Nonlinear Optical Response in Pseudo-Hermitian Systems at Steady State

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.