I obtain the dynamics of the continuous time quantum walk on a d-dimensional lattice, with periodic boundary conditions, as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of [8] which proved this limit for the case of the quantum walk on the infinite line and the quantum cellular automaton proposed in [1]. By highlighting the main features of the limiting procedure, I then extend it to general graphs. For a given discrete time quantum walk on a general graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics. 1 Generalities on quantum walks; continuous and discrete time Consider a graph G: = {V, E} with a set of vertices V of cardinality N and a set of edges E. We assume G to be undirected and without self-loops. In the interval of time ∆t a certain fraction γ∆t of a quantity pj leaves the location of the j-th vertex to move to a neighboring vertex k. The quantity pj, j = 1,..., N, at time t + ∆t is given by 1 pj(t + ∆t) = pj(t) − deg (j)γ∆tpj(t) + ∑ (k,j)∈E γ∆tpk(t). (1) Define ⃗p the N−vector whose components are the pj’s quantities, and L the Laplacian matrix defined by Ljj = − deg (j), Ljk = 1 if (j, k) ∈ E and Ljk = 0 otherwise. L is
Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of “fat graphs”—in other words, suitable families of shrinking manifolds—and discuss convergence of the spectra and resonances in such a setting.
Abstract. Quantum networks are often modelled using Schrödinger operators on metric graphs. To give meaning to such models one has to know how to interpret the boundary conditions which match the wave functions at the graph vertices. In this article we give a survey, technically not too heavy, of several recent results which serve this purpose. Specifically, we consider approximations by means of “fat graphs ” — in other words, suitable families of shrinking manifolds — and discuss convergence of the spectra and resonances in such a setting. 1.
Abstract In this paper, we discuss the concept of quantum graphs with transparent vertices by considering the case, where the graph interacts with an external time-independent field. In particular, we address the problem of transparent boundary conditions for quantum graphs, building on previous works on transparent boundary conditions for the stationary Schrödinger equation on a line. Physically relevant constraints that make the vertex transparent under boundary conditions in the form of (weight) continuity and Kirchhoff rules are derived using two approaches, where transparent boundary conditions for the time-independent Schrödinger equation are found. The latter is derived by extending the transparent boundary condition concept to the time-independent Schrödinger equation on driven quantum graphs. For the constant potential, these constraints are also obtained using the scattering approach. We also discuss how the eigenvalues and eigenfunctions of a quantum graph are influenced not only by its topology, but also by the shape/type of a potential when an external field is involved.
We consider the statics and dynamics of distinguishable spin-
21 systems on an arbitrary graph
G with
N vertices. In particular, we consider systems of quantum spins evolving according to one of two Hamiltonians: (i) the
XY Hamiltonian
HXY, which contains an
XY interaction for every pair of spins connected by an edge in
G; and (ii) the Heisenberg Hamiltonian
HHeis, which contains a Heisenberg interaction term for every pair of spins connected by an edge in
G. We find that the action of the
XY (respectively, Heisenberg) Hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial Laplacian) of a sequence
Gk,
k=0,…,N of graphs derived from
G (with
G1=G). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs
Gk. Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the
XY model in a magnetic field on the path consisting of
N vertices. We also use our approach to utilize results from spectral graph theory to solve new spin models: the
XY model and Heisenberg model in a magnetic field on the complete graph.