Targeting Eigenstates by "Simulated Measurements" using a Decoherence based Nonlinear Schr\"odinger Equation

Inspired by the idea of mimicking the measurement on a quantum system through a decoherence process to target specific eigenstates based on Born's law, i.e. the hiearchy of probabilities instead of the hierarchy of eigenvalues, we transform a Lindblad equation for the reduced density operator into a nonlinear Schr\"{o}dinger equation to obtain a computationally feasible simulation of the decoherent dynamics in the open quantum system. This gives the opportunity to target the eigenstates which have the largest $L^2$ overlap with an initial superposition state and hence more flexibility in the selection criteria. One can use this feature for instance to approximate eigenstates with certain localization or symmetry properties. As an application of the theory we discuss \textit{eigenstate towing}, which relies on the perturbation theory to follow the progression of an arbitrary subset of eigenstates along a sum of perturbation operators with the intention to explore for example the effect of interactions on these eigenstates. The easily parallelizable numerical method shows an exponential convergence and its computational costs scale linear for sparse matrix representations of the involved Hermitian operators.Comment: 12 pages, 11 figure

Landau levels in wrinkled and rippled graphene sheets

We study the discrete energy spectrum of curved graphene sheets in the presence of a magnetic field. The shifting of the Landau levels is determined for complex and realistic geometries of curved graphene sheets. The energy levels follow a similar square root dependence on the energy quantum number as for rippled and flat graphene sheets. The Landau levels are shifted towards lower energies proportionally to the average deformation and the effect is larger compared to a simple uni-axially rippled geometry. Furthermore, the resistivity of wrinkled graphene sheets is calculated for different average space curvatures and shown to obey a linear relation. The study is carried out with a quantum lattice Boltzmann method, solving the Dirac equation on curved manifolds.Comment: 6 pages, 4 figures, 27th International Conference on Discrete Simulation of Fluid Dynamic

Quantum spin-Hall effect on the M\"obius graphene ribbon

Topological phases of matter have revolutionized quantum engineering. Implementing a curved space Dirac equation solver based on the quantum Lattice Boltzmann method, we study the topological and geometrical transport properties of a M\"obius graphene ribbon. In the absence of a magnetic field, we measure a quantum spin-Hall current on the graphene strip, originating from topology and curvature, whereas a quantum Hall current is not observed. In the torus geometry a Hall current is measured. Additionally, a specific illustration of the equivalence between the Berry and Ricci curvature is presented through a travelling wave-packet around the M\"obius band.Comment: arXiv admin note: substantial text overlap with arXiv:1810.0210

Lattice Wigner equation

We present a numerical scheme to solve the Wigner equation, based on a lattice discretization of momentum space. The moments of the Wigner function are recovered exactly, up to the desired order given by the number of discrete momenta retained in the discretisation, which also determines the accuracy of the method. The Wigner equation is equipped with an additional collision operator, designed in such a way as to ensure numerical stability without affecting the evolution of the relevant moments of the Wigner function. The lattice Wigner scheme is validated for the case of quantum harmonic and anharmonic potentials, showing good agreement with theoretical results. It is further applied to the study of the transport properties of one and two dimensional open quantum systems with potential barriers. Finally, the computational viability of the scheme for the case of three- dimensional open systems is also illustrated