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

TL;DR

This paper introduces a Heisenberg-Langevin equation approach to analyze eigenfrequencies and degeneracies, including exceptional and diabolical points, in the Liouvillian spectra of open quantum systems with quadratic bosonic Hamiltonians.

Contribution

It presents a unified method to identify and analyze quantum Liouvillian exceptional and diabolical points in systems with quadratic Hamiltonians, including hidden and hybrid types.

Findings

Identified quantum Liouvillian exceptional and diabolical points for two bosonic modes.

Revealed hidden and hybrid exceptional points not visible in amplitude spectra.

Demonstrated the equivalence of Heisenberg-Langevin and operator moment approaches.

Abstract

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.