# Research Investigação

## Bose-Einstein condensate. Meanfield and quasi-classical theories

### Bose-Einstein condensation

An exciting physical phenomenon, called Bose-Einstein condensation (BEC), was predicted in 1924 and experimentally observed in 1995 in a series of remarkable experiments on vapors of rubidium and sodium atoms [1]. These experiments were considered a milestone in our understanding of Nature and three members of the above teams have been awarded by the Nobel Prize in Physics in 2001.

Those discoveries have attracted a great deal of attention of physicists almost everywhere in the World (for the most recent achievements of the theory, see http://colorado.edu/physics/2000/bec/, http://amo.phy.gasou.edu/bec.html. The phenomenon reveals essentially new physics at ultra-low temperatures- quantum phenomena on a macro-scale. In addition it opens new possibilities in creating novel devices, such as atomic lasers and interferometers, and is expected to play a prominent role in future of the nanotechnology.

The BEC behavior at low enough temperatures may be described by the so-called Gross-Pitaevskii equation which is based on a the mean-field approximation. The study of the properties of the BEC within the above approximation is one of the objectives of our group. This study is carried out in the context of an international collaboration with groups of the University of Castilla-La Mancha (Spain), University of Salerno (Italy), Tel Aviv Universty (Israel), University of Massachusetts (USA), Physico-Technical Institute of Uzbekistan Academy of Science (Tashkent, Uzbekistan), Lukin's Institute of Physical Problems (Zelenograd, Russia), and Institute of Spectroscopy (Troitsk, Russia).

More specific topics of the research are listed below.

### BEC in optical lattices

After pioneering experimental works by Anderson and Kasevich [2] where BEC was trapped in an optical lattice, i.e. the condensed atoms were collected in a periodic potential created by a standing wave originated by two counter-propagating laser beams, this system became the subject of rapidly increasing experimental and theoretical studies.

The interest of our group is concentrated on aspects of the BEC dynamics in optical lattices such as modulational instability, existence and generation of solitary matter waves in optical lattices, control of matter waves in optical lattices by means of Feshbach resonance (dynamically changing the scattering length), multi-component and atomic-molecular condensates in optical lattices, etc.

We pay special attention to the mathematical justification of different approaches and especially to the applicability of the tight-binding approximation, or in other words to intrinsic relations between partial differential equations with periodic coefficients and simplified lattice equations.

For the related publications of the group see http://alf1.cii.fc.ul.pt/~konotop/Publications.htm.

### BEC hydrodynamics and beyond in two and three dimensions

It turns out that the description of BEC at some stages of its evolution can be simplified and reduced to the hydrodynamics of a superfluid. This approach allows one to describe the asymmetry of expansion, shock wave formation, and some features of the collapse in BEC's (with attractive interactions, in the last case). Corresponding theoretical investigations are carried out by the group. We also consider more refined approximations going beyond simple hydrodynamics, as well as dynamics of the BEC in the presence of long-range interactions, leading to non-local terms in the Gross-Pitaevskii equation. Also, we are particularly interested in the long-term dynamics of solitary waves in BEC's, when the finite dimensions of a magnetic trap become important.

For the related publications of the group see http://alf1.cii.fc.ul.pt/~konotop/Publications.htm.

### WKB approach for nonlinear problems

A fundamental result of quantum mechanics - the Bohr-Sommerfeld quantization rule - is one of the main outputs of the quasi-classical approximation. From the mathematical point of view, a singular perturbation problem arises, when a natural small parameter of the problem multiplies the highest derivative. The Bohr-Sommerfeld rule is an example of the application of such a technique in the case of Schrödinger's equation. As such the quasi-classical approximation is of fundamental importance for many other branches of physics, such as nonlinear optics, acoustics, hydrodynamics and plasma physics, among others.

On the other hand, the developments in the physics of ultracold quantum gases naturally result in a new formulation of the quasi-classical approach. Indeed, the quantum-mechanical Schrödinger equation is linear while its mean-field counterpart in the many-body case, i.e., the so-called Gross-Pitaevskii equation, is essentially nonlinear; the nonlinearity results from the interactions among particles. In this context, aspects of the quasi-classical approach have been recently addressed where attention has been focused on the ground state. These can be viewed as generalizations of the well-known Thomas-Fermi (TF) approximation. We address the question of a generalization of the Bohr-Sommerfeld quantization rule which takes into account also higher levels of the spectrum: In other words we extract a "nonlinear WKB" heory, as well as the nonlinearity-induced corrections to the Bohr-Sommerfeld quantization rule.

### References

- M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, Science 269, 198 (1995); K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995)
- Science 282, 1686 (1998)