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Canadian Journal of Physics

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

Ethan Knox, P. T. Leung

Published Jan 22, 2013• 0 citations
Publisher

Abstract

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.

DOI
10.1139/cjp-2012-0459

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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.
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.
Generalized unitarity and reciprocity relations for PT\mathcal{P}\mathcal{T}PT-symmetric scattering potentials
We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, Rl/r(k)R^{l/r}(k)Rl/r(k) and T(k)T(k)T(k), of general PT{\cal PT}PT-symmetric scattering potentials. We use these identities to give a general proof of the relations, ∣T(−k)∣=∣T(k)∣|T(-k)|=|T(k)|∣T(−k)∣=∣T(k)∣ and ∣Rr(−k)∣=∣Rl(k)∣|R^r(-k)|=|R^l(k)|∣Rr(−k)∣=∣Rl(k)∣, conjectured in [Z. Ahmed, J. Phys. A 45 (2012) 032004], establish the generalized unitarity relation: Rl/r(k)Rl/r(−k)+∣T(k)∣2=1R^{l/r}(k)R^{l/r}(-k)+|T(k)|^2=1Rl/r(k)Rl/r(−k)+∣T(k)∣2=1, and show that it is a common property of both real and complex PT{\cal PT}PT-symmetric potentials. The same holds for T(−k)=T(k)∗T(-k)=T(k)^*T(−k)=T(k)∗ and ∣Rr(−k)∣=∣Rl(k)∣|R^r(-k)|=|R^l(k)|∣Rr(−k)∣=∣Rl(k)∣.
Abstract We consider a quantum particle with energy E incident upon a one-dimensional potential. We show that the probabilities of transmission and reflection are the same for incidence upon a general potential from either side (from ‘the left’ or ‘the right’). This equality holds true for any potential which goes to constant values as x → ± ∞ and is finite for all x . We present a remarkably simple proof that the probabilities are equal. The simplicity of our proof is the most important pedagogical result of our paper, and will be easily understood by undergraduate students in second to fourth year. We discuss several cases, including the step potential and the potential barrier.
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 concept of reciprocity symmetry for matter-wave propagation is established for nonrelativistic quantum mechanics with previous results in the literature extended to include nonlocal interactions. Examples are given for cases with both local and nonlocal potentials, where we show in particular that reciprocity can be violated for the motion of a charged particle in an external electromagnetic field. In addition, this symmetry is applied to interpret a recent analysis [Phys. Rev. A 64, 042716 (2001)] on the symmetry of transmission through one-dimensional complex potentials, with the emphasis that the validity of reciprocity can go beyond that of time-reversal symmetry, such as in the presence of absorption in which the latter symmetry breaks down.