Markovian embedding of non-Markovian quantum collisional models

A wide class of non-Markovian completely positive master equations can be formulated on the basis of quantum collisional models. In this phenomenological approach the dynamics of an open quantum system is modeled through an ensemble of stochastic realizations that consist in the application at random times of a (collisional) completely positive transformation over the system state. In this paper, we demonstrate that these kinds of models can be embedded in bipartite Markovian Lindblad dynamics consisting of the system of interest and an auxiliary one. In contrast with phenomenological formulations, here the stochastic ensemble dynamics an the inter-event time interval statistics are obtained from a quantum measurement theory after assuming that the auxiliary system is continuously monitored in time. Models where the system inter-collisional dynamics is non-Markovian [B. Vacchini, Phys. Rev. A 87, 030101(R) (2013)] are also obtained from the present approach. The formalism is exemplified through bipartite dynamics that leads to non-Markovian system effects such as an environment-to-system back flow of information.Comment: 12 pages, 2 figure

Post-Markovian quantum master equations from classical environment fluctuations

In this paper we demonstrate that two commonly used phenomenological post-Markovian quantum master equations can be derived without using any perturbative approximation. A system coupled to an environment characterized by self-classical configurational fluctuations, the latter obeying a Markovian dynamics, defines the underlying physical model. Both Shabani-Lidar equation [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(R) (2005)] and its associated approximated integro-differential kernel master equation are obtained by tracing out two different bipartite Markovian Lindblad dynamics where the environment fluctuations are taken into account by an ancilla system. Furthermore, conditions under which the non-Markovian system dynamics can be unravelled in terms of an ensemble of measurement trajectories are found. In addition, a non-Markovian quantum jump approach is formulated. Contrary to recent analysis [L. Mazzola, E. M. Laine, H. P. Breuer, S. Maniscalco, and J. Piilo, Phys. Rev. A 81, 062120 (2010)], we also demonstrate that these master equations, even with exponential memory functions, may lead to non-Markovian effects such as an environment-to-system backflow of information if the Hamiltonian system does not commutate with the dissipative dynamics.Comment: 13 pages, 4 figure

Non-Markovian quantum jumps from measurements in bipartite Markovian dynamics

The quantum jump approach allows to characterize the stochastic dynamics associated to an open quantum system submitted to a continuous measurement action. In this paper we show that this formalism can consistently be extended to non-Markovian system dynamics. The results rely in studying a measurement process performed on a bipartite arrangement characterized by a Markovian Lindblad evolution. Both a renewal and non-renewal extensions are found. The general structure of non-local master equations that admit an unravelling in terms of the corresponding non-Markovian trajectories are also found. Studying a two-level system dynamics, it is demonstrated that non-Markovian effects such as an environment-to-system flow of information may be present in the ensemble dynamics.Comment: 13 pages, 3 figure

Extended q-Gaussian and q-exponential distributions from Gamma random variables

The family of q-Gaussian and q-exponential probability densities fit the statistical behavior of diverse complex self-similar non-equilibrium systems. These distributions, independently of the underlying dynamics, can rigorously be obtained by maximizing Tsallis "non-extensive" entropy under appropriate constraints, as well as from superstatistical models. In this paper we provide an alternative and complementary scheme for deriving these objects. We show that q-Gaussian and q-exponential random variables can always be expressed as function of two statistically independent Gamma random variables with the same scale parameter. Their shape index determine the complexity q-parameter. This result also allows to define an extended family of asymmetric q-Gaussian and modified $q$-exponential densities, which reduce to the previous ones when the shape parameters are the same. Furthermore, we demonstrate that simple change of variables always allow to relate any of these distributions with a Beta stochastic variable. The extended distributions are applied in the statistical description of different complex dynamics such as log-return signals in financial markets and motion of point defects in fluid flows.Comment: 11 pages, 6 figure

Central limit theorem for a class of globally correlated random variables

The standard central limit theorem with a Gaussian attractor for the sum of independent random variables may lose its validity in presence of strong correlations between the added random contributions. Here, we study this problem for similar interchangeable globally correlated random variables. Under these conditions, a hierarchical set of equations is derived for the conditional transition probabilities. This result allows us to define different classes of memory mechanisms that depend on a symmetric way on all involved variables. Depending on the correlation mechanisms and single statistics, the corresponding sums are characterized by distinct statistical probability densities. For a class of urn models it is also possible to characterize their domain of attraction which, as in the standard case, is parametrized by the probability density of each random variable. Symmetric and asymmetric $q$-Gaussian attractors $(q<1)$ are a particular case of these models.Comment: 11 pages, 7 figures, Appendixes 5 page

Weak ergodicity breaking induced by global memory effects

We study the phenomenon of weak ergodicity breaking for a class of globally correlated random walk dynamics defined over a finite set of states. The persistence in a given state or the transition to another one depends on the whole previous temporal history of the system. A set of waiting time distributions, associated to each state, set the random times between consecutive steps. Their mean value is finite for all states. The probability density of time-averaged observables is obtained for different memory mechanisms. This statistical object explicitly shows departures between time and ensemble averages. While the mean residence time in each state may result divergent, we demonstrate that this condition is in general not necessary for breaking ergodicity. Hence, global memory effects are an alternative mechanism able to induce this property. Analytical and numerical calculations support these results.Comment: 11 pages, 3 figure