Alexander	Blumen -	Universität Freiburg (D) # 
Continuous time random walks and continuous time quantum walks #
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 known from the dynamics of charge carriers 
along polymer backbones and
of excitations in quantum gases. Hence, in modelling energy 
transfer one has to extend the
classical, master-equation-type formalism and incorporate 
quantum-mechanical aspects, while
still taking into account the complex network of molecules over 
which the transport takes
place.
<br/>
Interestingly, the continuous time random walk (CTRW) scheme, 
widely employed in
modelling transport in random environments, is mathematically 
very close to quantum-
mechanical Hamiltonians of tight-binding type[1]; a simple way 
to see it is to focus on the
time-evolution operators in statistical and in quantum 
mechanics: The transformation to the
quantal domain leads then to continuous-time quantum walks 
(CTQWs).
<br/>
Now, while the CTQW problem is then linear, and thus many 
results obtained in solving
CTRWs (such as eigenvalues and eigenfunctions) can be readily 
reutilized for CTQWs,
the physically relevant properties of the two models differ 
vastly: In the absence of traps
CTQWs 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 [1]. In this talk we will 
discuss this and additional features,
such as the topology dependence of CTQWs, ranging from very 
efficient transport on
regular lattices [2] to localization and trapping effects on 
small-world networks [3] and on
fractal and hyperbranched structures [4]. We will furthermore 
compare the CTQW results
to the corresponding CTRW results on topologically equivalent 
networks. This allows us to
systematically explore the similarities and differences between 
purely classical and purely
quantum-mechanical processes [1, 4].
<br/>
<br/>
[1] O. M&uuml;lken and A. Blumen; Phys. Rev. E 71, 036128 
(2005); 
Phys. Rev. E 73, 066117
(2006); Physics Reports 502, 37 (2011)<br/>
[2] O. M&uuml;lken, V. Bierbaum, and A. Blumen; J. Chem. Phys. 
124, 
124905 (2006).<br/>
[3] O. M&uuml;lken, V. Pernice, and A. Blumen; Phys. Rev. E 76, 
051125 (2007)<br/>
[4] E. Agliari, A. Blumen, and O. M&uuml;lken; J. Phys. A 41, 
445301 
(2008); Phys. Rev. A 82, 012305 (2010); Intern. J. Bifurc. 
Chaos, 20, 271 (2010) <br/>