Asymmetric scattering by non-hermitian potentials

The scattering of quantum particles by non-hermitian (generally nonlocal) potentials in one dimension may result in asymmetric transmission and/or reflection from left and right incidence. Eight generalized symmetries based on the discrete Klein's four-group (formed by parity, time reversal, their product, and unity) are used together with generalized unitarity relations to determine the possible and/or forbidden scattering asymmetries. Six basic device types are identified when the scattering coefficients (squared moduli of scattering amplitudes) adopt zero/one values, and transmission and/or reflection are asymmetric. They can pictorially be described as a one-way mirror, a one-way barrier (a Maxwell pressure demon), one-way (transmission or reflection) filters, a mirror with unidirectional transmission, and a transparent, one-way reflector. We design potentials for these devices and also demonstrate that the behavior of the scattering coefficients can be extended to a broad range of incident momenta.

The scattering of quantum particles by non-hermitian (generally nonlocal) potentials in one dimension may result in asymmetric transmission and/or reflection from left and right incidence. Eight generalized symmetries based on the discrete Klein's four-group (formed by parity, time reversal, their product, and unity) are used together with generalized unitarity relations to determine the possible and/or forbidden scattering asymmetries. Six basic device types are identified when the scattering coefficients (squared moduli of scattering amplitudes) adopt zero/one values, and transmission and/or reflection are asymmetric. They can pictorically be described as a one-way mirror, a one-way barrier (a Maxwell pressure demon), one-way (transmission or reflection) filters, a mirror with unidirectional transmission, and a transparent, one-way reflector. We design potentials for these devices and also demonstrate that the behavior of the scattering coefficients can be extended to a broad range of incident momenta.
PACS numbers: 03.65.Nk, 11.30.Er Introduction. The current interest to develop new quantum technologies is boosting applied and fundamental research on quantum phenomena and on systems with potential applications in logic circuits, metrology, communications or sensors. Robust basic devices performing elementary operations are needed to perform complex tasks when combined in a circuit.
In this paper we investigate the properties of potentials with asymmetric transmission or reflection for a quantum, spinless particle of mass m satisfying a onedimensional (1D) Schrödinger equation. If we restrict the analysis to transmission and reflection coefficients (squared moduli of the scattering complex amplitudes) being either zero or one, a useful simplification for quantum logic operations, there are six types of asymmetric devices. These devices cannot be constructed with Hermitian potentials. In fact for all (four) device types with transmission asymmetries, the potentials have to be also nonlocal. Therefore, nonlocal potentials play a major role in this paper. They appear naturally when applying partitioning techniques under similar conditions to the ones leading to non-hermitian potentials, namely, as effective interactions for a subsystem or component of the full wave-function, even if the interactions for the large system are hermitian and local [1].
Symmetries can be used, similarly to their standard application to determine allowed/forbidden transitions in atomic physics, to predict whether a certain potential may or may not lead to asymmetric scattering. The concept of symmetry, however, must be generalized when dealing with non-hermitian potentials.
Generalized symmetries. The detailed technical and formal background for the following can be found in a previous review on 1D scattering by complex potentials [1], a companion to this article for those readers willing to reproduce the calculations in detail. The Supplemental Material (Sec. I) provides also a minimal kit of scatter-ing theory formulae that may be read first to set basic concepts and notation. The notation is essentially as in [1], but it proves convenient to use for the potential matrix (or kernel function) in coordinate representation two different forms, namely x|V |y = V (x, y). "Local" potentials are those for which For hermitian Hamiltonians, symmetries are represented by the commutation of a symmetry operator with the Hamiltonian. In scattering theory, symmetry plays an important role as it implies relations among the Smatrix elements beyond those implied by its unitarity, see e.g. [2] and, for scattering in one dimension, Sec. 2.6 in [1].
Symmetries are also useful for non-hermitian Hamiltonians, but the mathematical and conceptual framework must be generalized. We consider that a unitary or antiunitary operator A represents a symmetry of H if it satisfies at least one of these relations, When the symmetry in Eq. (2) holds we say that H is A-pseudohermitian [3]. Parity-pseudohermiticity has played an important role as being equivalent to spacetime reflection (PT) symmetry for local potentials [3,4]. A large set of these equivalences will be discussed below. A relation of the form (2) has been also used with differential operators to get real spectra beyond PT-symmetry for local potentials [5,6].
