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arXiv:1701.06419 · Physical Review A

Coarse graining a non-Markovian collisional model

Nadja K. Bernardes, Andre R. R. Carvalho, Carlos Henrique Monken, Marcelo França Santos

Published Mar 16, 2017• 34 citations
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Abstract

The dynamics of systems subjected to noise is called Markovian in the absence of memory effects, i.e., when its immediate future depends only on its present. Time correlations in the noise source may generate non-Markovian effects that, sometimes, can be erased by appropriately coarse graining the time evolution of the system. In general, the coarse-graining time ${t}_{\text{CG}}$ is taken to be much larger than the correlation time $\ensuremath{\tau}$ but no direct relation between them is established. Here we analytically obtain a relation between ${t}_{\text{CG}}$ and $\ensuremath{\tau}$ for the dynamics of a qubit subjected to a time-correlated environment. Our results can be applied in principle to any distribution of the environmental correlations and can be tested through a collisional model where the qubit sequentially interacts with correlated qutrits.

DOI
10.1103/physreva.95.032117

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The dynamics of systems subjected to noise is called Markovian in the absence of memory effects, i.e., when its immediate future depends only on its present. Time correlations in the noise source may generate non-Markovian effects that, sometimes, can be erased by appropriately coarse graining the time evolution of the system. In general, the coarse-graining time tCG{t}_{\text{CG}}tCG​ is taken to be much larger than the correlation time \ensuremathτ\ensuremath{\tau}\ensuremathτ but no direct relation between them is established. Here we analytically obtain a relation between tCG{t}_{\text{CG}}tCG​ and \ensuremathτ\ensuremath{\tau}\ensuremathτ for the dynamics of a qubit subjected to a time-correlated environment. Our results can be applied in principle to any distribution of the environmental correlations and can be tested through a collisional model where the qubit sequentially interacts with correlated qutrits.
We consider the dynamics of a collisional model in which both the system and environment are embodied by spin-1/21/21/2 particles. In order to include non-Markovian features in our model, we introduce interactions among the environmental qubits and investigate the effect that different models of such interaction have on the degree of non-Markovianity of the system's dynamics. By extending that interaction beyond the nearest neighbor, we enhance the degree of non-Markovianity in the system's dynamics. A further significant increase can be observed if a collective interaction with the forthcoming environmental qubits is considered. However, the observed degree of non-Markovianity in this case is nonmonotonic with the increasing number of qubits included in the interaction. Moreover, one can establish a connection between the degree of non-Markovianity in the evolution of the system and the fading behavior of quantum coherence in its state as the number of collisions grows. We complement our study with an investigation of system-environment correlations and present an example of their importance on a physical upper bound on the trace distance derivative.
This MS thesis explores the effects and origins of a 'noise with memory' in the dynamics of an open quantum system. The system considered here is a multi-qubit register performing the Grover's quantum search algorithm. We show that a Markovian-correlated noise can enhance the efficiency of the algorithm over a time correlation-less noise. We also analytically find the set of necessary and sufficient conditions for the algorithm's success probability to remain invariant with respect to the number of noisy sites in the register and point out that these conditions hold irrespective of the presence of time-correlations in the noise. We then investigate the origins of the type of noise considered. In this regard, a 'collisional model' is constructed that exactly reproduces the noisy evolution of the open system. Non-Markovianity in the system's evolution is then assessed using two well-known measures and they are shown to be non-coincident. Our model is then slightly modified to accommodate an elementary thermal bath. There we show that increasing the bath's temperature increases information drainage from the system.
Accessing the physical mechanisms behind non-Markovian phenomena in open quantum dynamics requires the study of the statistical properties of the joint system-environment dynamics. This is impossible at the level of the reduced dynamics of the open system alone as the latter is obtained by suitably eliminating the environment. The task is instead made possible by considering multi-time correlation functions involving observables of the open system, only: the open system-environment interactions turn them into global ones thus building up correlations between the two systems. Multi-time correlations form the basis of both the theory of quantum stochastic processes and of the Alicki-Lindblad-Fannes dynamical entropy (ALF entropy for short). This latter quantity provides for quantum systems a measure of the dynamical entropy production as the Kolmogorov-Sinai entropy does for classical systems. In the case of a collisional model whereby the dissipative dynamics of a finite-level system is obtained by its coupling to an infinite classical spin chain, the ALF entropy can be explicitly computed. It turns out to depend on the parameters characterizing the statistical properties of the environment and can be related to the activation and super-activation of memory effects in the open quantum system.
This is a derivation of the quantum-optical master equation using coarse-graining of time, which aims at exploring Markovian evolution from a microscopic perspective. Here, coarse-graining refers to finding an approximate evolution operator by integrating over a time step Δ t , and considering the field states to be approximated by a similarly coarse-grained basis. By using these ideas and keeping the field states coarse-grained throughout the derivation, I explore the past-future independence of the quantum white noise formalism explicitly. Namely, that a quantum-optical system interacting with a vacuum field may be viewed as sequentially interacting with a series of initially uncorrelated and temporally coarse-grained vacuum states—this process leads to correlations only between the system and the time bins which have interacted with the system in the past, along with a Markovian evolution of the reduced system state. The derivation here may also be viewed in terms of a type of quantum-mechanical path integral.
We investigate the smallest set of requirements for inducing non-Markovian dynamics in a collisional model of open quantum systems. This is done by introducing correlations in the state of the environment and analyzing the divisibility of the quantum maps from consecutive time steps. Our model and results serve as a platform for the microscopic study of non-Markovian behavior as well as an example of a simple scenario of non-Markovianity with purely contractive maps, i.e., with no backflow of information between system and environment.
Environmental noise is a major source of decoherence and errors in quantum systems. In a non-Markovian environment, the noise has a finite correlation time, and the system can retain information about its previous interactions with the environment. Here, we investigate how such environmental memory can be observed through a finite-duration coherent quantum operation. Coherent control dynamically reshapes the stochastic coupling operator and thereby changes how two-time noise correlations affect the quantum process. For Ornstein-Uhlenbeck noise, we show analytically that two different limits reduce the information on the correlation time. In the short-memory limit, the leading dynamics depend on the integrated-noise combination σ/τc, where σ is the noise variance and τc is the correlation time, making these parameters locally degenerate. In the long-memory limit, the stochastic field becomes quasi-static during the operation, and sensitivity to the correlation time is suppressed. Exact stochastic simulations for a driven single-qubit Xπ rotation and a two-qubit exchange gate show that independent correlation-time information becomes largest between these limits, when the environmental and control timescales are comparable. Noncommuting controls enhance this information, whereas commuting controls do not. Simulations at different gate durations tg further show that the information window approximately follows the dimensionless correlation ratio τc/tg. These results provide a timescale-matching principle for probing finite-correlation non-Markovian noise with coherent quantum control.