Abstract
Cold atoms in dissipative optical lattices exhibit an unusual transport behaviour that cannot be described within Boltzmann–Gibbs statistical mechanics. New theoretical tools and concepts need thus be developed to account for their observable macroscopic properties. Here we review recent progress achieved in the study of these processes. We emphasize the generality of the findings for a broad class of physical, chemical and biological systems, and discuss open questions and perspectives for future work.
This is a preview of subscription content, access via your institution
Access options
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Dorfman, J. R. An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge Univ. Press, 1999).
Lebowitz, J. L. & Penrose, O. Modern ergodic theory. Phys. Today 26, 23–30 (February, 1973).
Bouchaud, J. P. & Georges, A. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990).
Shlesinger, M. F., Zaslavsky, G. M. & Klafter, J. Strange kinetics. Nature 363, 31–37 (1993).
Bardou, F., Bouchaud, J. P., Emile, O., Aspect, O. & Cohen-Tannoudji, C. Subrecoil laser cooling and Lévy flights. Phys. Rev. Lett. 72, 203–206 (1994).
Stefani, F. D., Hoogenboom, J. P. & Barkai, E. Beyond quantum jumps: Blinking nanoscale light emitters. Phys. Today 62, 34–39 (February, 2009).
Klafter, J., Shlesinger, M. F. & Zumofen, G. Beyond Brownian motion. Phys. Today 10, 33–39 (February, 1997).
Metzler, R. & Klafter, J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339, 1–77 (2000).
Sokolov, I. M., Klafter, J. & Blumen, A. Fractional kinetics. Phys. Today 55, 48–54 (November 2002).
Barkai, E., Garini, Y. & Metzler, R. Strange kinetics of single molecules in living cells. Phys. Today 65, 29–35 (August, 2012).
Grynberg, G. & Mennerat-Robilliard, C. Cold atoms in dissipative optical lattices. Phys. Rep. 355, 335–451 (2001).
Gardiner, C. W. & Zoller, P. Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, 2004).
Castin, Y., Dalibard, J. & Cohen-Tannoudji, C. in Light Induced Kinetic Effects on Atoms, Ions and Molecules (eds Moi, L. et al.) 5–24 (ETS Editrice, 1991).
Marksteiner, S., Ellinger, K. & Zoller, P. Anomalous diffusion and Lévy walks in optical lattices. Phys. Rev. A 53, 3409–3430 (1996).
Kondrashin, M. P. & Yakovlev, V. P. Bipotential Motion and Anomalous Transport in Optical Lattices. Laser Phys. 11, 486–492 (2001).
Lutz, E. Anomalous diffusion and Tsallis statistics in an optical lattice. Phys. Rev. A 67, 051402R (2003).
Breuer, H. P. & Petruccione, F. The Theory of Open Quantum Systems (Oxford Univ. Press, 2007).
Hodapp, T. W., Gerz, C., Furtlehner, C., Westbrook, C. I., Phillips, W. D. & Dalibard, J. Three-dimensional spatial diffusion in optical molasses. Appl. Phys. B 60, 135–143 (1995).
Katori, H., Schlipf, S. & Walther, H. Anomalous dynamics of a single ion in an optical lattice. Phys. Rev. Lett. 79, 2221–2224 (1997).
Schlipf, S., Katori, H., Perotti, L. & Walther, H. Diffusion of a single ion in a one-dimensional optical lattice. Opt. Express 3, 97–103 (1998).
Perotti, L., Alekeseev, V. & Walther, H. Transport of a single ion in an optical lattice: Spatial diffusion and energy. Opt. Commun. 183, 73–94 (2000).
Wickenbrock, A., Holz, P. C., Abdul Wahab, N. A., Phoonthong, P., Cubero, D. & Renzoni, F. Vibrational mechanics in an optical lattice: Controlling transport via potential renormalization. Phys. Rev. Lett. 108, 020603 (2012).
Douglas, P., Bergamini, S. & Renzoni, F. Tunable Tsallis distributions in dissipative optical lattices. Phys. Rev. Lett. 96, 110601 (2006).
Metcalf, H. J. & van der Straten, P. Laser Cooling and Trapping (Springer, 1999).
Sagi, Y., Brook, M., Almog, I. & Davidson, N. Observation of anomalous diffusion and fractional self-similarity in one dimension. Phys. Rev. Lett. 108, 093002 (2012).
Kessler, D. A. & Barkai, E. Theory of fractional Lévy kinetics for cold atoms diffusing in optical lattices. Phys. Rev. Lett. 108, 230602 (2012).
