Trace-class spectra of irreducible Gaussian quantum Markov semigroups

Abstract

We determine the spectrum of the predual generator of an irreducible Gaussian quantum Markov semigroup on trace-class operators, the natural space for density operators. The drift matrix governs the linear part of the Gaussian evolution and organizes the spectral classification. When the linear flow generated by this matrix is not periodic, the spectrum is the closed left half-plane. If all drift eigenvalues have negative real part, every interior point is an eigenvalue. There are no trace-class eigenvectors if some drift eigenvalue has positive real part or all have zero real part. For periodic flow, the spectrum consists of unions of horizontal half-lines or filled parabolic regions. Examples show that irreducibility cannot in general be omitted. We apply the results to oscillators, free particles, and boson chains.

arXiv comment69 pages. Revised and expanded version, including a complete trace-class spectral classification, new physical applications, and a revised discussion of previous results