Non-Markovianity and memory effects in quantum open systems

Although a number of measures for quantum non-Markovianity have been proposed recently, it is still an open question whether these measures directly characterize the memory effect of the environment, i.e., the dependence of a quantum state on its past in a time evolution. In this paper, we present a criterion and propose a measure for non-Markovianity with clear physical interpretations of the memory effect. The non-Markovianity is defined by the inequality $T(t_2,t_0)\ne T(t_2,t_1)T(t_1,t_0)$ in terms of memoryless dynamical map $T$ introduced in this paper. This definition is conceptually distinct from that based on divisibility used by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)], whose violation is manifested by noncomplete positivity of the dynamical map. We demonstrate via a typical quantum process that without Markovian approximation, nonzero memory effects (non-Markovianity) always exist even if the non-Markovianity is zero by the other non-Markovianity measures.

Figure 1

Schematic illustration of the non-Markovian inequality Eq. (20).

Figure 2

Evolutions of ${\rho }_{ee}$ with different starting time. The black solid line corresponds to the first evolution starting at 0, and the blue dashed line corresponds to the second evolution starting at $t_1$. In the evolutions, the system has the same state at $t_1$, however they became different at $t_2$, which is a manifestation of the memory effect. The phenomenon never happens in a (time-dependent) Markovian process.

Figure 3

Non-Markovianity as a function of $logR$ corresponding to the Lorentzian spectral density. It is shown that $N_M$ is always nonzero for $R>0$ by our measurement.

Figure 4

Non-Markovianity as a function of $log\alpha$ corresponding to the Ohmic spectral density with ${\omega }_0/{\omega }_c=1$. Similar to the result in Fig. 3, $N_M$ is always nonzero for $\alpha >0$.