Biorthogonal scattering and generalized unitarity in non-Hermitian systems

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.

💡 Research Summary

This paper investigates two‑port quantum scattering in non‑Hermitian dimer systems coupled to external leads. The authors focus on (i) a parity‑time (PT)‑symmetric dimer with balanced gain and loss and (ii) a non‑reciprocal dimer featuring asymmetric hopping. By explicitly constructing left‑ and right‑scattering eigenstates of the non‑Hermitian Hamiltonian, they build two scattering matrices, (S_R) (right‑handed) and (S_L) (left‑handed). In Hermitian systems the two matrices coincide and satisfy the usual unitarity condition (S^\dagger S = I). In contrast, for the non‑Hermitian dimers the right‑handed matrix alone violates unitarity: (|r_R|^2 + |t_R|^2 \neq 1). The key theoretical advance is the demonstration that the bi‑orthogonal combination obeys a generalized unitarity relation, \