Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems

TL;DR

This paper develops a comprehensive method to solve and analyze the dynamics of quasi-free and quadratic open quantum systems, including fermionic and bosonic types, using covariance matrices and third quantization.

Contribution

It extends previous work by treating fermionic and bosonic systems uniformly, completing derivations, and addressing non-diagonalizable Liouvillians for quadratic open systems.

Findings

Derived equations of motion for covariance matrices in quasi-free systems

Provided criteria for bosonic steady states and addressed odd-parity fermionic sectors

Discussed the structure of Liouvillian spectra and critical phenomena

Abstract

The dynamics of Markovian open quantum systems are described by Lindblad master equations. For fermionic and bosonic systems that are quasi-free, i.e., with Hamiltonians that are quadratic in the ladder operators and Lindblad operators that are linear in the ladder operators, we derive the equation of motion for the covariance matrix. This determines the evolution of Gaussian initial states and the steady states, which are also Gaussian. Using ladder super-operators (a.k.a. third quantization), we show how the Liouvillian can be transformed to a many-body Jordan normal form which also reveals the full many-body spectrum. Extending previous work by Prosen and Seligman, we treat fermionic and bosonic systems on equal footing with Majorana operators, shorten and complete some derivations, also address the odd-parity sector for fermions, give a criterion for the existence of bosonic steady states, cover non-diagonalizable Liouvillians also for bosons, and include quadratic systems. In extension of the quasi-free open systems, quadratic open systems comprise additional Hermitian Lindblad operators that are quadratic in the ladder operators. While Gaussian states may then evolve into non-Gaussian states, the Liouvillian can still be transformed to a useful block-triangular form, and the equations of motion for $k$-point Green's functions form a closed hierarchy. Based on this formalism, results on criticality and dissipative phase transitions in such models are discussed in a companion paper [arXiv:2204.05346].