arXiv subjects

Alexander Blumen

Publications and source records attributed to Alexander Blumen.

At least 19 recordsLinked to original sources

arXiv2017

Non-Markovian closure kinetics of flexible polymers with hydrodynamic interactions

This paper presents a theoretical analysis of the closure kinetics of a polymer with hydrodynamic interactions. This analysis, which takes into account the non-Markovian dynamics of the end-to-end vector and relies on the preaveraging of the mobility tensor (Zimm dynamics), is shown to reproduce very accurately the results of numerical simulations of the complete non linear dynamics. It is found that Markovian treatments, based on a Wilemski-Fixman approximation, significantly overestimate cyclization times (up to a factor 2), showing the importance of memory effects in the dynamics. In addition, this analysis provides scaling laws of the mean first cyclization time (MFCT) with the polymer size $N$ and capture radius $b$, which are identical in both Markovian and non-Markovian approaches. In particular, it is found that the scaling of the MFCT for large $N$ is given by $T\sim N^{3/2}\ln (N/b^2)$, which differs from the case of the Rouse dynamics where $T\sim N^{2}$. The extension to the case of the reaction kinetics of a monomer of a Zimm polymer with an external target in a confined volume is also presented.

cond-mat.stat-mech↗

arXiv2015

Contact Kinetics in Fractal Macromolecules

We consider the kinetics of first contact between two monomers of the same macromolecule. Relying on a fractal description of the macromolecule, we develop an analytical method to compute the Mean First Contact Time (MFCT) for various molecular sizes. In our theoretical description, the non-Markovian feature of monomer motion, arising from the interactions with the other monomers, is captured by accounting for the non-equilibrium conformations of the macromolecule at the very instant of first contact. This analysis reveals a simple scaling relation for the MFCT between two monomers, which involves only their equilibrium distance and the spectral dimension of the macromolecule, independently of its microscopic details. Our theoretical predictions are in excellent agreement with numerical stochastic simulations.

cond-mat.soft↗

arXiv2015

Directed transport in quantum star graphs

We study the quantum dynamics of Gaussian wave packets on star graphs whose arms feature each a periodic potential and an external time-dependent field. Assuming that the potentials and the field can be manipulated separately for each arm of the star, we show that it is possible to manipulate the direction of the motion of a Gaussian wave packet through the bifurcation point by a suitable choice of the parameters of the external fields. In doing so, one can achieve a transmission of the wave packet into the desired arm with nearly 70\% while also keeping the shape of the wave packet approximately intact. Since a star graph is the simplest element of many other complex graphs, the obtained results can be considered as the first step to wave packet manipulations on complex networks.

quant-ph↗

arXiv2014

Transport properties of continuous-time quantum walks on Sierpinski fractals

We model quantum transport, described by continuous-time quantum walks (CTQW), on deterministic Sierpinski fractals, differentiating between Sierpinski gaskets and Sierpinski carpets, along with their dual structures. The transport efficiencies are defined in terms of the exact and the average return probabilities, as well as by the mean survival probability when absorbing traps are present. In the case of gaskets, localization can be identified already for small networks (generations). For carpets, our numerical results indicate a trend towards localization, but only for relatively large structures. The comparison of gaskets and carpets further implies that, distinct from the corresponding classical continuous-time random walk, the spectral dimension does not fully determine the evolution of the CTQW.

quant-ph↗

arXiv2014

Slow Encounters of Particle Pairs in Branched Structures

On infinite homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent, i.e., a single random walker returns to its starting point with probability 1. However, on general inhomogeneous structures this property does not hold and, although a single random walker will certainly return to its starting point, two moving particles may never meet. This striking property has been shown to hold, for instance, on infinite combs. Due to the huge variety of natural phenomena which can be modeled in terms of encounters between two (or more) particles diffusing in comb-like structures, it is fundamental to investigate if and, if so, to what extent similar effects may take place in finite structures. By means of numerical simulations we evidence that, indeed, even on finite structures, the topological inhomogeneity can qualitatively affect the two-particle problem. In particular, the mean encounter time can be polynomially larger than the time expected from the related one particle problem.

