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Physical review. B, Condensed matter

Quantum Brownian motion on a lattice

Ping Sheng, Zhao-Qing Zhang

Published Nov 1, 1993• 6 citations
Publisher

Abstract

For a quantum particle on a one-dimensional lattice, the dynamics under the influence of a time-dependent, white-noise potential is shown to differ significantly from the classical diffusive behavior. In particular, the square of the distance displaced from origin is proportional to ${\mathit{t}}^{0.5--0.6}$. The origin of this dispersive behavior is traced to the special properties of the force-force correlation function, which integrates to zero and has a magnitude varying as ${\mathit{t}}^{\mathrm{\ensuremath{-}}0.5}$. These characteristics are attributed to the continuous spread of the quantum wave packet as well as its interaction with the discrete lattice.

DOI
10.1103/physrevb.48.12609

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For a quantum particle on a one-dimensional lattice, the dynamics under the influence of a time-dependent, white-noise potential is shown to differ significantly from the classical diffusive behavior. In particular, the square of the distance displaced from origin is proportional to t0.5−−0.6{\mathit{t}}^{0.5--0.6}t0.5−−0.6. The origin of this dispersive behavior is traced to the special properties of the force-force correlation function, which integrates to zero and has a magnitude varying as t\ensuremath−0.5{\mathit{t}}^{\mathrm{\ensuremath{-}}0.5}t\ensuremath−0.5. These characteristics are attributed to the continuous spread of the quantum wave packet as well as its interaction with the discrete lattice.
We address memory effects and diffusive properties of a continuous-time quantum walk on a one-dimensional percolation lattice affected by spatially correlated random telegraph noise. In particular, by introducing spatially correlated time-dependent fluctuations in nearest-neighbor hopping amplitudes, we describe random domains characterized by global noise. The resulting open dynamics of the walker is then unraveled by an ensemble average over all the noise realizations. Our results show that time-dependent noise assisted by spatial correlations leads to strong memory effects in the walker dynamics and to robust diffusive behavior against the detrimental action of uncorrelated noise. We also show that spatially correlated classical noise enhances localization breaking, thus making a quantum particle spread on longer distances across the lattice.
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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.
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A quantum-mechanical analysis of hyperfast (faster than ballistic) diffusion of a quantum wave packet in random optical lattices is presented. The main motivation of the presented analysis is experimental demonstrations of hyperdiffusive spreading of a wave packet in random photonic lattices [L. Levi et al., Nature Phys. 8, 912 (2012)]. A rigorous quantum-mechanical calculation of the mean probability amplitude is suggested, and it is shown that the power-law spreading of the mean-squared displacement (MSD) is $\ensuremath{\langle}{x}^{2}(t)\ensuremath{\rangle}\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$, where $2<\ensuremath{\alpha}\ensuremath{\le}3$. The values of the transport exponent $\ensuremath{\alpha}$ depend on the correlation properties of the random potential V(x,t)V(x,t)V(x,t), which describes random inhomogeneities of the medium. In particular, when the random potential is $\ensuremath{\delta}$ correlated in time, the quantum wave packet spreads according Richardson turbulent diffusion with the MSD $\ensuremath{\sim}{t}^{3}$. Hyperdiffusion with $\ensuremath{\alpha}=12/5$ is also obtained for arbitrary correlation properties of the random potential.