Biorthogonal scattering and generalized unitarity in non-Hermitian systems

Technical Summary: Biorthogonal Scattering and Generalized Unitarity in Non-Hermitian Systems

Problem Statement
The paper investigates the two-port scattering process in non-Hermitian dimer models, specifically addressing the breakdown of standard unitarity in quantum transport. In Hermitian systems, scattering matrices are unitary ($S^\dagger S = I$), ensuring the conservation of probability flux. However, non-Hermitian systems, characterized by complex eigenvalues and non-orthogonal eigenstates, exhibit non-unitary dynamics, leading to phenomena such as gain, loss, and transient amplification. A central challenge is determining how to consistently define scattering coefficients and probability conservation in these systems. The authors note that while physical observables can be formulated in either the right-right (RR) basis (using only right eigenstates) or the left-right (LR) basis (using biorthogonal pairs), these formulations do not necessarily yield equivalent physics. Specifically, it remains an open question how to reconcile the non-conservation of flux in the RR basis with the mathematical requirements of scattering theory in non-Hermitian contexts.

Methodology
The authors analyze two exemplary dimer models coupled to external Hermitian semi-infinite leads:

1. Parity-Time (PT) Symmetric Dimer: A model preserving PT symmetry via spatial gain-loss balance ($\epsilon_1 = -\epsilon_2 = i\gamma$) with symmetric hopping ($d_l = d_r = d$).
2. Non-Reciprocal (NR) Dimer: A model exhibiting directional hopping ($d_l \neq d_r$) with real onsite energies, leading to non-reciprocity.

The study employs a scattering matrix formalism constructed from the biorthogonality of the Hamiltonian's left ($\langle \phi_j|$) and right ($|\psi_j\rangle$) scattering states. The total Hamiltonian includes the system ($H_s$), the leads ($H_l$), and the coupling ($H_c$). The authors derive scattering coefficients ($r$ and $t$) for both right scattering states (solving $H|\psi\rangle = E|\psi\rangle$) and left scattering states (solving $\langle \phi|H = E\langle \phi|$).

Key analytical steps include:

• Constructing scattering matrices $S_R$ (right) and $S_L$ (left) as linear maps between incoming and outgoing amplitudes in the asymptotic leads.
• Calculating reflection and transmission probabilities in the RR basis ($R = |r_R|^2, T = |t_R|^2$) and the LR biorthogonal basis ($R_{bi} = r_L^* r_R, T_{bi} = t_L^* t_R$).
• Analyzing the pole and zero structures of the scattering coefficients to understand resonant behaviors and exceptional points (EPs).
• Investigating the "net flux imbalance" ($A = 1 - (R+T)$) to quantify absorption or emission in the measured channels.

Key Contributions and Results

1. Breakdown of Unitarity in the RR Basis:
   The analysis confirms that in non-Hermitian systems, the scattering process derived solely from right scattering states does not adhere to standard unitarity ($S_R^\dagger S_R \neq I$). Consequently, the sum of reflection and transmission probabilities ($R_R + T_R$) can deviate from unity.

   • In the PT-symmetric dimer, $R_R + T_R$ can exceed 1 (indicating net gain/amplification) or be less than 1 (net loss), depending on the gain-loss parameter $\gamma$ and incident energy.
   • In the NR dimer, the enhancement of transmission probabilities is observed even when eigenvalues are real, indicating that non-orthogonality alone can drive non-unitary transport.

2. Restoration of Generalized Unitarity via Biorthogonality:
   The paper demonstrates that the generalized unitarity condition $S_L^\dagger S_R = I$ holds for these systems. When reflection and transmission coefficients are combined using the biorthogonal formulation ($R_{bi} + T_{bi} = 1$), probability conservation is mathematically restored. This establishes that the "loss" of probability in the RR basis is compensated by the non-orthogonality of the eigenstates, which is captured in the LR framework.

3. Distinct Physical Origins of Enhanced Transport:
   The authors identify two independent mechanisms that cause reflection and transmission probabilities to exceed 1 (enhanced transport):

   • Complex Eigenvalues: In the PT-symmetric model, enhancement arises from the imaginary parts of the eigenenergies (gain/loss mechanisms).
   • Non-Orthogonality of Eigenstates: In the NR dimer (which has real eigenvalues), enhancement arises purely from the non-orthogonality of the eigenstates, leading to transient amplification effects.

4. Pole-Zero Structure and Exceptional Points:
   The study explicitly maps the poles and zeros of the scattering coefficients. In the PT dimer, poles and zeros form distinct quasi-elliptic trajectories in the complex energy plane, separating resonant and anti-resonant responses. In the NR dimer, the pole and zero loci are nearly identical in the real-energy limit, yet the non-orthogonality drives the transmission enhancement. The presence of Exceptional Points (EPs) is identified as the critical point where eigenvalues and eigenstates coalesce, marking the transition between different scattering regimes.

5. Lead Coupling Effects:
   The paper notes that the specific geometry of coupling to leads matters. For a vertically arranged PT dimer where the coupling preserves the total PT symmetry, the RR basis probabilities sum to unity ($R_R + T_R = 1$), contrasting with the horizontally arranged model where the coupling breaks the symmetry of the total Hamiltonian, leading to non-unitary RR scattering.

Significance
The paper claims to clarify the precise physical interpretation of choosing between RR and LR bases in non-Hermitian quantum transport. It establishes that while the RR basis provides operationally measurable flux ratios in Hermitian leads (which may show gain or loss), the LR biorthogonal basis provides the mathematically consistent extension of orthogonality required to restore a generalized unitary process.

The work distinguishes between two physical origins of non-unitary scattering: complex eigenvalues (associated with gain/loss) and non-orthogonal eigenstates (associated with non-normal operators). By demonstrating that biorthogonality restores generalized unitarity, the authors provide a unified framework for understanding quantum transport in non-Hermitian systems, offering new insights into how directional amplification and transient gain occur without violating fundamental conservation laws when viewed through the appropriate biorthogonal lens.