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arXiv:1508.05889 · Physical Review E

Hyperdiffusion of quantum waves in random photonic lattices

Alexander Iomin

Published Aug 24, 2015• 11 citations
arXivHTMLPDF

Abstract

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 $V(x,t)$, 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.

DOI
10.1103/physreve.92.022139

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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 ${\mathit{t}}^{\mathrm{\ensuremath{-}}0.5}$. These characteristics are attributed to the continuous spread of the quantum wave packet as well as its interaction with the discrete lattice.
Quantum diffusion (QD) is studied via dissipative Madelung hydrodynamics. Initially, the wave packet spreads ballistically, then passes for an instant through normal diffusion and later tends asymptotically to a sub-diffusive law. It is shown that the apparent QD coefficient is not a universal physical parameter because it depends on the initial wave packet preparation. The overdamped QD of an electron in the field of a periodic potential is also investigated; in this case, the wave packet spreads logarithmically in time. Thermo-QD of heavier particles such as hydrogen, deuterium and tritium atoms in periodic potentials is studied and a simple estimate of the tunneling effect is obtained in the framework of a quasi-equilibrium semiclassical approach. The effective thermo-quantum temperature is also discussed in relation to the known temperature dependence of muon diffusivity in solids.
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.
The quantum wave-function of a massive particle with small initial uncertainties (consistent with the uncertainty relation) is believed to spread very slowly, so that the dynamics is deterministic. This assumes that the classical motions for given initial data are unique. In fluid turbulence non-uniqueness due to "roughness" of the advecting velocity field is known to lead to stochastic motion of classical particles. Vanishingly small random perturbations are magnified by Richardson diffusion in a "nearly rough" velocity field so that motion remains stochastic as the noise disappears, or classical spontaneous stochasticity, . Analogies between stochastic particle motion in turbulence and quantum evolution suggest that there should be quantum spontaneous stochasticity (QSS). We show this for 1D models of a particle in a repulsive potential that is "nearly rough" with V(x)∼C∣x∣1+αV(x) \sim C|x|^{1+α}V(x)∼C∣x∣1+α at distances ∣x∣≫ℓ|x|\gg \ell∣x∣≫ℓ , for some UV cut-off ℓ\ellℓ, and for initial Gaussian wave-packet centered at 0. We consider the WKB limit with ℏ/m→0\hbar/m \to 0ℏ/m→0, then position-spread σ→0σ\to 0σ→0. The limit of the position density is non-deterministic, with equal probabilities of the particle following either of two, non-unique classical solutions and with "Richardson-like" spreading of the wave-packet. The splitting of the wave-packet occurs in a very short time t∼(ℓ1+α/C)1/2t\sim(\ell^{1+α}/C)^{1/2}t∼(ℓ1+α/C)1/2 for small ℓ\ellℓ. QSS also occurs in the scattering of a wave-packet off the rough potential, for careful fine-tuning. We also consider other semi-classical limits by a numerical solution of the Schrödinger equation, observing QSS both in position- and momentum-space. Although the wave-function remains split into two widely separated branches in the classical limit, rapid phase oscillations within each branch prevent any coherent superposition. These effects should be observable in laboratory experiments.
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.
We present a rigorous study of quantum diffusion of a relativistic particle subjected to a time-dependent random potential with δ correlation in time. We find that in the asymptotic time limit the particle wave packet spreads ballistically in contrast with the nonrelativistic case, which in the same situation exhibits superballistic diffusion. The relativistic suppression of wave packet diffusion is discussed in connection with statistical conservation laws that follow from relativistic dynamics.