Semiclassical expansion of quantum gases into a vacuum

In the framework of the Gross-Pitaevskii equation, we consider the problem of the expansion of quantum gases into a vacuum. For them, the chemical potential μ has a power-law dependence on the density n with the exponent ν = 2/D, whereD is the space dimension. For gas condensates of Bose atoms as the temperature T → 0, s scattering gives the main contribution to the interaction of atoms in the leading order in the gas parameter. Therefore, the exponent ν = 1 for an arbitrary D. In the three-dimensional case, ν = 2/3 is realized for condensates of Fermi atoms in the so-called unitary limit. For ν = 2/D, the Gross-Pitaevskii equation has an additional symmetry under Talanov transformations of the conformal type, which were first found for the stationary self-focusing of light. A consequence of this symmetry is the virial theorem relating the average size R of an expanding gas cloud to its Hamiltonian. The quantity R asymptotically increases linearly with time as t →∞. In the semiclassical limit, the equations of motion coincide with those of the hydrodynamics of an ideal gas with the adiabatic exponent γ = 1+2/D. In this approximations, self-similar solutions describe angular deformations of the gas cloud against the background of the expanding gas in the framework of equations of the Ermakov-Ray-Reid type.