• complex networks
• correlated dephasing
• noisy quantum walks
• open quantum system
• optimal quantum transport

During my master's thesis, entitled ''Spatially structured noisy environments for optimal quantum transport'', I have numerically analyzed how to exploit the environmental noise to control and drive quantum systems. I have shown how particular models of noisy environments affect the quantum dynamics of a single-particle moving over a complex network and how well-defined tasks, as optimal quantum transport or topological control, can be reached by manipulating the external source of noise.
In the first part of work, from the first to the fourth chapter, I introduced the fundamental tools to describe noisy quantum walks on general networks in the context of quantum transport phenomena. Specifically I presented the formalism of the so-called open quantum systems, describing in detail the Kossakowsky-Lindblad master equation and the Quantum Stochastic Walk model (QSW). I showed how the noise is encoded in these formalisms, which combine a fully quantum coherent dynamics with an incoherent hopping dynamics. Then I introduced elementary notions of networks theory and presented the topologies of networks, which I used during my work. Once established how to represent noisy quantum walks on networks I turned to describe how to model irreversible absorption processes to define the key concept of the efficiency of energy transport (EET).
Inspired by the beneficial role of noise for efficient photosynthetic energy transport in biological systems, I focused on the topic of the maximization of the efficiency of energy transport. I have shown, by using networks of different topologies, the well-known phenomenon of noise assisted transport (NAT), and I presented the existence of a costant noise parameter of QSW which maximizes the EET indipendently from the precise underlying topology. Then, I studied how to further optimize the temporal and spatial dependence of noise, in order to take advantage, as much as possible, of the environmental decoherence that affects the quantum walker.
In particular, I found the following main findings. Firstly I have shown that the global amount of noise, absorbed by the whole quantum dynamics, is a quite preserved quantity, regardless to the network's topology. Subsequently, after a detailed study on the relation between quantum transport and delocalization properties, by means of quantum walker entropy, I highlighted that a similar behaviour holds for all observed topologies. Then I introduced a measure of inhomogeneity of the populations distribution and I found an unexpected correlation between this quantity and the efficiency of energy transport. Specifically I have shown that the same noise, which maximizes the transport efficiency, also minimizes this new quantity encoding the average value of the inhomogeneity of the quantum walker distribution.
Therefore, I focused on this correlation and I introduced another quantity, namely the trace distance between the system's state and the maximally mixed state. I took evidence how the optimal noise forces the quantum walker in order to fastly reach a coherent and homogeneous delocalization on the whole network. This kind of delocalization is valid over time scales of the same order of those ones characterizing the transfer phenomenon. Indeed, we concluded that there exists a universal optimal quantum transport protocol based on the series of two stages: a first one fully coherent faster delocalization, followed by the emergence of a noise-induced homogeneous populations distribution, whose coherence slowly decays. In particular the coherence is sustained over a suitable time scale. So that excitation, being able to react more quickly to the trapping-site's back-action, can efficiently preserve the homogeneous distribution, which optimizes the quantum transport. In other words, after the dynamics of fully coherent delocalization, the optimal regime of noisy quantum transport can be realized by a noisy environment. This opens new paths towards the trapping site, which are coherently crossed by the particle until the absorption process is ended and the transfer efficiency maximized.
Finally, in the last chapter, I explored the behaviour of quantum walkers under the influence of spatially structured noisy environments. Various models of pure dephasing as well as of correlated dephasing have been presented and discussed. In particular, it has been treated the limit of high pure dephasing, which leads to Zeno's effect and which can be used to make transparent, for the quantum walkers, certain subsets of nodes. Then, great importance has been devoted to analyse how correlated dephasing affects quantum transport. I have shown that such models of correlated decoherence can be exploited to open new paths even between nodes of the network that may be unlinked. This forces the excitation to flow incoherently across these paths. In this case, the speed at which populations spread over the network is proportional to the strength of the non-zero terms of correlated decoherence. If their values are big enough, as within the regime of high correlated dephasing, a fast and uniform incoherent delocalization, among correlated nodes, can be reached. By following these ideas, one may actively modify the effective network topology just by suitably adjusting the spatial structure of the dephasing noise, in the low dephasing as well as in the high dephasing limit. I explored quantitatively all these phenomena and finally I discussed from a general perspective the theoretical possibility to use noise to manipulate the underlying network's topology, which could be very interesting from a technological point of view.
All the above findings establish a first step in the collective effort to understand how to drive quantum systems confined on networks by exploiting external noise. I think that this work could be useful in the fascinating perspective of exploiting the same quantum system to gain different technological functionalities, just by controlling the noisy environment which can interact with it.