# What is the gapless excitation for traditional Bose-Einstein condensates?

I want to know the properties and the behavior of the gapless excitation for the traditional BECs. Could you give me some idea or references about this?

Since the non-interacting condensate is a pathological situation (it is not a superfluid), I will assume that by "traditional" you mean a (perhaps extremely) weakly interacting condensate. I will denote the repulsive interaction strength (the T-matrix) by $g>0$. For simplicity, I will describe the situation at very low temperatures.
The elementary excitations are Bogoliubov quasiparticles. They are the Nambu-Goldstone modes, which appear due to spontaneous breaking of the global U(1) symmetry. This should be compared with the situation above critical temperature where the single-particle excitations have a gapped (Hartree-Fock) spectrum. The Bogoliubov quasiparticles have a linear dispersion at low momenta, namely, $E = c \hbar k$. The prefactor $$c=\sqrt{\frac{dp}{dn}\frac{1}{m}}=\sqrt{\frac{gn}{m}}$$ is the sound velocity, where $n$ is particle-density, $p$ is the pressure, and $m$ is the mass of the atom. Bogoliubov particles are coherent combinations of adding and removing particles (atoms and holes).