Biorthogonal scattering and generalized unitarity in non-Hermitian systems

TL;DR

The paper addresses quantum transport through non-Hermitian two-site dimers coupled to external leads, where spectra are generically complex and eigenstates non-orthogonal. It develops a biorthogonal scattering framework using left and right eigenstates to construct a scattering matrix, derives reflection/transmission coefficients, and shows that standard unitarity is broken for right-only scattering but restored in a generalized sense via ${S^L}^{\dagger}S^R=I$ and $R^{bi}+T^{bi}=1$. Through PT-symmetric and non-reciprocal dimers, the authors reveal transport enhancements arising from both complex eigenvalues and eigenstate non-orthogonality, with poles/zeros governing resonant features; a pole-zero mapping and EPs provide deeper insight. The vertically arranged PT dimer example demonstrates how PT symmetry can persist under lead coupling, keeping RR sums near unity while revealing distinct resonance structures. Overall, the work provides a biorthogonal, unitary-compatible framework for non-Hermitian quantum transport with implications for engineered gain/loss and directional coupling in quantum devices.

Abstract

We investigate the two-port scattering process in non-Hermitian dimer models via quantum measurements using external leads. We focus on two exemplary dimer models that preserve parity-time symmetry via spatial gain-loss balance and exhibit non-reciprocity due to directional hopping. The scattering matrix is constructed using the biorthogonality of the left and right scattering states of the Hamiltonian, allowing us to calculate the reflection and transmission probabilities. Our analysis compares the reflection and transmission coefficients derived from the left, right, and combined scattering states, revealing that, unlike in Hermitian systems, the non-Hermitian scattering process does not adhere to unitarity when considering only the right scattering states. Furthermore, non-Hermitian scattering can enhance the reflection and transmission probabilities, with distinct physical contributions arising independently from complex eigenvalues and the non-orthogonality of eigenstates. Our results clarify how biorthogonality restores generalized unitarity and identify distinct physical origins of enhanced transport in PT-symmetric and non-reciprocal dimers, providing new insights into quantum transport in non-Hermitian systems.

Figures (9)

• Figure 1: A dimer model with leads. Here, $a$ and $b$ represent the lattice labels for the system and positive and negative integer values denote the lattice labels for the right and left leads, respectively. Blue box represents a dimer model described by the system Hamiltonian $H_s$ of Eq. (\ref{['eq:Hamiltonian']}).
• Figure 2: (a) is a schematic figure for the right scattering process. (b) and (c) are reflection and transmission probabilities for the right scattering states of the single-site model indicated in the top panels with incident wave from a left lead as a function of ($E$, $\gamma$), respectively. (d)-(f) are the case of the left scattering process. The results with incident waves from left and right leads are the same. The left scattering states of $H$ correspond to the right scattering states of $H^\dagger$.
• Figure 3: (a) is a schematic figure for the right scattering process with incident wave from left leads attached to a PT dimer. (b) and (c) are reflection and transmission probabilities for the right scattering states as a function of ($E$, $\gamma$), respectively. (d)-(f) are the case of the right scattering process with incident wave from right leads. There is an EP when $\gamma = 1$. (g) Evolution of two eigenenergies connected via EP in a PT dimer model without leads. As $\gamma$ increases, two real eigenenergies (solid black) approach each other and then coalesce at an EP (large orange dot). They split into two complex conjugate eigenenergies from the EP. The three projected figures show the real (blue) and imaginary (red) parts of the complex eigenenergies as a function of $\gamma$ and complex energy (green).
• Figure 4: (a) is a schematic figure for the left scattering process with incident wave from left leads attached to a PT dimer. (b) and (c) are reflection and transmission probabilities for the left scattering states as a function of ($E$, $\gamma$), respectively. (d)-(f) are the case of the left scattering process with incident wave from right leads. Results are the same as those in Fig. \ref{['fig:fig_PT_re']} with opposite incident waves.
• Figure 5: (a) is a schematic figure for the right scattering process with incident wave from left leads attached to a PT dimer with additional overall loss or gain $\Gamma$. (b)-(d) are reflection probabilities for the right scattering states as a function of ($E$, $\Gamma$) for $\gamma=\{0.9, 1.0,\text{ and }1.1\}$. (e)-(g) are transmission probabilities. (h)-(n) are the case of the right scattering process with incident wave from right leads. In case of opposite incident waves, reflection probabilities are different but transmission probabilities are the same. There is an EP when $\gamma = 1$.