Here we consider A to be a member of the Klein 4group K 4 = {1, Π, Θ, ΠΘ} formed by unity, the parity operator Π, the antiunitary time-reversal operator Θ, and their product ΠΘ. This is a discrete, abelian group. We also assume that the Hamiltonian is of the form H = H 0 + V , with H 0 , the kinetic energy operator of the particle, hermitian and satisfying [H 0 , A] = 0 for all members of the group, whereas the potential V may arXiv:1709.07027v1 [quant-ph] 20 Sep 2017 The relations among potential matrix elements are given in coordinate and momentum representations in the third and fourth columns. The fifth column gives the relations they imply in the matrix elements of S and/or S matrices (S is for scattering by H and S for scattering by H † ). From them the next four columns set the relations implied on scattering amplitudes. Together with generalized unitarity relations (S8) they also imply relations for the moduli (tenth column), and phases (not shown). The last two columns indicate the possibility to achieve perfect asymmetric transmission or reflection: "P " means possible (but not necessary), "No" means impossible. In some cases "P " is accompanied by a condition that must be satisfied.
be non-local in position representation. The motivation to use Klein's group is that the eight relations implied by Eqs. (1) and (2) generate all possible symmetries of a non-local potential due to the identity, complex conjugation, transposition, and sign inversion, both in coordinate or momentum representation, see Table I, where each symmetry has been labeled by a roman number.
Interesting enough, in this classification hermiticity (II) may be regarded as 1-pseudohermiticity.
Examples on how to find the relations in the fifth column of Table I of S− and S-matrix elements (for scattering by H and H † respectively) are provided in ref. [1], where the symmetry types III, VI, and VII where worked out. Similar manipulations, making use of the action of unitary or antiunitary operators of Klein's group on Möller operators, help to complete the table. From this fifth column, equivalences among the amplitudes for left and right incidence for scattering by H, (T l,r , R l,r ) or H † ( T l,r , R l,r ), are deduced, see the Supplemental Material and the four columns for T l,r , and R l,r . Together with the generalized unitarity relations, see Eq. (S8), these relations between the amplitudes imply further consequences on the amplitudes' moduli (tenth column of Table I) and phases (not shown). The final two columns use the previous results to determine if perfect asymmetry is possible for transmission or reflection. This makes evident that hermiticity (II) and parity (III) preclude, independently, any asymmetry in the scattering coefficients; PT-symmetry (VII) or Θ-pseudohermiticity (VI) forbid transmission asymmetry, whereas time-reversal symmetry (i.e., a real potential in coordinate space) (V) or PT-pseudohermiticity (VIII) forbid reflection asymmetry. Note that all local potentials satisfy automatically Θ-pseudohermiticity (VI). Of course asymmetric effects forbidden by a certain symmetry in the linear (Schrödinger) regime considered in this paper might not be forbidden in a non-linear regime [7], which goes be-yond our present scope.
The occurrence of one particular symmetry in the potential (conventionally "first symmetry") does not exclude a second symmetry to be satisfied as well. When a double symmetry holds, excluding the identity, the "first" symmetry implies the equivalence of the second symmetry with a third symmetry. We have already mentioned that Π-pseudohermiticity (IV) is equivalent to P T -symmetry (VII) for local potentials. Being local is just one particular way to satisfy symmetry VI, namely Θ-pseudohermiticity. The reader may verify with the aid of the third column for x|V |y in Table I, that indeed, if symmetry VI is satisfied (first symmetry), symmetry IV has the same effect as symmetry VII. They become equivalent. Other well known example is the fact that for a local potential (symmetry VI is satisfied), a real potential in coordinate space is necessarily hermitian, so symmetries V and II become equivalent. These are just particular cases of the full set of equivalences given in Table II.
