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 ω̃₁ and ω̃₂ although with different weights. If the condensate n₀=0, the ω₂ mode only appears in the single-particle spectrum while ω₁ only appears in the density-fluctuation spectrum. For finite values of n₀, these modes are coupled and renormalized to ω̃₁ and ω̃₂. 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 (ω₂) are coupled to the zero-sound density fluctuations (ω₁) through the action of the condensate. In the SPA, the reducible self-energies have a pole at ω₁. The usual Bogoliubov, Hartree-Fock, and ordinary t-matrix approximations involve nonsingular self-energies and hence do not exhibit the ω̃₁ mode. Experimentally, it appears that the high-frequency ω̃₂ 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 ω̃₁ excitation corresponds to first sound and the ω̃₂ 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.