Anomalous diffusion of atoms in a 1D damped lattice

2010

We solve the Fokker-Planck equation for Brownian motion in a logarithmic potential. When the diffusion constant is below a critical value the solution approaches a non-normalizable scaling state, reminiscent of an infinite invariant density. With this non-normalizable density we obtain the phase diagram of anomalous diffusion for this important process. We briefly discuss the consequence for a range of physical systems including atoms in optical lattices and charges in vicinity of long polyelectrolytes. Our work explains in what sense the infinite invariant density and not Boltzmann's equilibrium describes the long time limit of these systems. PACS numbers: 05.40.-a,05.10.Gg

Physical Review A, 2012

Using the micro-canonical picture of transport -a framework ideally suited to describe the dynamics of closed quantum systems such as ultra-cold atom experiments -we show that the exact dynamics of non-interacting fermions and bosons exhibit very different transport properties when the system is initially in the ground state and set out of equilibrium by removing the particles from half of the lattice. We find that fermions, with or without a harmonic confining potential, rapidly develop a finite quasi steady-state current reminiscent of electronic transport in nanoscale systems, whereas the bosonic current exhibits strong oscillatory behavior that decays into a steady-state of zero current only in the thermodynamic limit. These differences appear most strikingly in the different particle number fluctuations on half of the lattice as a consequence of the spin statistics in the two cases. These predictions can be readily verified experimentally. PACS numbers: 72.10.-d, 67.10.Jn, 03.75.Mn Ultra-cold atoms loaded in various optical lattices help visualize and gain a deeper understanding of many exciting quantum phenomena. For instance, Hanbury-Brown-Twiss types of experiments measuring spatial correlations of density fluctuations in non-interacting bosons [1] and fermions [2] clearly demonstrate bunching and anti-bunching effects, which are definite signatures of the underlying spin statistics. On the other hand, bosonic and fermionic Mott insulators [3, 4] have been realized experimentally and the statistics do not seem to play an important role in their thermodynamic properties if magnetic phenomena due to multi-component fermions are not considered. It is then clear that the question of how and when statistics affect an atomic system is of great fundamental interest and worth further investigations.

First-Passage Phenomena and Their Applications, 2014

Physical Review A, 2009

We study the behaviour of an ultracold atomic gas of bosons in a bichromatic lattice, where the weaker lattice is used as a source of disorder. We numerically solve a discretized mean-field equation, which generalizes the one-dimensional Aubry-Andrè model for particles in a quasi-periodic potential by including the interaction between atoms. We compare the results for commensurate and incommensurate lattices. We investigate the role of the initial shape of the wavepacket as well as the interplay between two competing effects of the interaction, namely self-trapping and delocalization. Our calculations show that, if the condensate initially occupies a single lattice site, the dynamics of the interacting gas is dominated by self-trapping in a wide range of parameters, even for weak interaction. Conversely, if the diffusion starts from a Gaussian wavepacket, self-trapping is significantly suppressed and the destruction of localization by interaction is more easily observable.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2009

We show that the transition from Gaussian to the q-Gaussian distributions occurring in atomic transport in dissipative optical lattices can be interpreted as self-organization by recourse to a modified version of Klimontovich's S-theorem. As a result, we find that self-organization is possible in the transition regime, only where the second moment˙p 2¸i s finite. Therefore, the nonadditivity parameter q is confined within the range 1 < q < 5 3 , although whole spectrum of q values i.e., 1 < q < 3, is considered theoretically possible. The range of q values obtained from the modified S-theorem is also confirmed by the experiments carried out by Douglas et al. [Phys. Rev. Lett. 96, 110601 (2006)].

The European Physical Journal D, 2001

We present a detailed experimental study of a threedimensional lin⊥lin bright optical lattice. Measurements of the atomic temperature and spatial diffusion coefficients are reported for different angles between the lattice beams, i.e. for different lattice constants. The experimental findings are interpretated with the help of numerical simulations. In particular we show, both experimentally and theoretically, that the temperature is independent of the lattice constant.

Physical Review Letters, 2003

We demonstrate the phenomenon of directed diffusion in a symmetric periodic potential. This has been realized with cold atoms in a one-dimensional dissipative optical lattice. The stochastic process of optical pumping leads to a diffusive dynamics of the atoms through the periodic structure, while a zero-mean force which breaks the temporal symmetry of the system is applied by phase-modulating one of the lattice beams. The atoms are set into directed motion as a result of the breaking of the temporal symmetry of the system.

Physical review letters, 2011

We consider a cloud of fermionic atoms in an optical lattice described by a Hubbard model with an additional linear potential. While homogeneous interacting systems mainly show damped Bloch oscillations and heating, a finite cloud behaves differently: It expands symmetrically such that gains of potential energy at the top are compensated by losses at the bottom. Interactions stabilize the necessary heat currents by inducing gradients of the inverse temperature 1/T, with T&lt;0 at the bottom of the cloud. An analytic solution of hydrodynamic equations shows that the width of the cloud increases with t^{1/3} for long times consistent with results from our Boltzmann simulations.

