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arXiv:1412.7630 · Physical Review A

Transmission phase lapse in the non-Hermitian Aharonov-Bohm interferometer near the spectral singularity

Gang Zhang, Xuejing Li, X. Z. Zhang, Z. Song

Published Jan 26, 2015• 11 citations
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Abstract

We study the effect of $\mathcal{PT}$-symmetric imaginary potentials embedded in the two arms of an Aharonov-Bohm interferometer on the transmission phase by finding an exact solution for a concrete tight-binding system. It is observed that the spectral singularity always occurs at $k=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ for a wide range of fluxes and imaginary potentials. Critical behavior associated with the physics of the spectral singularity is also investigated. It is demonstrated that the quasispectral singularity corresponds to a transmission maximum and the transmission phase jumps abruptly by $\ensuremath{\pi}$ when the system is swept through this point. Moreover, we find that there exists a pulselike phase lapse when the imaginary potential approaches the boundary value of the spectral singularity.

DOI
10.1103/physreva.91.012116

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We study the effect of PT\mathcal{PT}PT-symmetric imaginary potentials embedded in the two arms of an Aharonov-Bohm interferometer on the transmission phase by finding an exact solution for a concrete tight-binding system. It is observed that the spectral singularity always occurs at k=\ifmmode±\else\textpm\fi\ensuremathπ/2k=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2k=\ifmmode±\else\textpm\fi\ensuremathπ/2 for a wide range of fluxes and imaginary potentials. Critical behavior associated with the physics of the spectral singularity is also investigated. It is demonstrated that the quasispectral singularity corresponds to a transmission maximum and the transmission phase jumps abruptly by \ensuremathπ\ensuremath{\pi}\ensuremathπ when the system is swept through this point. Moreover, we find that there exists a pulselike phase lapse when the imaginary potential approaches the boundary value of the spectral singularity.
Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a waveguide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic waveguide.
We study the possibility of asymmetric transmission induced by a non-Hermitian scattering center embedded in a one-dimensional waveguide, motivated by the aim of realizing quantum diodes in a non-Hermitian system. It is shown that a PT\mathcal{PT}PT-symmetric non-Hermitian scattering center always has symmetric transmission although the dynamics within the isolated center can be unidirectional, especially at its exceptional point. We propose a concrete scheme based on a flux-controlled non-Hermitian scattering center, which comprises a non-Hermitian triangular ring threaded by an Aharonov-Bohm flux. The analytical solution shows that such a complex scattering center acts as a diode at the resonant energy level of the spectral singularity, exhibiting perfect unidirectionality of the transmission. The connections between the phenomena of the asymmetric transmission and reflectionless absorption are also discussed.
We investigate a two-parametric family of one-dimensional non-Hermitian complex potentials with parity-time (PT\mathcal{PT}PT) symmetry. We find that there exist two distinct types of phase transitions, from an unbroken phase (characterized by a real spectrum) to a broken phase (where the spectrum becomes complex). The first type involves the emergence of a pair of complex eigenvalues bifurcating from the continuous spectrum. The second type is associated with the collision of such pairs at the bottom of the continuous spectrum. The first transition type is closely related to spectral singularities (SSs), at which point the transmission and reflection coefficients are divergent simultaneously. The second is associated with the emergence of bound states. In particular, under specific parameter conditions, we construct an exact bound state solution. By systematically exploring the parameter space, we establish a universal relationship governing the number of SSs in these potentials. These findings provide a fundamental theoretical framework for manipulating wave scattering in non-Hermitian systems, offering promising implications for designing advanced optical and quantum devices.
A non-Hermitian interferometer can realize asymmetric transmission in the presence of imaginary potential and magnetic flux. Here, we propose a non-Hermitian dimer with an unequal hopping rate by an interferometerlike cluster in the framework of a tight-binding model. The intriguing features of this design are the wave-vector independence and unidirectionality of scattering, which amplify the wave packet without distortion and absorb the incoherent wave without reflection. The absorption relates to the system spectral singularities. The dynamical behaviors of the spectral singularities are also investigated analytically and numerically.
Spectral singularities are predicted to occur in a non-Hermitian extension of the Friedrichs-Fano-Anderson model describing the decay of a discrete state ∣a⟩|a⟩∣a⟩ coupled to a continuum of modes. A physical realization of the model, based on electronic or photonic transport in a semi-infinite tight-binding lattice with an imaginary impurity site at the lattice boundary, is proposed. The occurrence of the spectral singularities is shown to correspond either to a diverging reflection probability (for an amplifying impurity) or to a vanishing reflection probability (for an absorbing impurity) from the lattice boundary. In the former case, the spectral singularity of the resolvent is also responsible for the nondecay of state ∣a⟩|a⟩∣a⟩ into the continuum, in spite of the absence of bound states.
The spectral singularity (SS) from a non-Hermitian potential is one of the most remarkable scattering feature of non-Hermitian quantum mechanics. At the spectral singular point, the scattering amplitudes diverge to infinite. This phenomena have been extensively studied over the last two decades. The previous studies have suggested the need of extremely fine control of the various system parameters for practical applications of SS. However no study have been carried out to understand the behavior of the scattering amplitude in the vicinity of SS. Such study would be important towards the practical application of SS where it is desired to maintain outgoing scattering amplitude to a specific value. The behavior of the loci of constant scattering amplitude in the neighborhood of SS are studied for a pure imaginary barrier potential iViViV, V∈R+V \in R^{+}V∈R+. We show that these loci are elliptical and self-similar to each other in E−VE-VE−V plane where EEE is energy of the wave. The orientations of these ellipses for reflection (RRR) and transmission (TTT) amplitude around a given SS are same. This is surprising due to the different mathematical forms of RRR and TTT. In the vicinity of SS, R≈TR \approx TR≈T.