Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2008.01702 · Physical Review A

Quantum-optical implementation of non-Hermitian potentials for asymmetric scattering

A. Ruschhaupt, Anthony Kiely, Miguel Angel Otero Simón, Juan Gonzalo Muga

Published Nov 5, 2020• 5 citations
arXivHTMLPDF

Abstract

Non-Hermitian, one-dimensional potentials which are also nonlocal, allow for scattering asymmetries, namely, asymmetric transmission or reflection responses to the incidence of a particle from left or right. The symmetries of the potential imply selection rules for transmission and reflection. In particular, parity-time (PT) symmetry or the symmetry of any local potential do not allow for asymmetric transmission. We put forward a feasible quantum-optical implementation of non-Hermitian, nonlocal, non-PT potentials to implement different scattering asymmetries, including transmission asymmetries.

DOI
10.1103/physreva.102.053705

Related papers

Asymmetric scattering by non-Hermitian potentials

A. Ruschhaupt, Tom Dowdall, Miguel Angel Otero Simón +1 more

The scattering of quantum particles by non-Hermitian (generally non-local) potentials in one dimension may result in asymmetric transmission and/or reflection from left and right incidence. After extending the concept of symmetry for non-Hermitian potentials, eight generalized symmetries based on the discrete Klein's four-group (formed by parity, time reversal, their product, and unity) are found. Together with generalized unitarity relations they determine selection rules for 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.

arXiv:1709.07027•Europhysics Letters (EPL)•Jan 2, 2018•20 citations

The Role of Symmetry in Non-Hermitian Scattering1

A. Ruschhaupt, Miguel Angel Otero Simón, Anthony Kiely +1 more

Abstract We review recent work on asymmetric scattering by Non-Hermitian (NH) Hamiltonians. Quantum devices with an asymmetric scattering response to particles incident from right or left in effective ID waveguides will be important to develop quantum technologies. They act as microscopic equivalents of familiar macroscopic devices such as diodes, rectifiers, or valves. The symmetry of the underlying NH Hamiltonian leads to selection rules which restrict or allow asymmetric response. NH-symmetry operations may be organized into group structures that determine equivalences among operations once a symmetry is satisfied. The NH Hamiltonian posseses a particular symmetry if it is invariant with respect to the corresponding symmetry operation, which can be conveniently expressed by a unitary or antiunitary superoperator. A simple group is formed by eight symmetry operations, which include the ones for Parity-Time symmetry and Hermiticity as specific cases. The symmetries also determine the structure of poles and zeros of the S matrix. The ground-state potentials for two-level atoms crossing properly designed laser beams realize different NH symmetries to achieve transmission or reflection asymmetries.

Journal of Physics Conference Series•Oct 1, 2021•2 citations

Scattering in PT-symmetric quantum mechanics

F. Cannata, J.-P. Dedonder, Alberto Ventura

A general formalism is worked out for the description of one-dimensional scattering in non-hermitian quantum mechanics and constraints on transmission and reflection coefficients are derived in the cases of P, T, or PT invariance of the Hamiltonian. Applications to some solvable PT-symmetric potentials are shown in detail. Our main original results concern the association of reflectionless potentials with asymptotic exact PT symmetry and the peculiarities of separable kernels of non-local potentials in connection with hermiticity, T invariance and PT invariance.

arXiv:quant-ph/0606129•Jun 15, 2006•73 citations

Constraints on the reciprocal propagation of a quantum particle through a one-dimensional localized complex potential

Ethan Knox, P. T. Leung

In the propagation of an electron through a one-dimensional asymmetric complex potential, it is known that while the conventional Green function reciprocity symmetry will ensure transmission to be symmetric between a “left-incident” and a “right-incident” beam, no such symmetry exists for the case of reflection. Here we derive generalized reciprocity relations for both the amplitude and phase of the reflected waves as constraints on the left- and right-incident beams, in complete analogy to what was established in optics. We further provide illustrations of these relations via direct analytical calculations in the case of a real potential, and via numerical studies in the case of a complex potential.

Canadian Journal of Physics•Jan 22, 2013•0 citations

New features of scattering from a one-dimensional non-Hermitian (complex) potential

Zafar Ahmed

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time reversal: R ( − k ) ≠ R ( k ) and T ( − k ) ≠ T ( k ), unless the potentials are real or PT-symmetric. For complex PT-symmetric scattering potentials, we propose that R left ( − k ) = R right ( k ) and T ( − k ) = T ( k ). So far, the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most 1. We present a new non-Hermitian parametrization of the Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies ( E * = α 2 , β 2 ) either in T ( k ) or in T ( − k ), when αβ > 0. Thirdly, when αβ < 0 it possesses one SS in T ( k ) and the other in T ( − k ). Fourthly, when the potential becomes PT-symmetric [(α + β) = 0], we obtain T ( k ) = T ( − k ), it possesses a unique SS at E = α 2 in both T ( − k ) and T ( k ). Lastly, for completeness, when α = iγ and β = iδ there are no SS, instead we get two real energies −γ 2 and −δ 2 of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound-state eigenvalues. We find them as E + M = −(γ − M ) 2 and E − N = −(δ − N ) 2 ; M ( N ) = 0, 1, 2, ... with 0 ⩽ M ( N ) < γ(δ).