Physical Review E, 2006

We study the persistent random walk of photons on a one-dimensional lattice of random transmittances. Transmittances at different sites are assumed independent, distributed according to a given probability density f͑t͒. Depending on the behavior of f͑t͒ near t = 0, diffusive and subdiffusive transports are predicted by the disorder expansion of the mean square-displacement and the effective medium approximation. Monte Carlo simulations confirm the anomalous diffusion of photons. To observe photon subdiffusion experimentally, we suggest a dielectric film stack for realization of a distribution f͑t͒.

Physical Review B, 1982

The diffusion process of particles in concentrated lattice gases is investigated, with the assumption of a fcc lattice and nearest-neighbor attraction (but double occupancy of sites being forbidden). Both the self-diffusion of tagged particles and the collective diffusion, by which concentration fluctuations decay, are studied. Apart from a mean-field treatment, the various diffusion coefficients are estimated by Monte Carlo techniques and interpreted in terms of static long-and short-range order (i.e. , unmixing) occurring in this model system. Collective diffusion is studied by direct simulation of linear response to wave-vector-dependent "fields. " Near the critical temperature, pronounced critical slowing down of the collective diffusion coefficient is observed. The self-diffusion constant stays finite but exhibits a singularity of its slope. The correlation factor for self-diffusion is found to be practically independent of temperature. A qualitative discussion of the behavior inside of the mixed-phase region is also given.

arXiv: Statistical Mechanics, 2018

Motivated by investigation into the mechanisms that yield directed diffusion in a symmetric periodic potential, Schiavoni et al. [1] have studied experimentally and numerically cold atoms in a one-dimensional dissipative optical lattice where directed motion appears as a result of the breaking of the system’s temporal symmetry after applying a biharmonic phase modulation to one of the lattice beams. In the accelerated reference frame, the atoms experience a stationary optical potential together with an inertial force

In this work, the effect of fluctuations in a disordered square lattice on diffusion of a test particle is studied using kinetic Monte Carlo simulations. Diffusion is relevant to a wide variety of problems, both within physics and outside of physics. Kinetic effects in diffusion are often hidden in a thermodynamical description of the problem. In this work, no assumptions based on energy are made, and diffusion occurs purely based on the attempt rate of the test particle and the occupation and fluctuation rate of the lattice. Although the average transition rate of the particle is the same for a static or fluctuating lattice with specific occupation, the diffusion constant is kinetically affected in a fluctuating, disordered lattice. If the lattice fluctuates faster than the attempt rate of the particle, diffusion is controlled by the attempt rate of the particle. However, if the lattice fluctuates slower than the attempt rate of the particle, diffusion is affected by the fluctuations. The slower the lattice fluctuates, the lower the diffusion constant. Furthermore, it is found that for fast fluctuating lattices, diffusion is due to Brownian motion. If the lattice fluctuates slower than the particle, diffusion becomes anomalous depending on the occupation of the lattice.

Journal of Statistical Mechanics: Theory and Experiment, 2017

We analyse diffusion at low temperature by bringing the fluctuation-dissipation theorem (FDT) to bear on a physically natural, viscous response-function R(t). The resulting diffusion-law exhibits several distinct regimes of time and temperature, each with its own characteristic rate of spreading. As with earlier analyses, we find logarithmic spreading in the quantum regime, indicating that this behavior is robust. A consistent R(t) must satisfy the key physical requirements of Wightman positivity and passivity, and we prove that ours does so. We also prove in general that these two conditions are equivalent when the FDT holds. Given current technology, our diffusion law can be tested in a laboratory with ultra cold atoms.

Physical Review E, 2015

A quantum-mechanical analysis of hyper-fast (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 hyper-diffusive 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 x 2 (t) ∼ t α , where 2 < α ≤ 3. The values of the transport exponent α 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 δ correlated in time, the quantum wave packet spreads according Richardson turbulent diffusion with the MSD ∼ t 3 . Hyper-diffusion with α = 12/5 is also obtained for arbitrary correlation properties of the random potential.

Fractional photon-assisted tunneling is investigated both numerically and analytically in a double-well lattice. While integer photon-assisted tunneling is a single-particle effect, fractional photon-assisted tunneling is an interaction-induced many-body effect. Double-well lattices with few particles in each double well are ideal to study this effect far from the mean-field effects. It is predicted that the 1/4-resonance is observable in such systems. Fractional photon-assisted tunneling provides a physically relevant model, for which N-th order time-dependent perturbation theory can be large although all previous orders are small. All predicted effects will be observable with an existing experimental setup.