Combining the information of the last two-columns in Table I with the additional condition that all scattering coefficients be 0/1 we elaborate Table III, which provides names for the six possible types of devices, a convenient letter code that summarizes the effect of left/right incidence, and the symmetries that do not allow the implementation of each device type. The complementary Table SI in the Supplemental Material gives instead the symmetries that allow, but do not necessarily imply, a given device type. The denominations in Table III are intended as short and meaningful, and do not necessarily coincide with some extended terminology, in part because the range of possibilities is broader here than those customarily considered, and because we use a 1 or 0 condition for the moduli. For example, a device with reflection asymmetry with T r = T l would in our case be a particular "transparent, one-way reflector", as full transmission occurs from both sides. This effect has however become popularized as "unidirectional invisibility" [8][9][10]. A debate on terminology is not our main concern here, and the use of a code system as the one proposed will be instrumental in avoiding misunderstandings.
Designing potentials for asymmetric devices. We will show how to design non-local potentials leading to the asymmetric devices. For simplicity we look for non-local potentials V (x, y) with local support that vanish for |x| > d and |y| > d.
Inverse scattering proceeds similarly to [11], by imposing an ansatz for the wavefunctions and the potential in the stationary Schrödinger equation The free parameters are fixed making use of the boundary conditions. The expected form of the wavefunction incident from the left is Our strategy is to assume polynomial forms for the two wavefunctions in the interval |x| < d, ψ l (x) = 5 j=0 c l,j x j and ψ r (x) = 5 j=0 c r,j x j , and also a polynomial ansatz of finite degree for the potential V (x, y) = i j v ij x i y j . Inserting these ansatzes in Eq. (3) and from the conditions that ψ l,r and their derivatives must be continuous, all coefficients c l,j , c r,j and v ij can be determined. Symmetry properties of the potential can also be imposed via additional conditions on the potential coefficients v ij . In the Supplemental Material, Sec. II, we use this strategy to implement potentials for different devices in Table III such that for a chosen k = k 0 the imposed boundary conditions (scattering amplitudes) are fulfilled exactly. They are also satisfied approximately in a neighborhood of k 0 .
Extending the asymmetry to a broad incidentmomentum domain. The inversion technique just described may be generalized to extend the range of incident momenta for which the potential works by imposing additional conditions and increasing correspondingly the number of parameters in the wavefunction ansatz, for example we may impose that the derivatives of the amplitudes, in one or more orders, vanish at k 0 , or 0/1 values for the coefficients not only at k 0 but at a series of grid points k 1 , k 2 , ... k N , as in [1,[11][12][13].
Here we put forward instead a method that provides a very broad working-window domain making use of the Born approximation. Specifically we apply the approach for a transparent one-way reflector T R/T . The goal is now to find a local PT-symmetric potential such that asymmetric reflection results, T l = T r = 1, R r = 0, |R l | = 1 for a broad range of incident momenta. A similar goal was pursued in [14] making use of a supersymmetric transformation, without imposing |R l | = 1.
In the Born approximation and for a local potential V (x), the reflection amplitudes take the simple form Defining the Fourier transform we get for k = p/ > 0: Assuming that the potential is local and PT-symmetrical, we calculate the transition coefficient from them using generalized unitarity as |T | 2 = 1 − R r * R l .
