Titolo della tesi: Experimental study of anomalous diffusion and quantum correlations in an all optical quantum walk
In recent years, quantum walks (QWs), the quantum counterpart of classical random walks (CRWs), have become increasingly attractive for quantum information fields, ranging from quantum computation, to quantum simulation, and quantum cryptography. Furthermore, QWs have been found to be a useful resource for the study of energy propagation and transport phenomena. Indeed, QWs are featured by a variance of the position probability distribution of the walker which grows quadratically with the number of the step, namely σ^2 = α*n^2. This behavior is usually referred to as ballistic. Several natural propagation phenomena follow an anomalous diffusion regime, for which it holds σ^2 = α*n^β with 1 < β < 2 (β < 1) in the superdiffusive (subdiffusive) case. Being it an intermediate regime between the ballistic and the diffusive one, which is typical of CRWs, it is natural to ask whether QWs allow to reproduce such a behavior.
The aim of this thesis work consists in studying this possibility by experimentally investigating anomalous diffusion within a QW framework, exploiting both single and two photons evolution. The superdiffusive dynamics has been investigated by adopting a novel bulk optics scheme, whose basic idea has been used in the study of non-Markovian dynamics. By adopting the so-called p-diluted disorder, we could investigate the average dynamics of a quantum particle moving in a random environment. We found that, in both cases of single and two walkers, the evolution is characterized by a superdiffusive behavior, with the value of the parameter β depending on the disorder level experienced by the walkers.
Subdiffusivity has been studied by means of a time-split QW. A static disorder, namely a disorder fixed in time but varying in space, can reproduce the so-called Anderson localization, consisting of an exponential localization of the walker around its starting position. By perturbing it through the p-diluted disordering technique, it is possible to break the static condition giving rise to Anderson localization. We found that in this case a clear subdiffusive behavior can be obtained, with a value of β increasing with the disorder level and reaching β = 1 when the disorder is maximum.
These results demonstrate that QWs provided with the p-diluted disorder are feasible resources to study anomalous diffusion processes, for both single and two photons cases, and that it can be helpful in shedding light on the physical mechanisms underlying anomalous diffusion phenomena.
Secondarily, by the same disordering technique, we studied how to quantify non-classical correlations due to indistinguishability between two walkers, by investigating the violation of a certain inequality, as a function of the disorder experienced by the photons during their evolution. Simulated and experimental results show that the disorder can control the non-classical correlation between the photons travelling along different pairs of modes. Moreover, we show that, by properly tuning the disorder, it is possible to retrieve a relevant amount of the non-classical correlation initially present between the walkers, which would be otherwise decreased by on ordered QW evolution. This suggests that this technique could be adopted as a useful tool in a quantum protocol scenario, such as for instance metrology.