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 $\mu$ has a power-law dependence on the density $n$ with the exponent $\nu=2/D$, where $D$ is the space dimension. For gas condensates of Bose atoms as the temperature $T\to0$, $s$ scattering gives the main contribution to the interaction of atoms in the leading order in the gas parameter. Therefore, the exponent $\nu=1$ for an arbitrary $D$. In the three-dimensional case, $\nu=2/3$ is realized for condensates of Fermi atoms in the so-called unitary limit. For $\nu=2/D$, 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 $t\to\infty$. In the semiclassical limit, the equations of motion coincide with those of the hydrodynamics of an ideal gas with the adiabatic exponent $\gamma=1+2/D$. 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.