To build a T R/T device we demand: V (k) = √ 2παk (k < 0) and V (k) = 0 (k ≥ 0). By inverse Fourier  II  III  IV  V  VI  VII  VIII  III=IV  II=IV II=III II=VI  II=V II=VIII II=VII  V=VI  V=VII V=VIII III=VII III=VIII III=V III=VI  VII=VIII VI=VIII VI=VII IV=VIII IV=VII IV=VI IV=V   TABLE II: Equivalences among symmetries for the potential elements. Given the symmetry of the upper row, the table provides the equivalent symmetries. For example, if II is satisfied, then III=IV holds. In words, if the potential is hermitian, parity symmetry amounts to parity pseudohermiticity. In terms of the matrix elements of the potential, if x|V |y = y|V |x * and also x|V |y = −x|V | − y , ∀(x, y), then x|V |y = −y|V | − x * holds as well. One may proceed similarly for all other relations. The commutation with the identity (I) is excluded as this symmetry is satisfied by all potentials.  The code summarizes the effect of left and right incidence, separated by a slash /. T or R on one side of the slash indicate a unit transmission or reflection coefficient for incidence from that side, whereas the absence of one or the other letter corresponds to zero coefficients. An A denotes "full absorption", i.e., both moduli of reflection and transmission amplitudes are zero for incidence from one side. For example, T R/A means unit modulus transmission and reflection from the left and total absorption from the right. The fifth column indicates the symmetries in Table I that forbid the device. transformation, this implies which is indeed a local, P T -symmetric potential for α real. α is directly related to the reflection coefficient, within the Born approximation, R l = 4πimα/ 2 . As the Born approximation may differ from exact results we shall keep α as an adjustable parameter in the following. In a possible physical implementation, the potential in Eq. (7) will be approximated by keeping a small finite > 0, see Fig. 1 (a). Then, its Fourier transform is V (k) = √ 2παke k (k < 0) and V (k) = 0 (k ≥ 0). In Figs. 1(b) and (c), the resulting coefficients for /d = 10 −4 and two different values of α are shown. These figures have been calculated by numerically solving the Schrödinger equation exactly and demonstrate that α can indeed be adjusted so that R l 2 ≈ 1. Fig. 1(c) demonstrates that the local PT-symmetric potential works as intended, i.e., as a transparent one-way reflector, for a broad range of k values.
Discussion. This paper brings to the fore the essential role of eight generalized symmetries to determine the transmission and reflection asymmetries by complex, and possibly nonlocal potentials. These symmetries are classified with the aid of the relations between unitary or antiunitary operators 1, Π, Θ, ΘΠ, which form Klein's 4-group, and H or its adjoint. The symmetries set equalities among the scattering amplitudes which, complemented by generalized unitarity relations, tell us which symmetries allow or disallow a certain device with asymmetric scattering. Simplifying the analysis by imposing 0 or 1 scattering coefficients, six possible device types exist. We show how to design potentials realising these devices and provide examples on how to extend the domain of incident momenta for which they work making use of Born's approximation. The theory is worked out for particles and the Schrödinger equation but it is clearly of relevance for optical devices due to the much exploited analogies and connections between Maxwell's equations and the Schrödinger equation, which were used, e.g., to propose the realization of PT-symmetric potentials in optics [15].
Interesting questions left for future work are the inclusion of other mechanisms for transmission and reflection asymmetries (for example nonlinearities [7,16], and time dependent potentials [17,18]), or a full discussion of the phases of the scattering amplitudes in addition to the moduli emphasized here. We shall also examine in a complementary paper the physical realization of complex nonlocal effective potentials. In a quantum optics scenario, simple examples were provided in [19] based on applying the partitioning technique [20,21] to the scattering of a particle with internal structure.
Supplemental Material: Asymmetric scattering by non-hermitian potentials

I. SCATTERING AMPLITUDES
We provide a lightning review of scattering amplitudes in 1D. For a more complete account, see [1]. We assume p > 0. The amplitudes for scattering by H = H 0 + V , may be calculated by where the l/r superscript indicates left or right incidence, and where E p = p 2 /(2m). To find Born-approximation expressions of the scattering coefficients (square moduli of the amplitudes), we take T op ≈ V in the expressions of R l , and R r . For T l and T r we also include the second order in V , which contributes to the square in second order due to the 1 in Eqs. (S2) and (S4). The on-shell S matrix, see [1], is formed as This on-shell matrix relates to the standard S-matrix elements in momentum representation, by factoring out a delta function, p|S|p = |p| m δ(E p − E p ) p|S|p . All the above formulae may be reproduced when the particle is scattered instead by H † = H 0 + V † , giving scattering amplitudes with a hat, T r , T l , R r , R l , and S. Hatted and unhatted amplitudes are not independent, they are linked by the generalized unitary relation S † S = S S † = 1, whose on-shell matrix elements lead to the four relations T l T l * + R l R l * = 1, T r T r * + R r R r * = 1, T l * R r + T r R l * = 0, T l R r * + T r * R l = 0. (S8)
For constructing examples of potentials for such devices, we fix the phases of the transmission amplitudes as T l = 1, T r = 0, and the reflection amplitudes will be specified in each case. We assume the form V (x, y) = 5 i=0 1 j=0 v ij x i y j for the potential, plug this ansatz in the Schrödinger equation (3), and equate equal powers of x. Moreover we demand that V (−d, y) = 0 = V (d, y) for all y such that the total potential (including the vanishing potential for x, y < −d and x, y > d) is continuous.
