Excitations in a Bose Gas at Finite Temperatures. II. Relation between Single-Particle and Density Fluctuations

By making a systematic analysis of correlation functions in terms of irreducible and reducible parts, Ma and others have developed a dielectric formulation of the dynamics of a condensed Bose system which fully includes the background (or excited) atoms. We show that this formulation quite generally leads to the density and single-particle-correlation functions both having two resonances ${\tilde{\omega }}_1$ and ${\tilde{\omega }}_2$ although with different weights. If the condensate $n_0=0\mathrm{}$, the ${\omega }_2$ mode only appears in the single-particle spectrum while ${\omega }_1$ only appears in the density-fluctuation spectrum. For finite values of $n_0$, these modes are coupled and renormalized to ${\tilde{\omega }}_1$ and ${\tilde{\omega }}_2$. As a specific illustration, we use the shielded-potential approximation (SPA) for the reducible self-energies. In this model, we find the free-particle excitations (${\omega }_2$) are coupled to the zero-sound density fluctuations (${\omega }_1$) through the action of the condensate. In the SPA, the reducible self-energies have a pole at ${\omega }_1$. The usual Bogoliubov, Hartree-Fock, and ordinary $t$-matrix approximations involve nonsingular self-energies and hence do not exhibit the ${\tilde{\omega }}_1$ mode. Experimentally, it appears that the high-frequency ${\tilde{\omega }}_2$ mode has never been detected, but this is probably owing to the fact that it is strongly damped as a result of its coupling to the zero-sound mode. Using the two-fluid hydrodynamic equations, one can argue that at low frequencies the ${\tilde{\omega }}_1$ excitation corresponds to first sound and the ${\tilde{\omega }}_2$ excitation corresponds to second sound. Finally, we briefly discuss the possibility of singularities in the irreducible self-energies at high frequencies and the resulting two-roton bound states.