Quantum Liouvillian exceptional and diabolical points for bosonic fields with quadratic Hamiltonians: The Heisenberg-Langevin equation approach

Jan Perina Jr1, Adam Miranowicz2, Grzegorz Chimczak2, and Anna Kowalewska-Kudlaszyk2

1Joint Laboratory of Optics of Palacký University and Institute of Physics of CAS, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic
2Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland

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Equivalent approaches to determine eigenfrequencies of the Liouvillians of open quantum systems are discussed using the solution of the Heisenberg-Langevin equations and the corresponding equations for operator moments. A simple damped two-level atom is analyzed to demonstrate the equivalence of both approaches. The suggested method is used to reveal the structure as well as eigenfrequencies of the dynamics matrices of the corresponding equations of motion and their degeneracies for interacting bosonic modes described by general quadratic Hamiltonians. Quantum Liouvillian exceptional and diabolical points and their degeneracies are explicitly discussed for the case of two modes. Quantum hybrid diabolical exceptional points (inherited, genuine, and induced) and hidden exceptional points, which are not recognized directly in amplitude spectra, are observed. The presented approach via the Heisenberg-Langevin equations paves the general way to a detailed analysis of quantum exceptional and diabolical points in infinitely dimensional open quantum systems.

Recently, a considerable interest in studying non-Hermitian systems has been focused on their exceptional points (EPs), which occur, e.g., at the phase transitions between the PT and non-PT regimes. Studies on EPs are usually limited to Hamiltonian EPs, which correspond to the degeneracies of the eigenvalues of non-Hermitian Hamiltonians associated with their coalescent eigenmodes (eigenvectors). Note that these EPs are semiclassical, because they are not affected by quantum jumps. Recently, quantum EPs (QEPs) have been defined as the degeneracies of the eigenvalues corresponding to coalescent eigenmatrices (eigenoperators) of the quantum Liouvillian superoperator for a Lindblad master equation. Unfortunately, the standard approach of finding QEPs via the eigenvalue problem of Liouvillians becomes quite inefficient for multi-qubit or multi-level quantum systems. For systems with infinitely-dimensional Hilbert spaces, the determination of EPs and QEPs is even more challenging. Here, we develop an efficient method based on the Heisenberg-Langevin equations for finding QEPs, and we show the equivalence of QEPs found by these two approaches.

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