Open quantum systems integrable by partial commutativity

Abstract

The article provides a framework to solve linear differential equations based on partial commutativity which is introduced by means of the Fedorov theorem. The framework is applied to specific types of three-level and four-level quantum systems. The efficiency of the method is evaluated and discussed. The Fedorov theorem appears to answer the need for methods which allow to study dynamical maps corresponding with time-dependent generators. By applying this method, one can investigate countless examples of dissipative systems such that the relaxation rates depend on time.

Entanglement dynamics governed by time-dependent quantum generators

“…In other words, the model describes a relaxation towards the ground state |1 . Let us assume that this process is governed by a time-local generator [10]:…”

“…The dynamics governed by Eq. 12 has a closed-form solution for such initial states that have zero probability of occupying the highest energy level [10]. Thus, the condition of partial commutativity allows one to precisely describe the evolution of two-level systems immersed within the 3−dimensional Hilbert space.…”

“…It can be demonstrated that the closed-form solution of a master equation with the generator Eq. 29 is legitimate for such initial states that does not involve the highest energy level [10].Therefore, the framework of partial commutativity allows one to study the dynamics of genuine qutrit states that are spanned by the vectors {|1 , |2 , |3 }.…”

“…To solve master equations with generators Eq. 3 we implement the condition of partial commutativity [9,10]. This method can be considered a generalization of functional and integral commutativity [11][12][13].…”

“…The proof of the Fedorov theorem can be found in Ref. [10]. The major limitation of the Fedorov theorem concerns the fact that the closed-form solution is admissible only for vectors α that satisfy the condition Eq.…”