Schrieffer-Wolff transformation for non-Hermitian systems: application for $\mathcal{PT}$-symmetric circuit QED

TL;DR

This paper develops a generalized Schrieffer-Wolff transformation for non-Hermitian systems and applies it to analyze phase transitions, Exceptional Points, and state mixing in a $ ext{PT}$-symmetric circuit QED setup.

Contribution

It introduces a new analytical method for non-Hermitian systems and demonstrates its application to $ ext{PT}$-symmetric circuit QED, revealing phase diagrams and state behaviors.

Findings

Identification of $ ext{PT}$-symmetry broken and unbroken phases

Formation of second and third order Exceptional Points

Non-Hermiticity causes mixing of dark and bright states

Abstract

Combining non-hermiticity and interactions yields novel effects in open quantum many-body systems. Here, we develop the generalized Schrieffer-Wolff transformation and derive the effective Hamiltonian suitable for various quasi-degenerate \textit{non-Hermitian} systems. We apply our results to an exemplary $\mathcal{PT}$--symmetric circuit QED composed of two non-Hermitian qubits embedded in a lossless resonator. We consider a resonant quantum circuit as $|\omega_r-\Omega| \ll \omega_r$, where $\Omega$ and $\omega_r$ are qubits and resonator frequencies, respectively, providing well-defined groups of quasi-degenerate resonant states. For such a system, using direct numerical diagonalization we obtain the dependence of the low-lying eigenspectrum on the interaction strength between a single qubit and the resonator, $g$, and the gain (loss) parameter $\gamma$, and compare that with the eigenvalues obtained analytically using the effective Hamiltonian of resonant states. We identify $\mathcal{PT}$--symmetry broken and unbroken phases, trace the formation of Exceptional Points of the second and the third order, and provide a complete phase diagram $g-\gamma$ of low-lying resonant states. We relate the formation of Exceptional Points to the additional $\mathcal{P}$-pseudo-Hermitian symmetry of the system and show that non-hermiticity mixes the "dark" and the "bright" states, which has a direct experimental consequence.