cond-mat.stat-mech↗

arXiv2013

Geometrical aspects of quantum walks on random two-dimensional structures

We study the transport properties of continuous-time quantum walks (CTQW) over finite two-dimensional structures with a given number of randomly placed bonds and with different aspect ratios (AR). Here, we focus on the transport from, say, the left side to the right side of the structure where absorbing sites are placed. We do so by analyzing the long-time average of the survival probability of CTQW. We compare the results to the classical continuous-time random walk case (CTRW). For small AR (landscape configurations) we observe only small differences between the quantum and the classical transport properties, i.e., roughly the same number of bonds is needed to facilitate the transport. However, with increasing AR (portrait configurations) a much larger number of bonds is needed in the CTQW case than in the CTRW case. While for CTRW the number of bonds needed decreases when going from small AR to large AR, for CTRW this number is large for small AR, has a minimum for the square configuration, and increases again for increasing AR. We corroborate our findings for large AR by showing that the corresponding quantum eigenstates are strongly localized in situations in which the transport is facilitated in the CTRW case.

quant-ph↗

arXiv2011

Enhancing the spreading of quantum walks on star graphs by additional bonds

We study the dynamics of continuous-time quantum walks (CTQW) on networks with highly degenerate eigenvalue spectra of the corresponding connectivity matrices. In particular, we consider the two cases of a star graph and of a complete graph, both having one highly degenerate eigenvalue, while displaying different topologies. While the CTQW spreading over the network - in terms of the average probability to return or to stay at an initially excited node - is in both cases very slow, also when compared to the corresponding classical continuous-time random walk (CTRW), we show how the spreading is enhanced by randomly adding bonds to the star graph or removing bonds from the complete graph. Then, the spreading of the excitations may become very fast, even outperforming the corresponding CTRW. Our numerical results suggest that the maximal spreading is reached halfway between the star graph and the complete graph. We further show how this disorder-enhanced spreading is related to the networks' eigenvalues.

quant-ph↗

arXiv2011

Transport efficiency in topologically disordered networks with environmentally induced diffusion

We study transport in topologically disordered networks that are subjected to an environment that induces classical diffusion. The dynamics is phenomenologically described within the framework of the recently introduced quantum stochastic walk, allowing to study the crossover between coherent transport and purely classical diffusion. We find that the coupling to the environment removes all effects of localization and quickly leads to classical transport. Furthermore, we find that on the level of the transport efficiency, the system can be well described by reducing it to a two-node network (a dimer).

quant-ph↗

arXiv2011

Continuous-Time Quantum Walks: Models for Coherent Transport on Complex Networks

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.

quant-ph↗

arXiv2010

Quantum walk approach to search on fractal structures

We study continuous-time quantum walks mimicking the quantum search based on Grover's procedure. This allows us to consider structures, that is, databases, with arbitrary topological arrangements of their entries. We show that the topological structure of the database plays a crucial role by analyzing, both analytically and numerically, the transition from the ground to the first excited state of the Hamiltonian associated with different (fractal) structures. Additionally, we use the probability of successfully finding a specific target as another indicator of the importance of the topological structure.

quant-ph↗

arXiv2009

Dissipative Dynamics with Trapping in Dimers

The trapping of excitations in systems coupled to an environment allows to study the quantum to classical crossover by different means. We show how to combine the phenomenological description by a non-hermitian Liouville-von Neumann Equation (LvNE) approach with the numerically exact path integral Monte-Carlo (PIMC) method, and exemplify our results for a system of two coupled two-level systems. By varying the strength of the coupling to the environment we are able to estimate the parameter range in which the LvNE approach yields satisfactory results. Moreover, by matching the PIMC results with the LvNE calculations we have a powerful tool to extrapolate the numerically exact PIMC method to long times.

