Master equations with and without time convolution

Toshihico Arimitsu, Tetsuo Tominaga

A laser master equation is derived microscopically by using the time-convolution-less formulation of the damping theory up to the fourth-order terms with respect to the coupling constant between the laser field and the atoms, in order to investigate the quantum effect and the system size (number of the active atoms) dependence of the kinetic operator . The master equation is compared with the equation given by the time-convolution formalism which is essentially equal to the master equation derived by Haake.

Michael G Cottam, Zahra Haghshenasfard

E. B. Davies

Sergey N. Filippov, Dariusz Chruściński

Convolutionless and convolution master equations are the two mostly used physical descriptions of open quantum systems dynamics. We subject these equations to time deformations: local dilations and contractions of time scale. We prove that the convolutionless equation remains legitimate under any time deformation (results in a completely positive dynamical map) if and only if the original dynamics is completely positive divisible. Similarly, for a specific class of convolution master equations we show that uniform time dilations preserve positivity of the deformed map if the original map is positive divisible. These results allow witnessing different types of non-Markovian behavior: the absence of complete positivity for a deformed convolutionless master equation clearly indicates that the original dynamics is at least weakly non-Markovian; the absence of positivity for a class of time-dilated convolution master equations is a witness of essentially non-Markovian original dynamics.

Umberto Lombardo, G. Giuliani, Yifei Niu

Mizuhiko Saeki

Time-convolutionless forms of the quantal master equations for driven open system are derived from the Liouville equation for the total system under an arbitrary initial condition for the two cases of weak external driving fields and of arbitrary ones. They have forms convenient for the perturbational expansions, and are respectively compared with the time-convolution forms of equations in the lowest Born approximation for the interaction H SB of the system with its surroundings. It is shown that the time-convolutionless forms of equations prevail over the time-convolution ones, and that the conventional Markoffian approximation for the latter in the narrowing limit leads to the incorrect conclusion. The time-convolutionless form of master equation is expanded up to n -th order in powers of the external driving fields in the lowest Born approximation for H SB .