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arXiv:quant-ph/0310125 · Physical Review B

Fractal noise in quantum ballistic and diffusive lattice systems

Ilias Amanatidis, D. E. Katsanos, S. N. Evangelou

Published May 21, 2004• 12 citations
arXivHTMLPDF

Abstract

We demonstrate fractal noise in the quantum evolution of wave packets moving either ballistically or diffusively in periodic and quasiperiodic tight-binding lattices, respectively. For the ballistic case with various initial superpositions we obtain a space-time self-affine fractal $\ensuremath{\Psi}(x,t)$ which verifies the predictions by Berry for ``a particle in a box,'' in addition to quantum revivals. For the diffusive case self-similar fractal evolution is also obtained. These universal fractal features of quantum theory might be useful in the field of quantum information, for creating efficient quantum algorithms, and can possibly be detectable in scattering from nanostructures.

DOI
10.1103/physrevb.69.195107

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We demonstrate fractal noise in the quantum evolution of wave packets moving either ballistically or diffusively in periodic and quasiperiodic tight-binding lattices, respectively. For the ballistic case with various initial superpositions we obtain a space-time self-affine fractal \ensuremathΨ(x,t)\ensuremath{\Psi}(x,t)\ensuremathΨ(x,t) which verifies the predictions by Berry for ``a particle in a box,'' in addition to quantum revivals. For the diffusive case self-similar fractal evolution is also obtained. These universal fractal features of quantum theory might be useful in the field of quantum information, for creating efficient quantum algorithms, and can possibly be detectable in scattering from nanostructures.
Quantum lattices are pivotal in the burgeoning fields of quantum materials and information science. Novel experimental techniques allow the preparation and monitoring of wave packet dynamics on quantum lattices with high spatiotemporal resolution. We present an analytical study of wave packet diffusivity and diffusion length on tight-binding quantum lattices subject to stochastic noise. Our analysis reveals the crucial role of spatial coherence and predicts a set of novel phenomena: (1) noise can enhance the transient diffusivity and diffusion length of spatially extended initial states; (2) standing or traveling initial states, with large momentum, spread faster than a localized initial state and exhibit a noise-induced peak in the transient diffusivity; (3) the differences in the diffusivity or diffusion length of extended and localized initial states have a universal dependence on initial width. These predictions suggest the possibility of controlling the wave packet dynamics by spatial manipulations, which will have implications for materials science and quantum technologies.
A quantum wave with probability density , confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state . For almost all positions , the graph of the evolution of P is a fractal curve with dimension . For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension . When D = 1, there are, in addition to these generic time and space fractals, infinitely many special `quantum revival' times when P is piecewise constant, and infinitely many special spacetime slices for which the dimension of P is 5/4. If the surface of the box is a fractal with dimension , simple arguments suggest that the dimension of the time fractal is , and that of the space fractal is .
Though classical random walks have been studied for many years, research concerning their quantum analogues, quantum random walks, has only come about recently. Numerous simulations of both types of walks have been run and analyzed, and are generally well-understood. Research pertaining to one of the more important properties of classical random walks, namely, their ability to build fractal structures in diffusion-limited aggregation, has been particularly noteworthy. However, only now has research begun in this area in regards to quantum random motion. The study of random walks and the structures they build has various applications in materials science. Since all processes are quantum in nature, it is important to consider the quantum variant of diffusion-limited aggregation. Recognizing that Schrödinger equation and a classical random walk are both diffusion equations, it is possible to connect and compare them. Using similar parameters for both equations, we ran various simulations aggregating particles. Our results show that particles moving according to Schrödinger equation can create fractal structures, much like the classical random walk. Furthermore, the fractal dimensions of these quantum diffusion-limited aggregates vary between 1.43 and 2, depending on the size of the initial wave packet.
We investigate the impact of decoherence and static disorder on the dynamics of quantum particles moving in a periodic lattice. Our experiment relies on the photonic implementation of a one-dimensional quantum walk. The pure quantum evolution is characterized by a ballistic spread of a photon's wave packet along 28 steps. By applying controlled time-dependent operations we simulate three different environmental influences on the system, resulting in a fast ballistic spread, a diffusive classical walk, and the first Anderson localization in a discrete quantum walk architecture.
The common perception is that strong coupling to the environment will always render the evolution of the system density matrix quasiclassical (in fact, diffusive) in the long time limit. We present here a counterexample, in which a particle makes quantum transitions between the sites of a ddd-dimensional hypercubic lattice while strongly coupled to a bath of two-level systems that ``record'' the transitions. The long-time evolution of an initial wave packet is found to be most unusual: the mean square displacement of the particle density matrix shows long-range ballistic behavior, with $⟨{n}^{2}⟩\ensuremath{\sim}{t}^{2}$, but simultaneously a kind of weakly localized behavior near the origin. This result may have important implications for the design of quantum computing algorithms, since it describes a class of quantum walks.
We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear potential, and free particle. The box-counting dimension of the probability density ${P}_{t}(x)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}|\ensuremath{\Psi}(x,t){|}^{2}$ is shown not to change during the time evolution. We prove a universal relation Dt=1+Dx/2{D}_{t}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1+D}_{x}/2Dt​=1+Dx​/2 linking the dimensions of space cross sections Dx{D}_{x}Dx​ and time cross sections Dt{D}_{t}Dt​ of the fractal quantum carpets.