Nonlinear Optical Response in Pseudo-Hermitian Systems at Steady State

Abstract

We establish a steady-state theory for nonlinear optical conductivity in pseudo-Hermitian systems. We derive compact formulas for the first and second order conductivity tensors in both the velocity and length gauges and prove their exact equivalence through generalized sum rules and Berry connection identities by formulating the nonlinear response in terms of a biorthogonal density matrix. Utilizing the formalism on parity-time symmetric two-level systems reveals nonlinear phenomena that are not present in Hermitian systems, such as extra terms in the conductivity, corrections to the velocity operator, photocurrent, and resonance structures with higher-order poles at one-photon transitions. These features yield qualitatively distinct harmonic generation responses like real second-order conductivities and nonzero DC limits. These results provide new insights into nonlinear light-matter interactions in active media characterized by balanced gain and loss, with implications for non-Hermitian photonics, dissipative topological systems, and quantum devices designed with engineered dissipation.

Figures (2)

• Figure 1: (Color online) (a) FOOC and (b),(c) SOOC of non-Hermitian qubit Hamiltonian (\ref{['hqubit']}) with parameters $\gamma=0.7$ (blue) and $\gamma=0.9$ (orange). Solid (dashed) lines depict the real (imaginary) parts. (d) SOOC as a function of $\gamma$ at one-photon resonance. According to (b),(d), as $\gamma$ increases toward the exceptional point ($\gamma = 1$), the SOOC resonance becomes sharper and more asymmetric, reflecting the approach to spectral degeneracy. The imaginary part undergoes the usual sign change at resonance, reminiscent of Hermitian absorption lines but modified by the non-Hermitian contributions to the velocity operator. Crucially, in (b) and (c), the one-photon resonance exhibits a second-order pole rather than the simple pole of Hermitian systems, producing sharp asymmetric line shapes and steep frequency dependence. The photocurrent in (c) remains finite, unlike the Hermitian qubit case, where it vanishes. This reflects the modified analytic structure of the conductivity tensor due to biorthogonal overlaps between left and right eigenstates. The second-harmonic generation is also real in the non-dissipative limit, in stark contrast with the purely imaginary Hermitian counterpart, implying a phase shift between fundamental and second-harmonic fields rather than absorption. This provides a clear experimental signature of pseudo-Hermiticity in NL optical measurements.
• Figure 2: (Color online) Panels (a) and (b) display the linear response: both real and imaginary parts show well-defined resonances at interband transition frequencies, with perfect agreement between gauges, confirming the gauge consistency of the formalism for non-Hermitian qubit Hamiltonian (\ref{['hqubit']}) with parameters $t=0.2, \gamma=0.7$. No low-frequency divergence occurs, as expected for a gapped, fully filled band structure. Panels (c) and (d) reveal the second-order response. Compared to the linear case, the number of resonant features doubles, corresponding to both one-photon and two-photon processes. Singularities appear, stemming from interband–interband contributions that have no Hermitian analog. Notably, a low-frequency divergence emerges in the SOOC due to these terms, which reflects the absence of conventional cancellation mechanisms in pseudo-Hermitian systems. This divergence is a direct illustration of the modified sum rules and could be probed in waveguide experiments via low-frequency harmonic generation. The excellent agreement between length and velocity gauge results across all frequencies demonstrates the robustness of the gauge-invariant formalism developed here.