Riassunto analitico
This work is devoted to the study of continuous-time quantum walks (QWs) of a charged particle on planar lattice graphs in the presence of a perpendicular uniform magnetic field. Specifically it addresses the origin of the oscillations observed in the variance of the space coordinates and then the generalization of QWs to planar lattice graphs. A planar lattice graph is a graph possessing a drawing whose embedding in a Euclidean plane forms a regular tiling, i.e. a regular tessellation. This can only be achieved with equilateral triangles, squares, and regular hexagons, which yield to triangular, square, and honeycomb lattice graphs, respectively. A previous study about QWs on square lattice in the presence of a magnetic field (vector potential in the symmetric gauge) showed clues of an oscillating variance of the space coordinates and we have carried on such study. As known, the original analytical Hamiltonian (before being discretized for the QW) turns out to be the Hamiltonian of a quantum harmonic oscillator, whose energy levels are the so-called Landau levels. Following what suggested in the literature, we develop an analytical approach to study the variance of the space coordinates and we numerically implement it. In this way we compare analytical results with those of the QW, so that we are not properly comparing the same system with two approaches. Indeed the QW comes from the discretization of the space and, consequently, of the Hamiltonian, whereas the analytical approach works in a continuum of space. In spite of this we carry on the analytical discussion because we believe it to be the only way of shedding some light on the origin of such oscillations, insisting on numerical discretizations would likely provide the same results of the QW without carrying further knowledge. The aperiodic non-regular oscillations of the variance of the space coordinates in the QW are found to be related to the periodic ones of the analytical approach, whose angular frequency turns out to be the expected cyclotron frequency. In particular the pseudo-angular frequency of such aperiodic oscillations shows a linear dependence on the magnetic field, just like the cyclotron frequency. To our knowledge, there are in the literature many works about QWs on arbitrary graphs (understood as mathematical structures, more general than lattice graphs), but they only involve the kinetic term (which is replaced by the graph Laplacian) and possible potential terms or interactions. Such QWs do not involve any gradient, since the linear momentum appears only at the second order within the kinetic term. The presence of a vector potential, instead, couples the position and the linear momentum, so that we have to find a method which generalizes both the Laplacian and the gradient or which avoids the gradient. To this end we explore three approaches: i) the introduction of the Peierls phase-factors, according to which the tunneling of the free particle becomes complex, accompanied by the Peierls phase due to the vector potential; ii) the generalization of the finite difference formulas obtained with Taylor expansion in the square lattice; iii) the use of conservative finite-difference methods on general grids. Each method, by itself, is not able to provide a coherent framework satisfying the constraints of a QW. Therefore, we suggest a hybrid method, which combines the best results of the above methods and provides a Laplacian analogous to the graph Laplacian and a gradient which involves all the nearest-neighbors of each vertex of the graph. In the end we study if and how the quantum Fisher information depends on the choice of the graph.
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