Perturbation theory of Lindbladians with multiple steady states

Lindbladians admitting multiple steady states are dissipative analogues of Hamiltonian systems with multiple ground states. As demonstrated with trapped-ion and circuit-QED quantum engineering, such systems can be used to store, protect, and process quantum information. We develop a perturbation expansion of a general Lindbladian whose unperturbed part can have multiple steady states. After making a few minor assumptions and dividing the expansion into terms that keep the initial state within the steady-state subspace and terms that take the state out, the expansion is solved exactly [1] and admits a diagrammatic interpretation. The number of diagrams in the expansion that are the same order in the perturbation is shown to be a Catalan number. With appropriate additional assumptions, the formalism reduces to open-system perturbation theory for a unique steady state [2], the effective operator formalism [3], and quantum Zeno dynamics [4].

[1] V. V. Albert, Ph.D. thesis, Yale University (2017).
[2] A. C. Y. Li, F. Petruccione, and J. Koch, Sci. Rep. 4, 4887 (2014).
[3] F. Reiter and A. S. Sorensen, Phys. Rev. A 85, 032111 (2012).
[4] P. Zanardi and L. Campos Venuti, Phys. Rev. Lett. 113, 240406 (2014)