Open quasi-free systems

I. Vakulchyk, Mikhail Ivanchenko, Sergej Flach A +1 more

Thomas Barthel, Yikang Zhang

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.

Esteban A. Calzetta, B. L. Hu

H. J. Carmichael

Hiroshi Tamura, Valentin Anatol'evich Zagrebnov

We consider a dynamical semigroup for unbounded Kossakowski-Lindblad-Davies generator corresponding to evolution of an open system for a tuned repeated harmonic perturbation. For this evolution, we prove the existence of uniquely determined minimal trace-preserving strongly continuous dynamical semigroups on the space of states. The corresponding dual W∗-dynamical system is shown to be unital quasi-free and completely positive automorphisms of the canonical commutation relation-algebra.

L. A. Lugiato