Normal and Anomalous Averages for Systems with Bose-Einstein Condensate

The comparative behaviour of normal and anomalous averages as functions of momentum or energy, at different temperatures, is analysed for systems with Bose-Einstein condensate. Three qualitatively distinct temperature regions are revealed: The critical region, where the absolute value of the anomalous average, for the main energy range, is much smaller than the normal average. The region of intermediate temperatures, where the absolute values of the anomalous and normal averages are of the same order. And the region of low temperatures, where the absolute value of the anomalous average, for practically all energies, becomes much larger than the normal average. This shows the importance of the anomalous averages for the intermediate and, especially, for low temperatures, where these anomalous averages cannot be neglected.Comment: Latex file, 17 pages, 6 figure

From Coherent Modes to Turbulence and Granulation of Trapped Gases

The process of exciting the gas of trapped bosons from an equilibrium initial state to strongly nonequilibrium states is described as a procedure of symmetry restoration caused by external perturbations. Initially, the trapped gas is cooled down to such low temperatures, when practically all atoms are in Bose-Einstein condensed state, which implies the broken global gauge symmetry. Excitations are realized either by imposing external alternating fields, modulating the trapping potential and shaking the cloud of trapped atoms, or it can be done by varying atomic interactions by means of Feshbach resonance techniques. Gradually increasing the amount of energy pumped into the system, which is realized either by strengthening the modulation amplitude or by increasing the excitation time, produces a series of nonequilibrium states, with the growing fraction of atoms for which the gauge symmetry is restored. In this way, the initial equilibrium system, with the broken gauge symmetry and all atoms condensed, can be excited to the state, where all atoms are in the normal state, with completely restored gauge symmetry. In this process, the system, starting from the regular superfluid state, passes through the states of vortex superfluid, turbulent superfluid, heterophase granular fluid, to the state of normal chaotic fluid in turbulent regime. Both theoretical and experimental studies are presented.Comment: Latex file, 25 pages, 4 figure