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

Barthel, Thomas; Zhang, Yikang

doi:10.1088/1742-5468/ac8e5c

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

Journal Article · Thu Nov 24 00:00:00 EST 2022 · Journal of Statistical Mechanics

DOI:https://doi.org/10.1088/1742-5468/ac8e5c· OSTI ID:2280985

Barthel, Thomas ^[1]; Zhang, Yikang ^[2]

Duke Univ., Durham, NC (United States); Duke University
Duke Univ., Durham, NC (United States)

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), here 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.

View Accepted Manuscript (DOE)

Research Organization:: Duke Univ., Durham, NC (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES)

Grant/Contract Number:: SC0019449

OSTI ID:: 2280985

Journal Information:: Journal of Statistical Mechanics, Journal Name: Journal of Statistical Mechanics Journal Issue: 11 Vol. 2022; ISSN 1742-5468

Publisher:: IOP PublishingCopyright Statement

Country of Publication:: United States

Language:: English

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