This paper reviews recent advances in continuous-time quantum walks (CTQW) and their application to transport in various systems. The introduction gives a brief survey of the historical background of CTQW. After a short outline of the theoretical ideas behind CTQW and of its relation to classical continuous-time random walks (CTRW) in Sec. 2, implications for the efficiency of the transport are presented in Sec. 3. The fourth section gives an overview of different types of networks on which CTQW have been studied so far. Extensions of CTQW to systems with long-range interactions and with static disorder are discussed in section V. Systems with traps, i.e., systems in which the walker's probability to remain inside the system is not conserved, are presented in section IV. Relations to similar approaches to the transport are studied in section VII. The paper closes with an outlook on possible future directions.

In the quest for signatures of coherent transport we consider exciton trapping in the continuous-time quantum walk framework. The survival probability displays different decay domains, related to distinct regions of the spectrum of the Hamiltonian. For linear systems and at intermediate times the decay obeys a power-law, in contrast to the corresponding exponential decay found in incoherent continuous-time random walk situations. To differentiate between the coherent and incoherent mechanisms, we present an experimental protocol based on a frozen Rydberg gas structured by optical dipole traps.PACS numbers: 05.60. Gg, 32.80.Rm, 34.20.Cf Recent years have seen an upsurge of interest in coherent energy transfer, given the experimental advances in manipulating and controlling quantum mechanical systems. From the theoretical side, such investigations are of long standing; see, e.g., [1]. Here, tight-binding models, which model coherent exciton transfer, are closely related to the quantum walks (QW An appropriate means to monitor transport is to follow the decay of the excitation due to trapping. The long time decay of chains with traps is a well studied problem for classical systems [8,9]: for an ensemble of chains of different length with traps at both ends the averaged exciton survival probability has a stretched exponential form exp(−bt λ ), with λ = 1/3 (see, e.g., [9]). In contrast, quantum mechanical tight-binding models lead to λ = 1/4 [10, 11]. However, up to now only little is known about the decay of the quantum mechanical survival probability at experimentally relevant intermediate times.Here we evaluate and compare the intermediate-time decays due to trapping for both RW and QW situations by employing the similarity of the CTRW and the CTQW formalisms. Without traps, the coherent dynamics of excitons on a graph of connected nodes is modeled by the CTQW, which is obtained by identifying the Hamiltonian H 0 of the system with the CTRW transfer matrix T 0 , i.e., H 0 = −T 0 ; see e.g. [3, 12] (we will set ≡ 1 in the following). For undirected graphs, T 0 is related to the connectivity matrix A 0 of the graph by T 0 = −A 0 , where (for simplicity) all transmission rates are taken to be equal. Thus, in the following we take H 0 = A 0 . The matrix A 0 has as non-diagonal elements A the Rydberg gases considered in the following, the coupling strength is roughly H (0) k,j / 1 MHz, i.e., the time unit for transfer between two nodes is of the order of a few hundred nanoseconds.The states |j associated with excitons localized at the nodes j (j = 1, . . . , N ) form a complete, orthonormal basis set (COBS) of the whole accessible Hilbert space, i.e., k|j = δ kj and k |k k| = 1. In general, the time evolution of a state |j starting at time t 0 = 0 is given by |j; t = exp(−iH 0 t)|j ; hence the transition amplitudes and the probabilities read α kj (t) ≡ k| exp(−iH 0 t)|j and π kj (t) ≡ |α kj (t)| 2 , respectively. In the corresponding classical CTRW case the transition probabilities follow from a master equation as ...

We present a classification scheme for phase transitions in finite systems like atomic and molecular clusters based on the Lee-Yang zeros in the complex temperature plane. In the limit of infinite particle numbers the scheme reduces to the Ehrenfest definition of phase transitions and gives the right critical indices. We apply this classification scheme to Bose-Einstein condensates in a harmonic trap as an example of a higher order phase transitions in a finite system and to small Ar clusters.

We model coherent exciton transport in dendrimers by continuous-time quantum walks. For dendrimers up to the second generation the coherent transport shows perfect recurrences when the initial excitation starts at the central node. For larger dendrimers, the recurrence ceases to be perfect, a fact which resembles results for discrete quantum carpets. Moreover, depending on the initial excitation site, we find that the coherent transport to certain nodes of the dendrimer has a very low probability. When the initial excitation starts from the central node, the problem can be mapped onto a line which simplifies the computational effort. Furthermore, the long time average of the quantum mechanical transition probabilities between pairs of nodes shows characteristic patterns and allows us to classify the nodes into clusters with identical limiting probabilities. For the (space) average of the quantum mechanical probability to be still or to be again at the initial site, we obtain, based on the Cauchy-Schwarz inequality, a simple lower bound which depends only on the eigenvalue spectrum of the Hamiltonian.

The propagation by continuous-time quantum walks (CTQWs) on one-dimensional lattices shows structures in the transition probabilities between different sites reminiscent of quantum carpets. For a system with periodic boundary conditions, we calculate the transition probabilities for a CTQW by diagonalizing the transfer matrix and by a Bloch function ansatz. Remarkably, the results obtained for the Bloch function ansatz can be related to results from (discrete) generalized coined quantum walks. Furthermore, we show that here the first revival time turns out to be larger than for quantum carpets.