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 [1], to quantum simulation [2], and quantum cryptography [3]. Furthermore, QWs have been found to be a useful resource for the study of energy propagation and transport phenomena [4]. 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 my 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 [5], whose basic idea has been used in the study of non-Markovian dynamics [6]. By adopting the so-called p-diluted disorder, we could investigate the average dynamics of a quantum particle moving in a random environment [7]. 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 mean 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 [8]. 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 between two walkers, by checking the violation of a certain inequality [9], as a function of the disorder experienced by the photons during their evolution. Preliminary results show that the disorder can control the non-classical correlation between the photons travelling along different pairs of modes, suggesting that it could be adopted as a useful instrument in a quantum protocol scenario, such as for instance metrology. References [1] A. M. Childs, “Universal computation by quantum walk," Physical review letters, vol. 102, no. 18, p. 180501, 2009. [2] S. E. Venegas-Andraca, “Quantum walks: a comprehensive review," Quantum Information Processing, vol. 11, no. 5, pp. 1015-1106, 2012. [3] C. Vlachou, W. Krawec, P. Mateus, N. Paunkovic, and A. Souto, “Quantum key distribution with quantum walks," Quantum Information Processing, vol. 17, no. 11, p. 288, 2018. [4] M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum walks in photosynthetic energy transfer," The Journal of chemical physics, vol. 129, no. 17, p. 11B603, 2008. [5] A. Geraldi, L. D. Bonavena, C. Liorni, P. Mataloni, and A. Cuevas, “A novel bulk-optics scheme for quantum walk with high phase stability," Condensed Matter, vol. 4, no. 1, p. 14, 2019. [6] A. Cuevas, A. Geraldi, C. Liorni, L. D. Bonavena, A. De Pasquale, F. Sciarrino, V. Giovannetti, and P. Mataloni, “All-optical implementation of collision-based evolutions of open quantum systems," Scientific reports, vol. 9, no. 1, pp. 1-8, 2019. [7] A. Geraldi, A. Laneve, L. D. Bonavena, L. Sansoni, J. Ferraz, A. Fratalocchi, F. Sciarrino, A. Cuevas, and P. Mataloni, “Experimental investigation of superdiffusion via coherent disordered quantum walks," Physical review letters, vol. 123, no. 14, p. 140501, 2019. [8] A. Geraldi, S. De, A. Laneve, S. Barkhofen, J. Sperling, P. Mataloni, and C. Silberhorn, “Paper in preparation," [9] Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices," Physical review letters, vol. 102, no. 25, p. 253904, 2009.