content_image/352454.jpg content_image/352455.jpg content_image/352457.jpg content_image/352456.jpg content_image/352453.jpg content_image/352452.jpg
result that second sound is the dominant excitation in S ( q, ω ) and mainly involves a pure oscillation of the condensate in the presence of a static thermal component [3, 4]. However, first and second sound in a strongly interacting Bose gas are similar to those in a Fermi gas near unitarity (see figure 7). 

## Acknowledgments 

We thank Lev Pitaevskii for useful comments. E.T. was supported by nsf-dmr 0907366 and ARO W911NF-08-1-0338. H.H. and X.-J.L. were supported by ARC and NSFC. S.S. acknowledges the support of the EuroQUAM FerMix program. A.G. was supported by NSERC and CIFAR. 

## Appendix A: Phonon thermodynamics at low temperatures 

At low temperatures, Goldstone phonons determine the thermodynamics of both super- fluid $^{4}$He and Fermi gases. For phonons with velocity c , the free energy F and normal fluid density ρ$_{n}$$_{0}$ are given by [41] (Recall that we have set ℏ = k$_{B}$ = 1) 

and 

Using (A1), it is straightforward to show that 

and 

<img src='content_image/352454.jpg'>

<img src='content_image/352455.jpg'>

<img src='content_image/352457.jpg'>

<img src='content_image/352456.jpg'>

<img src='content_image/352453.jpg'>

In arriving at these expressions, the temperature dependence of the Goldstone phonon ve- locity c has been ignored. P$_{0}$ is the pressure in the ground state. These results can be combined to give 

<img src='content_image/352452.jpg'>