quant-ph↗

arXiv2009

Continuous-Time Quantum Walks and Trapping

Recent findings suggest that processes such as the electronic energy transfer through the photosynthetic antenna display quantal features, aspects known from the dynamics of charge carriers along polymer backbones. Hence, in modeling energy transfer one has to leave the classical, master-equation-type formalism and advance towards an increasingly quantum-mechanical picture, while still retaining a local description of the complex network of molecules involved in the transport, say through a tight-binding approach. Interestingly, the continuous time random walk (CTRW) picture, widely employed in describing transport in random environments, can be mathematically reformulated to yield a quantum-mechanical Hamiltonian of tight-binding type; the procedure uses the mathematical analogies between time-evolution operators in statistical and in quantum mechanics: The result are continuous-time quantum walks (CTQWs). However, beyond these formal analogies, CTRWs and CTQWs display vastly different physical properties. In particular, here we focus on trapping processes on a ring and show, both analytically and numerically, that distinct configurations of traps (ranging from periodical to random) yield strongly different behaviours for the quantal mean survival probability, while classically (under ordered conditions) we always find an exponential decay at long times.

quant-ph↗

arXiv2008

Coherent Exciton Dynamics and Trapping in Topologically Disordered Systems

We analyze the coherent dynamics of excitons in three dimensional topologically disordered networks with traps. If the interactions between the nodes of the network are long ranged, i.e., algebraically decaying as a function of the distance between the nodes, the average survival probability of an exciton surprisingly shows a characteristic decay with features similar to the decay found for regular one-dimensional systems. We further show how this decay can be related to the eigenstates of the same system without a trap.

quant-ph↗

arXiv2008

Slow Excitation Trapping in Quantum Transport with Long-Range Interactions

Long-range interactions slow down the excitation trapping in quantum transport processes on a one-dimensional chain with traps at both ends. This is counter intuitive and in contrast to the corresponding classical processes with long-range interactions, which lead to faster excitation trapping. We give a pertubation theoretical explanation of this effect.

quant-ph↗

arXiv2007

Quantum transport on small-world networks: A continuous-time quantum walk approach

We consider the quantum mechanical transport of (coherent) excitons on small-world networks (SWN). The SWN are build from a one-dimensional ring of N nodes by randomly introducing B additional bonds between them. The exciton dynamics is modeled by continuous-time quantum walks and we evaluate numerically the ensemble averaged transition probability to reach any node of the network from the initially excited one. For sufficiently large B we find that the quantum mechanical transport through the SWN is, first, very fast, given that the limiting value of the transition probability is reached very quickly; second, that the transport does not lead to equipartition, given that on average the exciton is most likely to be found at the initial node.

quant-ph↗

arXiv2007

Universal Behavior of Quantum Walks with Long-Range Steps

Quantum walks with long-range steps $R^{-γ}$ ($R$ being the distance between sites) on a discrete line behave in similar ways for all $γ\geq2$. This is in contrast to classical random walks, which for $γ>3$ belong to a different universality class than for $γ\leq 3$. We show that the average probabilities to be at the initial site after time $t$ as well as the mean square displacements are of the same functional form for quantum walks with $γ=2$, 4, and with nearest neighbor steps. We interpolate this result to arbitrary $γ\geq2$.

quant-ph↗

arXiv2007

Survival Probabilities in Coherent Exciton Transfer with Trapping

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.

quant-ph↗

arXiv2007

Localization of coherent exciton transport in phase space

We study numerically the dynamics of excitons on discrete rings in the presence of static disorder. Based on continuous-time quantum walks we compute the time evolution of the Wigner function (WF) both for pure diagonal (site) disorder, as well as for diagonal and off-diagonal (site and transfer) disorder. In both cases, large disorder leads to localization and destroys the characteristic phase space patterns of the WF found in the absence of disorder.

quant-ph↗