a natural upper bound to the transmittance in a single-body Schr¨odinger equation picture. Recall the relation1 = R + T + A , where A corresponds to the absorption due to material properties. In case that T = 1 necessarily R = A = 0 and we consider the scenario of perfect transmission of a negatively refracted wave through the medium.In contrast in positive refraction indices materials usually waves decay - this is particularly shown by the graph ofFig. 1 for Re( n ) = 1 + φ/k >
0. Hence, in this setting complete transmission of a matter wave is feasible only withina negative refraction material. Furthermore it is known that negative refraction of waves allows localisation to asingle point given that n = − DISCUSSION
While negative refraction for electromagnetic waves was regarded a mathematical curiosity until the emergence ofmeta-materials, it now is an established and well-tested physical phenomenon. The results presented in this papershow the feasibility of negative refraction materials for matter waves due to the variety of free parameters of themany-body problem under consideration namely, the generally complex-valued scattering amplitude, the real-valuednumber density of local scatterers and the effective local field, which itself is a function of the former two and can beapproximated by a plane wave when neglecting multiple scattering processes at each scatterer but considering only theleading order. The general theoretical approach presented here extends to statistical ensembles, but trivially includesthe case of a definite choice of an ensemble configuration of scatterers. One striking feature of negative refraction isthe possibility of perfect transmission through the medium with n = − ρ with a certain fixed frequency dependence of the scatteringamplitude f ( x, k ) and scattering behaviour of their generating Green’s function G imply negative scattering. Furtherwhen considering statistical ensembles, instead of a definite choice, the considerations made only apply in the large N limit. METHODS
I utilized multiple scattering theory to analyse the feasibility of negative refraction for matter waves systems dueto multiple scattering events. After introducing a general widely accepted and experimentally well-tested approachanalytical expressions for the effective local field, Green’s functions, density and the frequency behaviour of thescatterers were stated, that consequently yield an outgoing counter-propagating/negatively refracted wave in analogyto the scenario observed for EM waves. The deductive reasoning/mathematical analysis is based on standard analyticaltools, e.g. approximations in the large N limit, utilizing continuity properties of the wave function at the surfacesof the negative refractive material - an essential property in quantum mechanics. The latter implied the completetransmission of the quantum particle for particular n <
0. Finally the proposed negative refractive materials forsub-wavelength localisation of matter waves are an extension from light waves and are justified by analog behaviourof waves in materials with refraction index n independent of their physical nature. ACKNOWLEDGEMENTS
I acknowledge financial support through a Schr¨odinger Fellowship (Austrian Science Fund (FWF): J3675) at theUniversity of Oxford and I am grateful for funding through the NQIT project (EP/M013243/1). I thank AlexanderDreismann (Cavendish Laboratory) for critical discussions and for helping to state the problem. Furthermore I wouldlike to thank the referee for his/her very useful comments.
ADDITIONAL INFORMATION
There are no competing financial Interests. ∗ fl[email protected][1] Veselago, V. G., The electrodynamics of substances with simultaneously negative values of (cid:15) and µ , Sov. Phys. Usp. ,509-514 (1968).[2] Pendry, J. B., Negative Refraction Makes a Perfect Lens, Phys. Rev. Lett. , 3966 (2000).[3] Pendry, J. B., Negative Refraction, Contemporary Physics , , Issue 3, 191-202 (2004).[4] Pendry, J. B. & Smith, D. R., Reversing Light with Negative Refraction, Physics Today , Issue 6, 37-43 (2004).[5] Cubukcu, E. et al., Electromagnetic waves: Negative refraction by photonic crystals, Nature , 604-605 (2003).[6] Aydin, K., Bulu, I. & Ozbay, E., Electromagnetic wave focusing from sources inside a two-dimensional left-handed materialsuperlens,
New Journal of Physics
221 (2006).[7] Champenois, C. et al., Index of refraction of gases for matter waves: Effect of the motion of the gas particles on thecalculation of the index,
Phys. Rev. A , 013621 (2008).[8] Schmiedmayer, J. et al., The Matter Wave Index of Refraction of a Gas Measured by an Atom Interferometer, Phys. Rev.Lett. , 1043 (1995). [9] Hammond, T. D. et al., Matter-Wave Index of Refraction, Inertial Sensing, and Quantum Decoherence in an AtomInterferometer, Braz. Jour. Phys. , 193-213 (1997).[10] Lax, M., Multiple Scattering of Waves, Rev. Mod. Phys. , 287 (1951).[11] Baudon, J., et al., Negative-Index Media for Matter-Wave Optics, Phys. Rev. Lett. , 140403 (2009).[12] Vogt, T. & Li, W., Negative refraction for incoherent atomic matter waves,
Phys. Rev. A , 033634 (2015).[13] Leder, M., Grossert, C. & Weitz, M., Veselago lensing with ultracold atoms in an optical lattice, Nature Communications , 3327 (2014).[14] Foldy, L. L., The Multiple Scattering of Waves. I. General Theory of Isotropic Scattering by Randomly DistributedScatterers, Phys. Rev. , 107 (1945).[15] Sears, V.S., Fundamental aspects of neutron optics, Physics Reports , , Issue 1, 1-29 (1982).[16] Forrey, R. C., You, L., Kharchenko, V. & Dalgarno, A., Index of refraction of noble gases for sodium matter waves, Phys.Rev. A , 2180 (1996).[17] De Broglie, L., Recherches sur la th´eorie des quanta (Researches on the quantum theory), Doctoral thesis, Univ. Paris,France (1924); Ann. Phys. , 22 (1925).[18] Wang, S., Xia, J., v. d. Hoop, M. & Li X., Anisotropic ‘Helmholtz’ equations: Massively parallel structured multofrontalsolver using nested dissection based domain decomposition with separators of variable thickness, . Proceedings of the ProjectReview, Geo-Mathematical Imaging Group (Purdue University, West Lafayette IN) , , 175-192 (2011).[19] Operto, S., Virieux, J., Ribodetti, A. & Anderson, J.E., Finite-Difference Frequency-Domain Modeling of ViscoacousticWave Propagation in 2D Tilted Transversely Isotropic (TTI) Media, Geophysics ,74