content_image/331452.jpg content_image/331451.jpg content_image/331444.jpg content_image/331446.jpg content_image/331449.jpg content_image/331450.jpg
potential only varies spatially along the z -axis (and q = ˆ z q ), for t > 0, one obtains [10] 

<img src='content_image/331452.jpg'>

Equation (6) reveals an important feature: in contrast to Bragg scattering, where the re- sponse is proportional to Im χ$_{nn}$ ( q , ω ), the response involved in the excitation of density pulses in proportional to Im χ$_{nn}$ ( q , ω ) /ω . As we discuss in section III, for uniform super- fluids, this factor of 1 /ω leads to an enhancement in the signal of second sound in density pulse experiments as compared to its signal in Bragg spectroscopy. In table I, we review the quantities measured by the experimental probes of interest in both dilute gases and superfluid $^{4}$He. 

Before discussing the form of the density response function in the two-fluid hydrodynamic region in section III, we first review below some general properties of the dynamic structure factor that will be of use later on in understanding the contributions from first and second sound in experimental probes of superfluid atomic gases. 

Quite generally, S ( q , ω ) satisfies various frequency moment sum rules. Two are of special particular interest in the two-fluid domain [23]. The f -sum rule (valid for all q ) is 

<img src='content_image/331451.jpg'>

The compressibility sum rule arises from the exact Kramers-Kronig identity (also valid for all q ) ∞ ′ ′ 

which involves an inverse frequency moment. (Note that χ$_{nn}$ ( q , ω = 0) = Re χ$_{nn}$ ( q , ω = 0) is purely real.) Using (2), one can show that (8) is equivalent to 

<img src='content_image/331444.jpg'>

<img src='content_image/331446.jpg'>

The usefulness of (8) and (9) is due to the fact that in the long wavelength limit ( q → 0) , the density response function χ$_{nn}$ ( q = 0 , ω = 0) describes static thermodynamic fluctuations which can be related to the equilibrium isothermal compressibility [23, 24] 

<img src='content_image/331449.jpg'>

Thus we arrive at the so-called compressibility sum rule for the dynamic structure factor, 

<img src='content_image/331450.jpg'>

