Closed dynamical recursion equations for correlation functions and the application on the construction of Liouvillian spectrum in Lindbladian systems

TL;DR

This paper develops a recursive method to compute correlation functions and construct the Liouvillian spectrum for open quantum systems described by Lindblad equations, extending previous approaches to more complex models.

Contribution

It derives closed-form dynamical recursion equations for all even-order correlations in quadratic Lindbladian systems, enabling Liouvillian spectrum reconstruction.

Findings

Derived recursion equations for even-order correlations

Established conditions for closure of second-order correlations

Proposed a method to construct Liouvillian spectrum in extended models

Abstract

For an open quantum system described by the Lindblad equation, full characterization of its dynamics typically needs the knowledge of the Liouvillian spectrum and correlation functions. Solving the Liouvillian spectrum and correlation functions are usually formidable tasks, and most previous studies are constrained to simple models and lower-order correlations. In this work, we derive a closed form of dynamical recursion equations for all even-order correlation functions associated with the quadratic Liouvillian superoperator, thus extending the Wick's theorem and enabling the reconstruction of the corresponding Liouvillian spectrum. Furthermore, we study the quartic Liouvillian superoperator, establishing the necessary and sufficient conditions for the closure of second-order correlation functions. Building on this, the dynamical expressions for even-order correlation functions under quadratic dissipation are derived, culminating in a method for constructing the Liouvillian spectrum in this extended framework.