Bose condensation to quantum gravity and back

We account for the interaction of the Bose-condensed fraction with the normal phase in an effective dynamical equation such as the Gross-Pitaevskii equation. We show that the low-energy excitations can be treated as sound waves with speed dependent on the condensate density. This allows us to reduce the problem to the calculation of the determinant of the Laplace operator on a curved space and apply standard methods of quantum gravity to get the leading logarithmic contribution of the determinant. This produces the first quantum correction due to the noncondensed fraction to the Gross-Pitaevskii equation for the condensate. The correction describes an additional quantum pressure in the condensate and evaporation-condensation effects. 1. Last years there was a new growth of the interest in Bose-Einstein condensation (in particular, for spatially inhomogeneous situations) due to the recent success of the experimental observation of Bose-Einstein condensation for systems of spin polarized magnetically trapped atomic gases at ultra-low temperatures [1, 2, 3] and further investigation of their collective properties [4]. These experimental results stimulated the development of a theory of a nonuniform Bose condensate and its collective excitations. Moreover, new exciting problems such as a description of the evolution of the Bose condensate from a relaxed trap, stability and dynamics of the collapse of the condensate for Li 7 atoms, collective excitations, heating-cooling phenomena, various coherence effects for the condensate and so on are being posed both theoretically and experimentally. All the problems listed above require an accurate account of the interaction of the condensate with the non-condensed fraction. Formally, the problem can be tackled using the Bogoliubov-Popov equations which represent a set of coupled linear and nonlinear Schrödinger equations; the nonlinear Schrödinger equation (for the condensate fraction) coupled via a potential term with a pair of Schrödinger