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? 
 A: 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).
What I have described above are low-temperature quasiparticle-like excitations. One can also take a hydrodynamic point of view. In that case, there are two types of collective modes - first and second sound - which are both gapless and have a linear dispersion. They correspond to combinations of density and temperature waves in the system. In many aspects they are similar to the sound waves that we are all familiar with.
More details are given in a book by Pethick and Smith, Bose–Einstein Condensation in Dilute Gases. Another reference, where it is explicitly shown that the phase fluctuations cause gapless excitations is arXiv:1306.1104 (see Eq. 14 there for the gapless dispersion).
