Infinite density for cold atoms in shallow optical lattices

Holz, P.P.; Dechant, Andreas; Lutz, Eric

doi:10.1209/0295-5075/109/23001

Cited by 28 publications

(32 citation statements)

References 41 publications

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“…Previous work [23,[38][39][40][42][43][44][45] on applications of the infinite density in physics, dealt with bounded systems which attain an equilibrium. For example the momentum distribution of cold atoms where the Gibbs-measure is finite the infinite density describes the large rare fluctuations of the kinetic energy [23,45]. Or intermittent maps with unstable fixed points, e.g., the Pomeau-Manneville transformation on the unit interval [38,40].…”

Section: Introductionmentioning

confidence: 99%

Infinite densities for Lévy walks

et al. 2014

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Motion of particles in many systems exhibits a mixture between periods of random diffusive like events and ballistic like motion. In many cases, such systems exhibit strong anomalous diffusion, where low order moments |x(t)| q with q below a critical value qc exhibit diffusive scaling while for q > qc a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent α and the diffusion coefficient Kα. We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.

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Section: Introductionmentioning

confidence: 99%

Infinite densities for Lévy walks

et al. 2014

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“…This density is not normalizable, despite being a limit of a properly normalized probability law. While such objects play a key role in the mathematical field of infinite ergodic theory [27][28][29], their usage is less common in physical models, such as subdiffusion on intermittent maps [30][31][32][33], diffusion in logarithmic potentials [34][35][36] and recently strong anomalous diffusion [37]. Our goal is to showcase the scope and applicability of infinite densities and familiarize the reader with its peculiar properties.…”

Section: Introductionmentioning

confidence: 99%

Fluctuations around equilibrium laws in ergodic continuous-time random walks

Schulz

Barkai

2015

Phys. Rev. E

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We study occupation time statistics in ergodic continuous-time random walks. Under thermal detailed balance conditions, the average occupation time is given by the Boltzmann-Gibbs canonical law. But close to the non-ergodic phase, the finite-time fluctuations around this mean are large and nontrivial. They exhibit dual time scaling and distribution laws: the infinite density of large fluctuations complements the Lévy-stable density of bulk fluctuations. Neither of the two should be interpreted as a stand-alone limiting law, as each has its own deficiency: the infinite density has an infinite norm (despite particle conservation), while the stable distribution has an infinite variance (although occupation times are bounded). These unphysical divergences are remedied by consistent use and interpretation of both formulas. Interestingly, while the system's canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion, they also control the infinite and Lévy-stable densities of fluctuations. The duality of stable and infinite densities is in fact ubiquitous for these dynamics, as it concerns the time averages of general physical observables.

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“…An intriguing system to look at in this context is that of anomalous dynamics for which the mean square displacement (MSD) scales as x 2 ∼ t 2α , with α = 1/2. This type of dynamics, found in a wide variety of systems in nature ranging from dynamics of "bubbles" in denaturing DNA molecules [1], through fluctuations in the stockmarket [2] to models describing brief awakenings in the course of a night's sleep [3], is generally non-universal and system-dependent [4][5][6].A uniquely interesting model system for the study of anomalous diffusion is that of cold atoms diffusing in a dissipative 1D lattice, closely related to Lévy walks and motion in logarithmic potentials, displaying such phenomena as the breakdown of ergodicity and of equipartition, memory effects and slow relaxation to equilibrium [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The major advantage of such a system is the high degree of control it enables over the physical parameters governing the dynamics.…”

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confidence: 99%

Observing Power-Law Dynamics of Position-Velocity Correlation in Anomalous Diffusion

et al. 2017

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In this letter we present a measurement of the phase-space density distribution (PSDD) of ultracold 87 Rb atoms performing 1D anomalous diffusion. The PSDD is imaged using a direct tomographic method based on Raman velocity selection. It reveals that the position-velocity correlation function Cxv(t) builds up on a timescale related to the initial conditions of the ensemble and then decays asymptotically as a power-law. We show that the decay follows a simple scaling theory involving the power-law asymptotic dynamics of position and velocity. The generality of this scaling theory is confirmed using Monte-Carlo simulations of two distinct models of anomalous diffusion.The phase-space density distribution (PSDD) contains information concerning the degrees of freedom of a system and allows calculation of any observable. An intriguing system to look at in this context is that of anomalous dynamics for which the mean square displacement (MSD) scales as x 2 ∼ t 2α , with α = 1/2. This type of dynamics, found in a wide variety of systems in nature ranging from dynamics of "bubbles" in denaturing DNA molecules [1], through fluctuations in the stockmarket [2] to models describing brief awakenings in the course of a night's sleep [3], is generally non-universal and system-dependent [4][5][6].A uniquely interesting model system for the study of anomalous diffusion is that of cold atoms diffusing in a dissipative 1D lattice, closely related to Lévy walks and motion in logarithmic potentials, displaying such phenomena as the breakdown of ergodicity and of equipartition, memory effects and slow relaxation to equilibrium [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The major advantage of such a system is the high degree of control it enables over the physical parameters governing the dynamics. One of the fundamental insights that can be obtained from the PSDD of such a system is the phase-space cross correlation between position and velocity C xv . C xv can reveal the fingerprint of the underlying model and in particular is essential for understanding concepts and techniques such as adiabatic cooling in lattices [24], stochastic cooling [25], point source atom interferometry [26,27] and enhanced velocity resolution [28,29], alongside elementary notions in quantum mechanics [30]. These correlations have been surprisingly overlooked in both theory and experiment, perhaps due to the lack of a direct method for imaging the phasespace of atomic clouds, which does not require cumbersome mathematical tools or a specific potential [31][32][33][34][35]. No analysis of the dynamics of the correlations has been reported to the best of our knowledge.In this letter we analyze and measure the dynamics of the position-velocity correlation of an ensemble of classical particles, originating from a point-like source and undergoing one-dimensional anomalous super-diffusion. The measurement is done using a new tomographic method for direct phase-space imaging, utilizing a combination of the straightforward tools of absorpt...

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Infinite density for cold atoms in shallow optical lattices

Cited by 28 publications

References 41 publications

Infinite densities for Lévy walks

Infinite densities for Lévy walks

Fluctuations around equilibrium laws in ergodic continuous-time random walks

Observing Power-Law Dynamics of Position-Velocity Correlation in Anomalous Diffusion

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