Hydrodynamic collective modes for cold trapped gases

We suggest that collective oscillation frequencies of cold trapped gases can be used as high precision observables for quantum many-body physics. Our motivation lies both in rigid experimental tests of theoretical calculations and accuracy improvements of measurements of particle number, chemical potential or temperature. We calculate the effects of interaction, dimensionality and thermal fluctuations on the collective modes of a dilute Bose gas. The underlying equation of state is provided by non-perturbative Functional Renormalization Group or by Lee--Yang theory. The spectrum of oscillation frequencies could be measured by response techniques. Our findings are generalized to an arbitrary quantum gas in the two-fluid hydrodynamic regime. The collective oscillation frequencies in a d-dimensional isotropic harmonic potential for an equation of state P(\mu,T) and normal fluid density n_n(\mu,T) are found to be the eigenvalues of an ordinary differential operator. We suggest a method of numerical solution and discuss the zero-temperature limit. Exact results are provided in certain cases.