Abstract
In non-Hermitian scattering problems the behavior of the transmission probability is very different from its Hermitian counterpart; it can exceed unity or even be divergent, since the non-Hermiticity can add or remove the probability to and from the scattering system. In the present paper, we consider the scattering problem of a PT-symmetric potential and find a counter-intuitive behavior. In the usual PT-symmetric non-Hermitian system, we would typically find stationary semi-Hermitian dynamics in a regime of weak non-Hermiticity but observe instability once the non-Hermiticity goes beyond an exceptional point. Here, in contrast, the behavior of the transmission probability is strongly non-Hermitian in the regime of weak non-Hermiticity with divergent peaks, while it is superficially Hermitian in the regime of strong non-Hermiticity, recovering the conventional Fabry-Perot-type peak structure. We show that the unitarity of the S-matrix is generally broken in both of the regimes, but is recovered in the limit of infinitely strong non-Hermiticity.
Highlights
Non-Hermitian quantum mechanics has attracted a great deal of attention recently; for a recent review, see, e.g., Ref. [1]
We present a brief review in Appendix A for the point-spectral complex eigenvalues in open quantum systems, and details of the analytic calculations are presented in Appendix B
We have considered the non-Hermitian scattering problem for a Fabry-Pérot-type PT -symmetric model
Summary
Non-Hermitian quantum mechanics has attracted a great deal of attention recently; for a recent review, see, e.g., Ref. [1]. One of our motivations here is to see in an infinite open quantum system the physics of the exceptional point typical in the two-site PT -symmetric model (1) It manifests itself in the following simplest non-Hermitian scattering problem. For the eigenvalue k = π /2 + iκ, the probability amplitude starts to stay in the vicinity of the potential site, leading to an exponentially decaying wave function e−κ|x|, i.e., a sort of pseudo-bound-state is formed The formation of such a bound state (or a peculiar form of resonant state) is specific to non-Hermitian systems. A Breit-Wigner-type formula [62] for the transmission coefficient holds, leading to a finite, broadened transmission peak; the width of the peak is determined by the imaginary part of k or E [cf Eq (56)] This is contrasting to the case of the small-γ regime in which the resonant poles are on the real axis, leading to a sharp divergent peak.
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