We consider first an ideal one-way mirror (T R/A) with amplitudes R l = −1, R r = 0. Waves sent from the left are fully reflected, but there is also perfect transmission, whereas waves sent from the right are absorbed. The potential that achieves this for k = k 0 = 1/d is shown in One-way T-filters (T /A) and the mirror&1-way transmitters (T R/R) can be also constructed using the method described in the previous subsection. Nevertheless, unlike the two devices in the previous subsection, these devices can fulfill symmetry VIII. We assume now the form V (x, y) = To simplify the potential, we also demand v 4,4 = v 4,5 = v 5,4 = v 5,5 = 0. Moreover we demand that V (−d, y) = 0 = V (d, y) for all y such that the total potential (including the vanishing potential for x, y < −d and x, y > d) is continuous. It is also required that R l = R r = R, consistent with Table I. In Fig. S4, the potential for the one-way T-filter (T /A), with R = 0, T l = 1, is shown, and the potential for the mirror&1-way transmitter (T R/R), calcu- lated for R = −1, T l = 1, is shown in Fig. S5 where we have chosen k 0 = 1/d. The transmission and reflection coefficients around k 0 are also depicted.
For the first three devices (T R/A, T /R and T /A), it follows from the generalized unitarity relations (S9) that one or more of the transmission and reflection amplitudes of the corresponding adjoint Hamiltonian will diverge at k = k 0 = 1/d (if the numerator on the righthand side of these relations stays finite while the corresponding denominator T l T r − R l R r = −R l R r → 0). In the mirror&1-way transmitter, it follows from (S9) that T l = 0, R l = −1, T r = −1, R r = −1, and therefore the adjoint Hamiltonian provides a mirror&1-way transmitter device with l ↔ r.

IIc. Devices with asymmetric reflection
In the previous subsections we have already considered two device types with asymmetric reflection coefficients, namely, the one-way mirror (T R/A), and the one  way-barrier (T /R). These are the only two device types which are simultaneously asymmetrical for transmission and reflection. Two more types are possible which have only reflection asymmetry, namely, the one-way R-filter (R/A), and the transparent one-way reflector (T R/T ). Both are compatible with symmetry type VI, in particular with local potentials. A one-way R-filter R/A acts as a perfect absorber from one side and as a perfect reflector from the other side. It may thus be constructed by adding an infinite barrier with its edge touching the end of known-perfect absorbers for one-sided incidence [1,[11][12][13]. Local, perfect absorbers can be worked out for one or more incident momenta, or for a momentum window. According to Table III, a R/A device cannot have PT-symmetry. Indeed experimental realizations in optics imply local non-PT-symmetric potentials [S1].
The remaining device is a one-way reflector (T R/T ). Specifically, if we set T l = 1, R l = 1, T r = −1, R r = 0, i.e. T l = T r but T l 2 = |T r | 2 = 1, it can be achieved with a PT-symmetric potential, but it must be nonlocal, see Table I. (If we set T l = T r = 1, local forms of the potential are also possible, as demonstrated in the main text.) For a nonlocal PT-symmetric potential, V (x, y) = V (−x, −y) * for all x, y. We assume the form ij , in other words, v ij must be real for i + j even and purely imaginary for i + j odd. We also require that V (−d, y) = V (d, y) = 0 for all y and follow the same procedure described in previous subsections. The nonlocal PT-potential found can be seen in Fig. S6 (a),(b) for k = k 0 = 1/d. The transmission and reflection coefficients around k 0 are depicted in Fig. S6 (c),(d).