From Continuous-Time Random Walks to Continuous-Time Quantum Walks: Disordered Networks

Abstract

Recent years have seen a growing interest in dynamical quantum processes; thus it was found that the electronic energy transfer through photosynthetic antennae displays quantum features, aspects also known from the dynamics of charge carriers along polymer backbones. Hence, in modeling energy transfer one has to extend the classical, master-equation-type formalism and incorporate quantum-mechanical aspects, while still aiming to describe complex networks of molecules over which the transport takes place. The continuous time random walk (CTRW) scheme is widely employed in modeling transport in random environments (Sokolov et al, Phys Today 55:48, 2002) and is mathematically akin to quantum-mechanical Hamiltonians of tight-binding type (Mülken and Blumen, Phys Rep 502:37, 2011; Mülken and Blumen, Phys Rev E 73:066117, 2006); a simple way to see it is to focus on the time-evolution operators in statistical and in quantum mechanics: The transition to the quantal domain leads then to continuous-time quantum walks (CTQW). In this way the CTQW problem stays linear, and thus many results obtained in solving CTRW (such as eigenvalues and eigenfunctions) can be readily reutilized for CTQW. However, the physically relevant properties of the two models differ vastly: In the absence of traps CTQW are time-inversion symmetric and no energy equipartition takes place at long times. Also, the quantum system keeps memory of the initial conditions, a fact exemplified by the occurrence of quasi-revivals (Mülken and Blumen, Phys Rep 502:37, 2011). Here we will exemplify the vastly different behaviors of CTQW and CTRW on disordered networks, namely on small-world networks (Mülken et al, Phys Rev E 76:051125, 2007) and on star-graphs with randomly added bonds (Anishchenko et al, Quantum Inf Process 11:1273, 2012).