Chin. Phys. Lett.  2022, Vol. 39 Issue (3): 037302    DOI: 10.1088/0256-307X/39/3/037302
CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES |
Unitary Scattering Protected by Pseudo-Hermiticity
L. Jin*
School of Physics, Nankai University, Tianjin 300071, China
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L. Jin 2022 Chin. Phys. Lett. 39 037302
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Abstract 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.
Received: 07 December 2021      Express Letter Published: 26 January 2022
PACS:  73.50.Bk (General theory, scattering mechanisms)  
  03.65.-w (Quantum mechanics)  
  11.80.Gw (Multichannel scattering)  
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https://cpl.iphy.ac.cn/10.1088/0256-307X/39/3/037302       OR      https://cpl.iphy.ac.cn/Y2022/V39/I3/037302
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[104]For a rigorous proof, please see Eq. (12) in Ref.[103] and compare the scattering coefficients of the scattering centers $H$ and $H^{\rm T}$. Notice that the symbols $t_{\scriptscriptstyle{\rm L}}$, $t_{\scriptscriptstyle{\rm R}}$, $ r_{\scriptscriptstyle{\rm L}}$ and $r_{\scriptscriptstyle{\rm R}}$ in Ref.[103] are $s_{nm}$, $s_{mn}$, $s_{mm}$ and $s_{nn}$ of the scattering matrix with our current notations for any pair of ports $m$ and $n$. The fact $(A^T)^{-1}=(A^{-1})^T$ is also used in the proof for any square matrix $A$.
[105]From the off-diagonal term of $SS^{† }$, we obtain $r_{\scriptscriptstyle{\rm L}}=-t_{\scriptscriptstyle{\rm R}}r_{\scriptscriptstyle{\rm R}}^{\ast }/t_{\scriptscriptstyle{\rm L}}^{\ast }$. From the diagonal terms of $SS^{† }$, we obtain $1=r_{\scriptscriptstyle{\rm L}}r_{\scriptscriptstyle{\rm L}}^{\ast }+t_{\scriptscriptstyle{\rm R}}t_{\scriptscriptstyle{\rm R}}^{\ast }=\left( r_{\scriptscriptstyle{\rm R}}^{\ast }r_{\scriptscriptstyle{\rm R}}/t_{\scriptscriptstyle{\rm L}}^{\ast }t_{\scriptscriptstyle{\rm L}}+1\right) t_{\scriptscriptstyle{\rm R}}t_{\scriptscriptstyle{\rm R}}^{\ast }=t_{\scriptscriptstyle{\rm R}}t_{\scriptscriptstyle{\rm R}}^{\ast }/t_{\scriptscriptstyle{\rm L}}^{\ast }t_{\scriptscriptstyle{\rm L}}$.
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