Fabrizio Minganti

Theoretical Quantum Physics Laboratory,
RIKEN
Japan

Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians

Exceptional points (EPs) correspond to degeneracies of open systems attracting much interest in optics, optoelectronics, plasmonics, and condensed matter physics. In the classical and semiclassical approaches, Hamiltonian EPs (HEPs) are usually defined as degeneracies of non-Hermitian Hamiltonians, such that at least two eigenfrequencies are identical and the corresponding eigenstates coalesce. HEPs result from continuous, mostly slow, non-unitary evolution without quantum jumps. Clearly, quantum jumps should be included in a fully quantum approach to make it equivalent to, e.g., the Lindblad master-equation approach. Thus, we suggest to define EPs via degeneracies of a Liouvillian superoperator (including the full Lindbladian term: LEPs), and we clarify the relations between HEPs and LEPs. We prove two main Theorems about differences and analogies between HEPs and LEPs [2]. We explore the consequences of these theorems in several different examples. In particular, we apply both formalisms, based on an NHH and a Liouvillian within the Scully-Lamb laser theory, to determine and compare the corresponding HEPs and LEPs in the semiclassical and quantum regimes [3].

[1] Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, "Parity-time symmetry and exceptional points in photonics", Nat. Mater. 18, 783 (2019).>
[2] F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, “Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps,” (2019), arxiv:1909.11619.
[3] I. I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, "Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory," (2019), arXiv:1909.12276.