Lévy, P. Théorie de l’Addition des Variables Aléatoires (Gauthier-Villars, 1953).
Dechant, A. & Lutz, E. Anomalous spatial diffusion and multifractality in optical lattices. Phys. Rev. Lett. 108, 230601 (2012).
Stanley, H. E. & Meakin, P. Multifractal phenomena in physics and chemistry. Nature 335, 405–409 (1988).
Lutz, E. Power-law tail distributions and nonergodicity. Phys. Rev. Lett. 93, 190602 (2004).
Dechant, A., Lutz, A., Kessler, D. A. & Barkai, E. Super-aging correlation function and ergodicity breaking in logarithmic potentials. Phys. Rev. E 84, 051124 (2012).
Dechant, A., Lutz, A., Kessler, D. A. & Barkai, E. Fluctuations of time averages for Langevin dynamics in a binding force field. Phys. Rev. Lett. 107, 240603 (2011).
Aaronson, J. An Introduction to Infinite Ergodic Theory (American Mathematical Society, 1997).
Korabel, N. & Barkai, E. Infinite invariant density determines statistics of time averages for weak chaos. Phys. Rev. Lett. 108, 060604 (2012).
Akimoto, T. Distributional response to biases in deterministic superdiffusion. Phys. Rev. Lett. 108, 164101 (2012).
Kessler, D. A. & Barkai, E. Infinite covariant density for diffusion in logarithmic potentials and optical lattices. Phys. Rev. Lett. 105, 120602 (2010).
Dechant, A., Lutz, A., Kessler, D. A. & Barkai, E. Solution of the Fokker–Planck equation with a logarithmic potential. J. Stat. Phys. 145, 1524–1545 (2011).
Dechant, A., Lutz, A., Kessler, D. A. & Barkai, E. Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials. Phys. Rev. E 85, 051124 (2012).
Manning, G. S. Limiting laws and counterion condensation in polyelectrolyte solutions I. Colligative properties. J. Chem. Phys. 51, 924–933 (1969).
Bouchet, F. & Dauxois, T. Prediction of anomalous diffusion and algebraic relaxations for long-range interacting systems, using classical statistical mechanics. Phys. Rev. E 72, 045103(R) (2005).
Chavanis, P.H. & Lemou, M. Kinetic theory of point vortices in two dimensions: Analytical results and numerical simulations. Eur. Phys. J. B 59, 217–247 (2007).
Sire, C. & Chavanis, P. H. Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions. Phys. Rev. E 66, 046133 (2002).
Chavanis, P. H. Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions. Eur. Phys. J. B 57, 391–409 (2007).
Fogedby, H. C. & Metzler, R. DNA bubble dynamics as a quantum Coulomb problem. Phys. Rev. Lett. 98, 070601 (2007).
Bar, A., Kafri, Y. & Mukamel, D. Loop dynamics in DNA denaturation. Phys. Rev. Lett. 98, 038103 (2007).
Levine, E., Mukamel, D. & Schütz, G.M. Long-range attraction between probe particles mediated by a driven fluid. Europhys. Lett. 70, 565–571 (2005).
Dalibard, J. & Cohen-Tannoudji, C. Laser cooling below the Doppler limit by polarization gradients: Simple theoretical models. J. Opt. Soc. Am. B 6, 2023–2045 (1989).
Cohen-Tannoudji, C. & Phillips, W. D. New mechanisms for laser cooling. Phys. Today 43, 33–40 (October, 1990).
Cohen-Tannoudji, C. Manipulating atoms with photons. Rev. Mod. Phys. 70, 707–719 (1998).
Acknowledgements
We thank the DFG (Contract No 1382/4-1) and the Leverhulme Trust for financial support.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
About this article
Cite this article
Lutz, E., Renzoni, F. Beyond Boltzmann–Gibbs statistical mechanics in optical lattices. Nature Phys 9, 615–619 (2013). https://doi.org/10.1038/nphys2751
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys2751
This article is cited by
-
Distribution of energy in the ideal gas that lacks equipartition
Scientific Reports (2023)
-
Neural complexity through a nonextensive statistical–mechanical approach of human electroencephalograms
Scientific Reports (2023)
-
A generalization of the standard map and its statistical characterization
Scientific Reports (2022)
-
Initial measurement of ion nonextensive parameter with geodesic acoustic mode theory
Scientific Reports (2022)
-
Connecting complex networks to nonadditive entropies
Scientific Reports (2021)