Nakajima-Zwanzig versus time-convolutionless master equation for the non-Markovian dynamics of a two-level system

摘要

We consider the exact reduced dynamics of a two-level system coupled to abosonic reservoir, further obtaining the exact time-convolutionless andNakajima-Zwanzig non-Markovian equations of motion. The considered systemincludes the damped and undamped Jaynes-Cummings model. The result is obtainedby exploiting an expression of quantum maps in terms of matrices and a simplerelation between the time evolution map and time-convolutionless generator aswell as Nakajima-Zwanzig memory kernel. This non-perturbative treatment showsthat each operator contribution in Lindblad form appearing in the exacttime-convolutionless master equation is multiplied by a different timedependent function. Similarly, in the Nakajima-Zwanzig master equation eachsuch contribution is convoluted with a different memory kernel. It appears thatdepending on the state of the environment the operator structures of the twoset of equations of motion can exhibit important differences.