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arXiv:1109.2187 · Physical Review A

Hermitian scattering behavior for a non-Hermitian scattering center

L. Jin, Z. Song

Published Jan 30, 2012• 31 citations
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Abstract

We study the scattering problem for the non-Hermitian scattering center, which consists of two Hermitian clusters with anti-Hermitian couplings between them. Counterintuitively, it is shown that it acts as a Hermitian scattering center, that is, the $S$ matrix is unitary, or the Dirac probability current is conserved, ${|r|}^{2}+{|t|}^{2}=1$, when one of the two clusters is embedded in the waveguides. This conclusion can be applied to an arbitrary parity-symmetric real Hermitian graph with additional $\mathcal{PT}$-symmetric potentials, which is more feasible in the experiment. An exactly solvable model is presented to illustrate the theory. Bethe ansatz solution indicates that the transmission spectrum of such a cluster displays a peculiar feature arising from the non-Hermiticity of the scattering center.

DOI
10.1103/physreva.85.012111

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We study the scattering problem for the non-Hermitian scattering center, which consists of two Hermitian clusters with anti-Hermitian couplings between them. Counterintuitively, it is shown that it acts as a Hermitian scattering center, that is, the SSS matrix is unitary, or the Dirac probability current is conserved, ∣r∣2+∣t∣2=1{|r|}^{2}+{|t|}^{2}=1∣r∣2+∣t∣2=1, when one of the two clusters is embedded in the waveguides. This conclusion can be applied to an arbitrary parity-symmetric real Hermitian graph with additional PT\mathcal{PT}PT-symmetric potentials, which is more feasible in the experiment. An exactly solvable model is presented to illustrate the theory. Bethe ansatz solution indicates that the transmission spectrum of such a cluster displays a peculiar feature arising from the non-Hermiticity of the scattering center.
We explore a way of finding the link between a non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of Bethe ansatz solutions for a class of non-Hermitian Hamiltonians and the scattering problems for the corresponding Hermitian Hamiltonians, it is shown that a scattering state of an arbitrary Hermitian lattice embedded in a chain as the scattering center shares the same wavefunction with the corresponding non-Hermitian tight binding lattice, which consists of the Hermitian lattice with two additional on-site complex potentials, no matter whether the non-Hermitian is broken symmetry or even non- symmetry. Exactly solvable models are presented to demonstrate the main points of this paper.
We investigate the scattering properties of a parity-time-antisymmetric non-Hermitian system. The parity-time-antisymmetric scattering center possesses imaginary nearest-neighbor hoppings and real on-site potentials; it has been experimentally realized through dissipative coupling and frequency detuning between atomic spin waves. We find that such parity-time-antisymmetric system displays three salient features: First, the reflection and transmission are both reciprocal. Second, the reflection and transmission probabilities satisfy $R\ifmmode\pm\else\textpm\fi{}T=1$, which depends on the parity of the scattering center size. Third, the scattering matrix satisfies $\left(S{\ensuremath{\sigma}}_{\mathrm{z}}\right){\left(S{\ensuremath{\sigma}}_{\mathrm{z}}\right)}^{*}=I$ for a scattering center with an even site; for a scattering center with an odd site, the dynamics exhibits Hermitian scattering behavior, possessing unitary scattering matrix $S{S}^{\ifmmode\dagger\else\textdagger\fi{}}=I$.
We investigate the properties of two anti-parity-time (anti-PT\mathcal{PT}PT)-symmetric four-site scattering centers. The anti-PT\mathcal{PT}PT-symmetric scattering center may have imaginary couplings, real couplings, and real on-site potentials. The only difference between the two scattering centers is the coupling between two central sites of the scattering center, which plays a crucial role in determining the parity of anti-PT\mathcal{PT}PT symmetry and significantly affects the scattering properties. For the imaginary coupling, the even-parity anti-PT\mathcal{PT}PT-symmetric scattering center possesses nonunitary scattering and the difference between the reflection and transmission is unity; for the real coupling, the odd-parity anti-PT\mathcal{PT}PT-symmetric scattering center possesses unitary scattering and the sum of the reflection and transmission is unity. The coupling-induced different scattering behaviors are verified in the numerical simulations. Our findings reveal that a significant difference in the dynamics can be caused by a slight difference between two similar anti-PT\mathcal{PT}PT-symmetric non-Hermitian scattering centers.
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.
Local non-Hermitian potentials $V(x)\ensuremath{\ne}{V}^{*}(x)$ can, sometimes, generate stable bound states $\ensuremath{\psi}(x)$ at real energies. Unfortunately, the idea [based on the use of a non-Dirac ad hoc metric $\ensuremath{\Theta}(x,{x}^{\ensuremath{'}})\ensuremath{\ne}\ensuremath{\delta}(x\ensuremath{-}{x}^{\ensuremath{'}})$ in Hilbert space] cannot directly be transferred to scattering due to the related loss of the asymptotic observability of xxx [cf. H. F. Jones, Phys. Rev. D 78, 065032 (2008)]. We argue that for smeared (typically, nonlocal or momentum-dependent) potentials $V\ensuremath{\ne}{V}^{\ifmmode\dagger\else\textdagger\fi{}}$ this difficulty may be circumvented. A return to the usual (i.e., causal and unitary) quantum scattering scenario is then illustrated via an exactly solvable multiple-scattering example. In it, the anomalous loss of observability of the coordinate remains restricted to a small vicinity of the scattering centers.
Hermitian systems possess unitary scattering. However, the Hermiticity is unnecessary for a unitary scattering although the scattering under the influence of non-Hermiticity is mostly non-unitary. Here we prove that the unitary scattering is protected by certain type of pseudo-Hermiticity and unaffected by the degree of non-Hermiticity. The energy conservation is violated in the scattering process and recovers after scattering. The subsystem of the pseudo-Hermitian scattering center including only the connection sites is Hermitian. These findings provide fundamental insights on the unitary scattering, pseudo-Hermiticity, and energy conservation, and are promising for light propagation, mesoscopic electron transport, and quantum interference in non-Hermitian systems.