Spectral and diffusive properties of silver-mean quasicrystals in one, two, and three dimensions

Cerovski, Viktor Z.; Schreiber, Michael; Grimm, Uwe

doi:10.1103/physrevb.72.054203

Cited by 21 publications

(28 citation statements)

References 57 publications

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“…The dynamics of quantum particles on non-regular structures has been investigated in several works meant to analyze the quantum dynamics of tight-biding electrons in quasicrystals, in aperiodic and quasi-periodic chains and in random environments [20,21,36]. There, the highlighted dramatic deviations from the ballistic behaviour (expected for regular, infinite lattices) range from anomalous to superdiffusion, to decoherence, and even to Anderson localization [37].…”

Section: Average Displacementmentioning

confidence: 99%

Dynamics of continuous-time quantum walks in restricted geometries

Agliari¹,

Blumen²,

Muelken³

2008

J. Phys. A: Math. Theor.

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Abstract. We study quantum transport on finite discrete structures and we model the process by means of continuous-time quantum walks. A direct and effective comparison between quantum and classical walks can be attained based on the average displacement of the walker as a function of time. Indeed, a fast growth of the average displacement can be advantageously exploited to build up efficient search algorithms. By means of analytical and numerical investigations, we show that the finiteness and the inhomogeneity of the substrate jointly weaken the quantum walk performance. We further highlight the interplay between the quantum-walk dynamics and the underlying topology by studying the temporal evolution of the transfer probability distribution and the lower bound of long time averages.

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Section: Average Displacementmentioning

confidence: 99%

Dynamics of continuous-time quantum walks in restricted geometries

Agliari¹,

Blumen²,

Muelken³

2008

J. Phys. A: Math. Theor.

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“…In the case of a single particle in vacuum ( 0) b  the solution of Eq. (7) is the known expression (Cerovski et al 2005) show linear dependence of  on the hopping strength, spanned between 0.25 and the ballistic 1 (the Einstein law corresponds to 0.5  …”

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confidence: 99%

Nonlinear Theory of Quantum Brownian Motion

Tsekov

2008

Int J Theor Phys

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A nonlinear theory of quantum Brownian motion in classical environment is developed based on a thermodynamically enhanced nonlinear Schrödinger equation. The latter is transformed via the Madelung transformation into a nonlinear quantum Smoluchowski-like equation, which is proven to reproduce key results from quantum and classical physics. The application of the theory to a free quantum Brownian particle results in a nonlinear dependence of the position dispersion on time, being quantum generalization of the Einstein law of Brownian motion. It is shown that the time of decoherence for the transition from quantum to classical diffusion is proportional to the square of the thermal de Broglie wavelength divided by the Einstein diffusion constant.Brownian motion is the permanent irregular movement of a particle immersed in a medium. Its rigorous description requires joint consideration of the coupled dynamics of the Brownian particle and the medium, usually referred as thermal bath. Fundamental problems appear in the Brownian motion theory if the quantum effects become important. For instance, at time larger than the classical momentum relaxation time the Brownian motion of a quantum particle in a classical environment is regularly described by the classical Einstein law

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“…(1) the position dispersion of a free quantum Brownian particle obeys the classical Einstein law 2 2Dt  . Numerical simulations [2] have shown, however, that the width of the wave packet of electrons exhibits anomalous diffusion with t   , where  depends strongly on the hopping strength. A decade ago a Letter to the Editor of the present journal was published [3], where the following quantum Smoluchowski equation is derived…”

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confidence: 99%

Comment on ‘Semiclassical Klein–Kramers and Smoluchowski equations for the Brownian motion of a particle in an external potential’

Tsekov¹

2007

J. Phys. A: Math. Theor.

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The range of validity of the semiclassical Smoluchowski equation derived recently by Coffey et al is discussed. The analysis is based on the quantum Smoluchowski equation derived by the present author before. A quantum generalization of the Einstein law of Brownian motion is also obtained.Recently, Coffey et al [1] have proposed a heuristic method for determination of the effective diffusion coefficient of a quantum Brownian particle. Starting from the master equation for the Wigner distribution function they have derived a semiclassical Smoluchowski equationHere P is the probability density of the Brownian particle,  is the friction coefficient, V is an arbitrary external potential and ]Here  is the Hamiltonian of the quantum Brownian particle. It is proven that the equilibrium solution of Eq. (2) is the quantum canonical Gibbs distribution. In contrast to Eq.

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Spectral and diffusive properties of silver-mean quasicrystals in one, two, and three dimensions

Cited by 21 publications

References 57 publications

Dynamics of continuous-time quantum walks in restricted geometries

Dynamics of continuous-time quantum walks in restricted geometries

Nonlinear Theory of Quantum Brownian Motion

Comment on ‘Semiclassical Klein–Kramers and Smoluchowski equations for the Brownian motion of a particle in an external potential’

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