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EN
We review a progress in understanding of statistical properties of a quantum degenerate Bose gas. We show how the Maxwell demon ensemble helps to compute fluctuations of the Bose-Einstein condensate of an ideal Bose gas according to the microcanonical ensemble. Then, we review a method of measuring these fluctuations. Using a soluble model of interacting Bose gas we also stress the importance of higher-order correlation functions. Finally, we review our novel computational method of studying an interacting Bose gas near its critical temperature.
EN
We study an ultracold dilute gas of bosonic atoms in an optical lattice induced by intersecting laser beams. As a first approximation we neglect confining potential and atom-atom interactions. In this case the Gross-Pitaevskij equation reduces to simple Mathieu equation. Upon choosing periodic boundary conditions this problem has well known periodic solution. This simple picture allows to demonstrate localization of the wave packet and formation of the band structure. We calculate spectrum of the excited states as a function of the strength of modulating potential and using a standard adiabaticity criterion we predict the most efficient way to ramp up optical lattice, without higher state excitation. Finally, we discuss the influence of the atom-atom interaction (nonlinearity) on the adiabaticity of the process.
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Quantum Optics of Atomic Wave-Packets

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EN
Recent progress in production and control of the atomic wave-packets calls for re-examination of the most fundamental quantum optical processes. We have examined the evolution of the atomic wave-packet resonantly driven by the monochromatic laser beam and the spontaneous emission from an extended atom. We stress the role of Doppler shifts in the dephasing of the atomic dipole. New effects of single laser pulse photon echo and homogeneous Doppler modification of the emission line are discussed.
EN
We study one- and two-dimensional systems of two interacting particles in a time dependent harmonic potential. In a case of one-dimensional geometry a frequency of the potential varies periodically, while in the two-dimen- sional~case the harmonic potential rotates with a constant angular velocity. We show that depending on the driving frequency the distance between the particles can either explode or stay bound. Repulsive interaction can prevent the explosion, which seems quite counter-intuitive. Our work is related to Ecole Normale Supérieure experiment and shows that the effect found there is purely classical.
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