Continuous time quantum walks in two-dimensional networks

>Antonio Volta

University of Freiburg


Antonio Volta, Oliver Mülken and Alexander Blumen Theoretical Polymer Physics, University Freiburg
The quantum transport is a topic of keen interest, especially in the last few years for the development of quantum information theory and the applications to potential quantum computers. In the present work we study the transport by continuous time quantum walks (CTQW) on networks which are topologically equivalent to two-dimensional lattices. Placing the excitation at one corner, one can observe at short times a directed very fast transport to the opposite corner through the network. The propagation gives rise to a wave interference pattern in contrast to a classical diffusive process. We present the results for CTQW on tori, cylinders, and finite squares. In the second and third case, the long time average of the transition probability distribution shows asymmetric features depending on the starting point and, remarkably, on the lattice's size. This asymmetry of the limiting probability manifests itself clearly when looking at the initial corner and its opposite,i.e. the "mirror" node with respect to the center of the structure. We focus also on the probability to be still or again at the initial site, for which in the quantum case we can give a lower bound, which is rather close to the exact,numerical result. The lower bound depends only on the eigenvalue spectrum of the Hamiltonian. For the above-mentioned two-dimensional structures, we are able, by applying methods from solid state- and polymer-physics, to obtain analytically the eigenvalues.
(*)Mülken O., Volta A. and Blumen A., Phys. Rev. A (2005) in press description: The central topic of my research is the transport on networks. This mathematical problem has several applications, see for instance [2,3] for the energy tranfer on polymers, or in quantum information theory [1]. The structures domain includes regular d-dimensional lattices, and hyperbranched structures, such as dendrimers or regular hyperbranched fractals (RHF). The used techniques are based on the continuous time random walks (CTRW), or its quantum mechanical equivalent, i.e. the continuous time quantum walks (CTQW). For both, we are able to provide analytical results, which permit to investigate what happens in case of big lattices too.
1) Mülken O., Volta A. and Blumen A., Phys. Rev. A (2005) in press
2) Blumen A., Volta A., Jurjiu A. and Koslowski Th., J. of Lumin. 111 (2005) 327
3) Blumen A., Volta A., Jurjiu A. and Koslowski Th., Physica A 356 (2005) 12