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

• URL: http://arxiv.org/abs/2309.09829v1
• Date: Mon, 18 Sep 2023 14:50:29 GMT
• Title: Schrieffer-Wolff transformation for non-Hermitian systems: application for $\mathcal{PT}$-symmetric circuit QED
• Authors: Grigory A. Starkov, Mikhail V. Fistul, Ilya M. Eremin
• Abstract summary: We develop the generalized Schrieffer-Wolff transformation and derive the effective Hamiltonian suitable for various quasi-degenerate textitnon-Hermitian systems. We show that non-hermiticity mixes the "dark" and the "bright" states, which has a direct experimental consequence.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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