From 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.