arXiv:1110.4485•Journal of Physics A Mathematical and Theoretical•Dec 19, 2011•49 citations

Exact scattering states in one-dimensional Hermitian and non-Hermitian potentials*

Ruo-Lin Chai, Qiongtao Xie, Xiaoliang Liu

The scattering states in one-dimensional Hermitian and non-Hermitian potentials are investigated. An analytical solution for the scattering states is presented in terms of Heun functions. It is shown that for some specially chosen parameter conditions, an infinite number of the exact scattering states is obtained. In the Hermitian potentials, they correspond to the reflectionless states. In the non-Hermitian complex potentials with parity-time symmetry, they are the unidirectionally reflectionless states.

Chinese Physics B•May 13, 2020•2 citations
Non-Hermitian, one-dimensional potentials which are also nonlocal, allow for scattering asymmetries, namely, asymmetric transmission or reflection responses to the incidence of a particle from left or right. The symmetries of the potential imply selection rules for transmission and reflection. In particular, parity-time (PT) symmetry or the symmetry of any local potential do not allow for asymmetric transmission. We put forward a feasible quantum-optical implementation of non-Hermitian, nonlocal, non-PT potentials to implement different scattering asymmetries, including transmission asymmetries.
The scattering of quantum particles by non-Hermitian (generally non-local) potentials in one dimension may result in asymmetric transmission and/or reflection from left and right incidence. After extending the concept of symmetry for non-Hermitian potentials, eight generalized symmetries based on the discrete Klein's four-group (formed by parity, time reversal, their product, and unity) are found. Together with generalized unitarity relations they determine selection rules for 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.
Abstract We review recent work on asymmetric scattering by Non-Hermitian (NH) Hamiltonians. Quantum devices with an asymmetric scattering response to particles incident from right or left in effective ID waveguides will be important to develop quantum technologies. They act as microscopic equivalents of familiar macroscopic devices such as diodes, rectifiers, or valves. The symmetry of the underlying NH Hamiltonian leads to selection rules which restrict or allow asymmetric response. NH-symmetry operations may be organized into group structures that determine equivalences among operations once a symmetry is satisfied. The NH Hamiltonian posseses a particular symmetry if it is invariant with respect to the corresponding symmetry operation, which can be conveniently expressed by a unitary or antiunitary superoperator. A simple group is formed by eight symmetry operations, which include the ones for Parity-Time symmetry and Hermiticity as specific cases. The symmetries also determine the structure of poles and zeros of the S matrix. The ground-state potentials for two-level atoms crossing properly designed laser beams realize different NH symmetries to achieve transmission or reflection asymmetries.
A general formalism is worked out for the description of one-dimensional scattering in non-hermitian quantum mechanics and constraints on transmission and reflection coefficients are derived in the cases of P, T, or PT invariance of the Hamiltonian. Applications to some solvable PT-symmetric potentials are shown in detail. Our main original results concern the association of reflectionless potentials with asymptotic exact PT symmetry and the peculiarities of separable kernels of non-local potentials in connection with hermiticity, T invariance and PT invariance.
In the propagation of an electron through a one-dimensional asymmetric complex potential, it is known that while the conventional Green function reciprocity symmetry will ensure transmission to be symmetric between a “left-incident” and a “right-incident” beam, no such symmetry exists for the case of reflection. Here we derive generalized reciprocity relations for both the amplitude and phase of the reflected waves as constraints on the left- and right-incident beams, in complete analogy to what was established in optics. We further provide illustrations of these relations via direct analytical calculations in the case of a real potential, and via numerical studies in the case of a complex potential.
For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time reversal: R ( − k ) ≠ R ( k ) and T ( − k ) ≠ T ( k ), unless the potentials are real or PT-symmetric. For complex PT-symmetric scattering potentials, we propose that R left ( − k ) = R right ( k ) and T ( − k ) = T ( k ). So far, the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most 1. We present a new non-Hermitian parametrization of the Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies ( E * = α 2 , β 2 ) either in T ( k ) or in T ( − k ), when αβ > 0. Thirdly, when αβ < 0 it possesses one SS in T ( k ) and the other in T ( − k ). Fourthly, when the potential becomes PT-symmetric [(α + β) = 0], we obtain T ( k ) = T ( − k ), it possesses a unique SS at E = α 2 in both T ( − k ) and T ( k ). Lastly, for completeness, when α = iγ and β = iδ there are no SS, instead we get two real energies −γ 2 and −δ 2 of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound-state eigenvalues. We find them as E + M = −(γ − M ) 2 and E − N = −(δ − N ) 2 ; M ( N ) = 0, 1, 2, ... with 0 ⩽ M ( N ) < γ(δ).
The scattering states in one-dimensional Hermitian and non-Hermitian potentials are investigated. An analytical solution for the scattering states is presented in terms of Heun functions. It is shown that for some specially chosen parameter conditions, an infinite number of the exact scattering states is obtained. In the Hermitian potentials, they correspond to the reflectionless states. In the non-Hermitian complex potentials with parity-time symmetry, they are the unidirectionally reflectionless states.